Leskille/code_chall

1prob_desc_description,clean_tags,labels2"Today, Wet Shark is given n bishops on a 1000 by 1000 grid. Both rows and columns of the grid are numbered from 1 to 1000. Rows are numbered from top to bottom, while columns are numbered from left to right.Wet Shark thinks that two bishops attack each other if they share the same diagonal. Note, that this is the only criteria, so two bishops may attack each other (according to Wet Shark) even if there is another bishop located between them. Now Wet Shark wants to count the number of pairs of bishops that attack each other.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"3"Galya is playing one-dimensional Sea Battle on a 1 × n grid. In this game a ships are placed on the grid. Each of the ships consists of b consecutive cells. No cell can be part of two ships, however, the ships can touch each other.Galya doesn't know the ships location. She can shoot to some cells and after each shot she is told if that cell was a part of some ship (this case is called ""hit"") or not (this case is called ""miss"").Galya has already made k shots, all of them were misses.Your task is to calculate the minimum number of cells such that if Galya shoot at all of them, she would hit at least one ship.It is guaranteed that there is at least one valid ships placement.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"4"You are given two arrays A and B, each of size n. The error, E, between these two arrays is defined . You have to perform exactly k1 operations on array A and exactly k2 operations on array B. In one operation, you have to choose one element of the array and increase or decrease it by 1.Output the minimum possible value of error after k1 operations on array A and k2 operations on array B have been performed.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"5"There are $$$n$$$ students at your university. The programming skill of the $$$i$$$-th student is $$$a_i$$$. As a coach, you want to divide them into teams to prepare them for the upcoming ICPC finals. Just imagine how good this university is if it has $$$2 \cdot 10^5$$$ students ready for the finals!Each team should consist of at least three students. Each student should belong to exactly one team. The diversity of a team is the difference between the maximum programming skill of some student that belongs to this team and the minimum programming skill of some student that belongs to this team (in other words, if the team consists of $$$k$$$ students with programming skills $$$a[i_1], a[i_2], \dots, a[i_k]$$$, then the diversity of this team is $$$\max\limits_{j=1}^{k} a[i_j] - \min\limits_{j=1}^{k} a[i_j]$$$).The total diversity is the sum of diversities of all teams formed.Your task is to minimize the total diversity of the division of students and find the optimal way to divide the students.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"6"You are given an array $$$a$$$ of length $$$n$$$, and an integer $$$x$$$. You can perform the following operation as many times as you would like (possibly zero): replace two adjacent elements of the array by their sum. For example, if the initial array was $$$[3, 6, 9]$$$, in a single operation one can replace the last two elements by their sum, yielding an array $$$[3, 15]$$$, or replace the first two elements to get an array $$$[9, 9]$$$. Note that the size of the array decreases after each operation.The beauty of an array $$$b=[b_1, \ldots, b_k]$$$ is defined as $$$\sum_{i=1}^k \left\lceil \frac{b_i}{x} \right\rceil$$$, which means that we divide each element by $$$x$$$, round it up to the nearest integer, and sum up the resulting values. For example, if $$$x = 3$$$, and the array is $$$[4, 11, 6]$$$, the beauty of the array is equal to $$$\left\lceil \frac{4}{3} \right\rceil + \left\lceil \frac{11}{3} \right\rceil + \left\lceil \frac{6}{3} \right\rceil = 2 + 4 + 2 = 8$$$.Please determine the minimum and the maximum beauty you can get by performing some operations on the original array.","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"7"Polycarp found on the street an array $$$a$$$ of $$$n$$$ elements.Polycarp invented his criterion for the beauty of an array. He calls an array $$$a$$$ beautiful if at least one of the following conditions must be met for each different pair of indices $$$i \ne j$$$:   $$$a_i$$$ is divisible by $$$a_j$$$;  or $$$a_j$$$ is divisible by $$$a_i$$$. For example, if:   $$$n=5$$$ and $$$a=[7, 9, 3, 14, 63]$$$, then the $$$a$$$ array is not beautiful (for $$$i=4$$$ and $$$j=2$$$, none of the conditions above is met);  $$$n=3$$$ and $$$a=[2, 14, 42]$$$, then the $$$a$$$ array is beautiful;  $$$n=4$$$ and $$$a=[45, 9, 3, 18]$$$, then the $$$a$$$ array is not beautiful (for $$$i=1$$$ and $$$j=4$$$ none of the conditions above is met); Ugly arrays upset Polycarp, so he wants to remove some elements from the array $$$a$$$ so that it becomes beautiful. Help Polycarp determine the smallest number of elements to remove to make the array $$$a$$$ beautiful.","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"8"You are given a string $$$s$$$ consisting of lowercase letters of the English alphabet. You must perform the following algorithm on $$$s$$$:  Let $$$x$$$ be the length of the longest prefix of $$$s$$$ which occurs somewhere else in $$$s$$$ as a contiguous substring (the other occurrence may also intersect the prefix). If $$$x = 0$$$, break. Otherwise, remove the first $$$x$$$ characters of $$$s$$$, and repeat. A prefix is a string consisting of several first letters of a given string, without any reorders. An empty prefix is also a valid prefix. For example, the string ""abcd"" has 5 prefixes: empty string, ""a"", ""ab"", ""abc"" and ""abcd"".For instance, if we perform the algorithm on $$$s =$$$ ""abcabdc"",   Initially, ""ab"" is the longest prefix that also appears somewhere else as a substring in $$$s$$$, so $$$s =$$$ ""cabdc"" after $$$1$$$ operation.  Then, ""c"" is the longest prefix that also appears somewhere else as a substring in $$$s$$$, so $$$s =$$$ ""abdc"" after $$$2$$$ operations.  Now $$$x=0$$$ (because there are no non-empty prefixes of ""abdc"" that also appear somewhere else as a substring in $$$s$$$), so the algorithm terminates. Find the final state of the string after performing the algorithm.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"9"There is a conveyor with $$$120$$$ rows and $$$120$$$ columns. Each row and column is numbered from $$$0$$$ to $$$119$$$, and the cell in $$$i$$$-th row and $$$j$$$-th column is denoted as $$$(i, j)$$$. The top leftmost cell is $$$(0, 0)$$$. Each cell has a belt, and all belts are initially facing to the right.Initially, a slime ball is on the belt of $$$(0, 0)$$$, and other belts are empty. Every second, the state of the conveyor changes as follows:   All slime balls on the conveyor move one cell in the direction of the belt at the same time. If there is no cell in the moved position, the slime gets out of the conveyor, and if two slime balls move to the same cell, they merge into one.  All belts with slime ball in the previous second change direction at the same time: belts facing to the right become facing to the down, and vice versa.  A new slime ball is placed on cell $$$(0, 0)$$$. There are $$$q$$$ queries, each being three integers $$$t$$$, $$$x$$$, and $$$y$$$. You have to find out if there is a slime at the cell $$$(x, y)$$$ after $$$t$$$ seconds from the start. Can you do it?",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"10"Calculate the minimum number of characters you need to change in the string s, so that it contains at least k different letters, or print that it is impossible.String s consists only of lowercase Latin letters, and it is allowed to change characters only to lowercase Latin letters too.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"11"For given n, l and r find the number of distinct geometrical progression, each of which contains n distinct integers not less than l and not greater than r. In other words, for each progression the following must hold: l ≤ ai ≤ r and ai ≠ aj , where a1, a2, ..., an is the geometrical progression, 1 ≤ i, j ≤ n and i ≠ j.Geometrical progression is a sequence of numbers a1, a2, ..., an where each term after first is found by multiplying the previous one by a fixed non-zero number d called the common ratio. Note that in our task d may be non-integer. For example in progression 4, 6, 9, common ratio is .Two progressions a1, a2, ..., an and b1, b2, ..., bn are considered different, if there is such i (1 ≤ i ≤ n) that ai ≠ bi.","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"12"Andrew skipped lessons on the subject 'Algorithms and Data Structures' for the entire term. When he came to the final test, the teacher decided to give him a difficult task as a punishment.The teacher gave Andrew an array of n numbers a1, ..., an. After that he asked Andrew for each k from 1 to n - 1 to build a k-ary heap on the array and count the number of elements for which the property of the minimum-rooted heap is violated, i.e. the value of an element is less than the value of its parent.Andrew looked up on the Wikipedia that a k-ary heap is a rooted tree with vertices in elements of the array. If the elements of the array are indexed from 1 to n, then the children of element v are elements with indices k(v - 1) + 2, ..., kv + 1 (if some of these elements lie outside the borders of the array, the corresponding children are absent). In any k-ary heap every element except for the first one has exactly one parent; for the element 1 the parent is absent (this element is the root of the heap). Denote p(v) as the number of the parent of the element with the number v. Let's say that for a non-root element v the property of the heap is violated if av < ap(v).Help Andrew cope with the task!",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"13"Bob is a farmer. He has a large pasture with many sheep. Recently, he has lost some of them due to wolf attacks. He thus decided to place some shepherd dogs in such a way that all his sheep are protected.The pasture is a rectangle consisting of R × C cells. Each cell is either empty, contains a sheep, a wolf or a dog. Sheep and dogs always stay in place, but wolves can roam freely around the pasture, by repeatedly moving to the left, right, up or down to a neighboring cell. When a wolf enters a cell with a sheep, it consumes it. However, no wolf can enter a cell with a dog.Initially there are no dogs. Place dogs onto the pasture in such a way that no wolf can reach any sheep, or determine that it is impossible. Note that since you have many dogs, you do not need to minimize their number. ",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"14"After rejecting $$$10^{100}$$$ data structure problems, Errorgorn is very angry at Anton and decided to kill him.Anton's DNA can be represented as a string $$$a$$$ which only contains the characters ""ANTON"" (there are only $$$4$$$ distinct characters). Errorgorn can change Anton's DNA into string $$$b$$$ which must be a permutation of $$$a$$$. However, Anton's body can defend against this attack. In $$$1$$$ second, his body can swap $$$2$$$ adjacent characters of his DNA to transform it back to $$$a$$$. Anton's body is smart and will use the minimum number of moves.To maximize the chance of Anton dying, Errorgorn wants to change Anton's DNA the string that maximizes the time for Anton's body to revert his DNA. But since Errorgorn is busy making more data structure problems, he needs your help to find the best string $$$B$$$. Can you help him?","['math', 'strings']","[0, 0, 0, 1, 0, 0, 1, 0]"15"Limak is a little bear who learns to draw. People usually start with houses, fences and flowers but why would bears do it? Limak lives in the forest and he decides to draw a tree.Recall that tree is a connected graph consisting of n vertices and n - 1 edges.Limak chose a tree with n vertices. He has infinite strip of paper with two parallel rows of dots. Little bear wants to assign vertices of a tree to some n distinct dots on a paper so that edges would intersect only at their endpoints — drawn tree must be planar. Below you can see one of correct drawings for the first sample test.  Is it possible for Limak to draw chosen tree?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"16"After a successful year of milk production, Farmer John is rewarding his cows with their favorite treat: tasty grass!On the field, there is a row of $$$n$$$ units of grass, each with a sweetness $$$s_i$$$. Farmer John has $$$m$$$ cows, each with a favorite sweetness $$$f_i$$$ and a hunger value $$$h_i$$$. He would like to pick two disjoint subsets of cows to line up on the left and right side of the grass row. There is no restriction on how many cows must be on either side. The cows will be treated in the following manner:   The cows from the left and right side will take turns feeding in an order decided by Farmer John.  When a cow feeds, it walks towards the other end without changing direction and eats grass of its favorite sweetness until it eats $$$h_i$$$ units.  The moment a cow eats $$$h_i$$$ units, it will fall asleep there, preventing further cows from passing it from both directions.  If it encounters another sleeping cow or reaches the end of the grass row, it will get upset. Farmer John absolutely does not want any cows to get upset. Note that grass does not grow back. Also, to prevent cows from getting upset, not every cow has to feed since FJ can choose a subset of them. Surprisingly, FJ has determined that sleeping cows are the most satisfied. If FJ orders optimally, what is the maximum number of sleeping cows that can result, and how many ways can FJ choose the subset of cows on the left and right side to achieve that maximum number of sleeping cows (modulo $$$10^9+7$$$)? The order in which FJ sends the cows does not matter as long as no cows get upset. ",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"17"You are given an array of positive integers. While there are at least two equal elements, we will perform the following operation. We choose the smallest value $$$x$$$ that occurs in the array $$$2$$$ or more times. Take the first two occurrences of $$$x$$$ in this array (the two leftmost occurrences). Remove the left of these two occurrences, and the right one is replaced by the sum of this two values (that is, $$$2 \cdot x$$$).Determine how the array will look after described operations are performed.For example, consider the given array looks like $$$[3, 4, 1, 2, 2, 1, 1]$$$. It will be changed in the following way: $$$[3, 4, 1, 2, 2, 1, 1]~\rightarrow~[3, 4, 2, 2, 2, 1]~\rightarrow~[3, 4, 4, 2, 1]~\rightarrow~[3, 8, 2, 1]$$$.If the given array is look like $$$[1, 1, 3, 1, 1]$$$ it will be changed in the following way: $$$[1, 1, 3, 1, 1]~\rightarrow~[2, 3, 1, 1]~\rightarrow~[2, 3, 2]~\rightarrow~[3, 4]$$$.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"18"An agent called Cypher is decrypting a message, that contains a composite number $$$n$$$. All divisors of $$$n$$$, which are greater than $$$1$$$, are placed in a circle. Cypher can choose the initial order of numbers in the circle.In one move Cypher can choose two adjacent numbers in a circle and insert their least common multiple between them. He can do that move as many times as needed.A message is decrypted, if every two adjacent numbers are not coprime. Note that for such constraints it's always possible to decrypt the message.Find the minimal number of moves that Cypher should do to decrypt the message, and show the initial order of numbers in the circle for that.","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"19Let's define the following recurrence: $$$$$$a_{n+1} = a_{n} + minDigit(a_{n}) \cdot maxDigit(a_{n}).$$$$$$Here $$$minDigit(x)$$$ and $$$maxDigit(x)$$$ are the minimal and maximal digits in the decimal representation of $$$x$$$ without leading zeroes. For examples refer to notes.Your task is calculate $$$a_{K}$$$ for given $$$a_{1}$$$ and $$$K$$$.,['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"20"Recently Vova found $$$n$$$ candy wrappers. He remembers that he bought $$$x$$$ candies during the first day, $$$2x$$$ candies during the second day, $$$4x$$$ candies during the third day, $$$\dots$$$, $$$2^{k-1} x$$$ candies during the $$$k$$$-th day. But there is an issue: Vova remembers neither $$$x$$$ nor $$$k$$$ but he is sure that $$$x$$$ and $$$k$$$ are positive integers and $$$k > 1$$$.Vova will be satisfied if you tell him any positive integer $$$x$$$ so there is an integer $$$k>1$$$ that $$$x + 2x + 4x + \dots + 2^{k-1} x = n$$$. It is guaranteed that at least one solution exists. Note that $$$k > 1$$$.You have to answer $$$t$$$ independent test cases.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"21"On the board, Bob wrote $$$n$$$ positive integers in base $$$10$$$ with sum $$$s$$$ (i. e. in decimal numeral system). Alice sees the board, but accidentally interprets the numbers on the board as base-$$$11$$$ integers and adds them up (in base $$$11$$$).What numbers should Bob write on the board, so Alice's sum is as large as possible?",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"22"AquaMoon has $$$n$$$ friends. They stand in a row from left to right, and the $$$i$$$-th friend from the left wears a T-shirt with a number $$$a_i$$$ written on it. Each friend has a direction (left or right). In the beginning, the direction of each friend is right.AquaMoon can make some operations on friends. On each operation, AquaMoon can choose two adjacent friends and swap their positions. After each operation, the direction of both chosen friends will also be flipped: left to right and vice versa.AquaMoon hopes that after some operations, the numbers written on the T-shirt of $$$n$$$ friends in the row, read from left to right, become non-decreasing. Also she wants, that all friends will have a direction of right at the end. Please find if it is possible.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"23"The position of the leftmost maximum on the segment $$$[l; r]$$$ of array $$$x = [x_1, x_2, \ldots, x_n]$$$ is the smallest integer $$$i$$$ such that $$$l \le i \le r$$$ and $$$x_i = \max(x_l, x_{l+1}, \ldots, x_r)$$$.You are given an array $$$a = [a_1, a_2, \ldots, a_n]$$$ of length $$$n$$$. Find the number of integer arrays $$$b = [b_1, b_2, \ldots, b_n]$$$ of length $$$n$$$ that satisfy the following conditions:   $$$1 \le b_i \le m$$$ for all $$$1 \le i \le n$$$;  for all pairs of integers $$$1 \le l \le r \le n$$$, the position of the leftmost maximum on the segment $$$[l; r]$$$ of the array $$$b$$$ is equal to the position of the leftmost maximum on the segment $$$[l; r]$$$ of the array $$$a$$$. Since the answer might be very large, print its remainder modulo $$$10^9+7$$$.","['math', 'trees']","[0, 0, 0, 1, 0, 0, 0, 1]"24"On February 14 Denis decided to give Valentine to Nastya and did not come up with anything better than to draw a huge red heart on the door of the length $$$k$$$ ($$$k \ge 3$$$). Nastya was very confused by this present, so she decided to break the door, throwing it on the mountains.Mountains are described by a sequence of heights $$$a_1, a_2, \dots, a_n$$$ in order from left to right ($$$k \le n$$$). It is guaranteed that neighboring heights are not equal to each other (that is, $$$a_i \ne a_{i+1}$$$ for all $$$i$$$ from $$$1$$$ to $$$n-1$$$).Peaks of mountains on the segment $$$[l,r]$$$ (from $$$l$$$ to $$$r$$$) are called indexes $$$i$$$ such that $$$l < i < r$$$, $$$a_{i - 1} < a_i$$$ and $$$a_i > a_{i + 1}$$$. It is worth noting that the boundary indexes $$$l$$$ and $$$r$$$ for the segment are not peaks. For example, if $$$n=8$$$ and $$$a=[3,1,4,1,5,9,2,6]$$$, then the segment $$$[1,8]$$$ has only two peaks (with indexes $$$3$$$ and $$$6$$$), and there are no peaks on the segment $$$[3, 6]$$$.To break the door, Nastya throws it to a segment $$$[l,l+k-1]$$$ of consecutive mountains of length $$$k$$$ ($$$1 \le l \le n-k+1$$$). When the door touches the peaks of the mountains, it breaks into two parts, after that these parts will continue to fall in different halves and also break into pieces when touching the peaks of the mountains, and so on. Formally, the number of parts that the door will break into will be equal to $$$p+1$$$, where $$$p$$$ is the number of peaks on the segment $$$[l,l+k-1]$$$.Nastya wants to break it into as many pieces as possible. Help her choose such a segment of mountains $$$[l, l+k-1]$$$ that the number of peaks on it is maximum. If there are several optimal segments, Nastya wants to find one for which the value $$$l$$$ is minimal.Formally, you need to choose a segment of mountains $$$[l, l+k-1]$$$ that has the maximum number of peaks. Among all such segments, you need to find the segment that has the minimum possible value $$$l$$$.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"25"At the first holiday in spring, the town Shortriver traditionally conducts a flower festival. Townsfolk wear traditional wreaths during these festivals. Each wreath contains exactly $$$k$$$ flowers.The work material for the wreaths for all $$$n$$$ citizens of Shortriver is cut from the longest flowered liana that grew in the town that year. Liana is a sequence $$$a_1$$$, $$$a_2$$$, ..., $$$a_m$$$, where $$$a_i$$$ is an integer that denotes the type of flower at the position $$$i$$$. This year the liana is very long ($$$m \ge n \cdot k$$$), and that means every citizen will get a wreath.Very soon the liana will be inserted into a special cutting machine in order to make work material for wreaths. The machine works in a simple manner: it cuts $$$k$$$ flowers from the beginning of the liana, then another $$$k$$$ flowers and so on. Each such piece of $$$k$$$ flowers is called a workpiece. The machine works until there are less than $$$k$$$ flowers on the liana.Diana has found a weaving schematic for the most beautiful wreath imaginable. In order to weave it, $$$k$$$ flowers must contain flowers of types $$$b_1$$$, $$$b_2$$$, ..., $$$b_s$$$, while other can be of any type. If a type appears in this sequence several times, there should be at least that many flowers of that type as the number of occurrences of this flower in the sequence. The order of the flowers in a workpiece does not matter.Diana has a chance to remove some flowers from the liana before it is inserted into the cutting machine. She can remove flowers from any part of the liana without breaking liana into pieces. If Diana removes too many flowers, it may happen so that some of the citizens do not get a wreath. Could some flowers be removed from the liana so that at least one workpiece would conform to the schematic and machine would still be able to create at least $$$n$$$ workpieces?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"26"After you had helped George and Alex to move in the dorm, they went to help their friend Fedor play a new computer game «Call of Soldiers 3».The game has (m + 1) players and n types of soldiers in total. Players «Call of Soldiers 3» are numbered form 1 to (m + 1). Types of soldiers are numbered from 0 to n - 1. Each player has an army. Army of the i-th player can be described by non-negative integer xi. Consider binary representation of xi: if the j-th bit of number xi equal to one, then the army of the i-th player has soldiers of the j-th type. Fedor is the (m + 1)-th player of the game. He assume that two players can become friends if their armies differ in at most k types of soldiers (in other words, binary representations of the corresponding numbers differ in at most k bits). Help Fedor and count how many players can become his friends.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"27"As Will is stuck in the Upside Down, he can still communicate with his mom, Joyce, through the Christmas lights (he can turn them on and off with his mind). He can't directly tell his mom where he is, because the monster that took him to the Upside Down will know and relocate him.   Thus, he came up with a puzzle to tell his mom his coordinates. His coordinates are the answer to the following problem.A string consisting only of parentheses ('(' and ')') is called a bracket sequence. Some bracket sequence are called correct bracket sequences. More formally:  Empty string is a correct bracket sequence.  if s is a correct bracket sequence, then (s) is also a correct bracket sequence.  if s and t are correct bracket sequences, then st (concatenation of s and t) is also a correct bracket sequence. A string consisting of parentheses and question marks ('?') is called pretty if and only if there's a way to replace each question mark with either '(' or ')' such that the resulting string is a non-empty correct bracket sequence.Will gave his mom a string s consisting of parentheses and question marks (using Morse code through the lights) and his coordinates are the number of pairs of integers (l, r) such that 1 ≤ l ≤ r ≤ |s| and the string slsl + 1... sr is pretty, where si is i-th character of s.Joyce doesn't know anything about bracket sequences, so she asked for your help.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"28"There is a weighted tree with $$$n$$$ nodes and $$$n-1$$$ edges. The nodes are conveniently labeled from $$$1$$$ to $$$n$$$. The weights are positive integers at most $$$100$$$. Define the distance between two nodes to be the sum of edges on the unique path between the nodes. You would like to find the diameter of the tree. Diameter is the maximum distance between a pair of nodes.Unfortunately, the tree isn't given to you, but you can ask some questions about it. In one question, you can specify two nonempty disjoint sets of nodes $$$p$$$ and $$$q$$$, and the judge will return the maximum distance between a node in $$$p$$$ and a node in $$$q$$$. In the words, maximum distance between $$$x$$$ and $$$y$$$, where $$$x \in p$$$ and $$$y \in q$$$. After asking not more than $$$9$$$ questions, you must report the maximum distance between any pair of nodes.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"29"You are given an array of integers $$$a_1, a_2, \ldots, a_n$$$.You can do the following operation any number of times (possibly zero):   Choose any index $$$i$$$ and set $$$a_i$$$ to any integer (positive, negative or $$$0$$$). What is the minimum number of operations needed to turn $$$a$$$ into an arithmetic progression? The array $$$a$$$ is an arithmetic progression if $$$a_{i+1}-a_i=a_i-a_{i-1}$$$ for any $$$2 \leq i \leq n-1$$$.","['graphs', 'math']","[0, 0, 1, 1, 0, 0, 0, 0]"30"You are given a directed acyclic graph (a directed graph that does not contain cycles) of $$$n$$$ vertices and $$$m$$$ arcs. The $$$i$$$-th arc leads from the vertex $$$x_i$$$ to the vertex $$$y_i$$$ and has the weight $$$w_i$$$.Your task is to select an integer $$$a_v$$$ for each vertex $$$v$$$, and then write a number $$$b_i$$$ on each arcs $$$i$$$ such that $$$b_i = a_{x_i} - a_{y_i}$$$. You must select the numbers so that:  all $$$b_i$$$ are positive;  the value of the expression $$$\sum \limits_{i = 1}^{m} w_i b_i$$$ is the lowest possible. It can be shown that for any directed acyclic graph with non-negative $$$w_i$$$, such a way to choose numbers exists.","['graphs', 'math']","[0, 0, 1, 1, 0, 0, 0, 0]"31"One day Vasya got hold of information on the Martian dollar course in bourles for the next n days. The buying prices and the selling prices for one dollar on day i are the same and are equal to ai. Vasya has b bourles. He can buy a certain number of dollars and then sell it no more than once in n days. According to Martian laws, one can buy only an integer number of dollars. Which maximal sum of money in bourles can Vasya get by the end of day n?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"32"During the ""Russian Code Cup"" programming competition, the testing system stores all sent solutions for each participant. We know that many participants use random numbers in their programs and are often sent several solutions with the same source code to check.Each participant is identified by some unique positive integer k, and each sent solution A is characterized by two numbers: x — the number of different solutions that are sent before the first solution identical to A, and k — the number of the participant, who is the author of the solution. Consequently, all identical solutions have the same x.It is known that the data in the testing system are stored in the chronological order, that is, if the testing system has a solution with number x (x > 0) of the participant with number k, then the testing system has a solution with number x - 1 of the same participant stored somewhere before.During the competition the checking system crashed, but then the data of the submissions of all participants have been restored. Now the jury wants to verify that the recovered data is in chronological order. Help the jury to do so.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"33"3R2 as DJ Mashiro - Happiness Breeze Ice - DJ Mashiro is dead or aliveNEKO#ΦωΦ has just got a new maze game on her PC!The game's main puzzle is a maze, in the forms of a $$$2 \times n$$$ rectangle grid. NEKO's task is to lead a Nekomimi girl from cell $$$(1, 1)$$$ to the gate at $$$(2, n)$$$ and escape the maze. The girl can only move between cells sharing a common side.However, at some moments during the game, some cells may change their state: either from normal ground to lava (which forbids movement into that cell), or vice versa (which makes that cell passable again). Initially all cells are of the ground type.After hours of streaming, NEKO finally figured out there are only $$$q$$$ such moments: the $$$i$$$-th moment toggles the state of cell $$$(r_i, c_i)$$$ (either from ground to lava or vice versa).Knowing this, NEKO wonders, after each of the $$$q$$$ moments, whether it is still possible to move from cell $$$(1, 1)$$$ to cell $$$(2, n)$$$ without going through any lava cells.Although NEKO is a great streamer and gamer, she still can't get through quizzes and problems requiring large amount of Brain Power. Can you help her?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"34"The ship crashed into a reef and is sinking. Now the entire crew must be evacuated. All n crew members have already lined up in a row (for convenience let's label them all from left to right with positive integers from 1 to n) and await further instructions. However, one should evacuate the crew properly, in a strict order. Specifically:The first crew members to leave the ship are rats. Then women and children (both groups have the same priority) leave the ship. After that all men are evacuated from the ship. The captain leaves the sinking ship last.If we cannot determine exactly who should leave the ship first for any two members of the crew by the rules from the previous paragraph, then the one who stands to the left in the line leaves the ship first (or in other words, the one whose number in the line is less).For each crew member we know his status as a crew member, and also his name. All crew members have different names. Determine the order in which to evacuate the crew.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"35"In some country there are exactly n cities and m bidirectional roads connecting the cities. Cities are numbered with integers from 1 to n. If cities a and b are connected by a road, then in an hour you can go along this road either from city a to city b, or from city b to city a. The road network is such that from any city you can get to any other one by moving along the roads.You want to destroy the largest possible number of roads in the country so that the remaining roads would allow you to get from city s1 to city t1 in at most l1 hours and get from city s2 to city t2 in at most l2 hours.Determine what maximum number of roads you need to destroy in order to meet the condition of your plan. If it is impossible to reach the desired result, print -1.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"36"Monocarp is playing Minecraft and wants to build a wall of cacti. He wants to build it on a field of sand of the size of $$$n \times m$$$ cells. Initially, there are cacti in some cells of the field. Note that, in Minecraft, cacti cannot grow on cells adjacent to each other by side — and the initial field meets this restriction. Monocarp can plant new cacti (they must also fulfil the aforementioned condition). He can't chop down any of the cacti that are already growing on the field — he doesn't have an axe, and the cacti are too prickly for his hands.Monocarp believes that the wall is complete if there is no path from the top row of the field to the bottom row, such that:   each two consecutive cells in the path are adjacent by side;  no cell belonging to the path contains a cactus. Your task is to plant the minimum number of cacti to build a wall (or to report that this is impossible).",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"37"Recently Vasya learned that, given two points with different $$$x$$$ coordinates, you can draw through them exactly one parabola with equation of type $$$y = x^2 + bx + c$$$, where $$$b$$$ and $$$c$$$ are reals. Let's call such a parabola an $$$U$$$-shaped one.Vasya drew several distinct points with integer coordinates on a plane and then drew an $$$U$$$-shaped parabola through each pair of the points that have different $$$x$$$ coordinates. The picture became somewhat messy, but Vasya still wants to count how many of the parabolas drawn don't have any drawn point inside their internal area. Help Vasya.The internal area of an $$$U$$$-shaped parabola is the part of the plane that lies strictly above the parabola when the $$$y$$$ axis is directed upwards.","['geometry', 'math']","[0, 1, 0, 1, 0, 0, 0, 0]"38"A picture can be represented as an $$$n\times m$$$ grid ($$$n$$$ rows and $$$m$$$ columns) so that each of the $$$n \cdot m$$$ cells is colored with one color. You have $$$k$$$ pigments of different colors. You have a limited amount of each pigment, more precisely you can color at most $$$a_i$$$ cells with the $$$i$$$-th pigment.A picture is considered beautiful if each cell has at least $$$3$$$ toroidal neighbors with the same color as itself.Two cells are considered toroidal neighbors if they toroidally share an edge. In other words, for some integers $$$1 \leq x_1,x_2 \leq n$$$ and $$$1 \leq y_1,y_2 \leq m$$$, the cell in the $$$x_1$$$-th row and $$$y_1$$$-th column is a toroidal neighbor of the cell in the $$$x_2$$$-th row and $$$y_2$$$-th column if one of following two conditions holds:  $$$x_1-x_2 \equiv \pm1 \pmod{n}$$$ and $$$y_1=y_2$$$, or  $$$y_1-y_2 \equiv \pm1 \pmod{m}$$$ and $$$x_1=x_2$$$. Notice that each cell has exactly $$$4$$$ toroidal neighbors. For example, if $$$n=3$$$ and $$$m=4$$$, the toroidal neighbors of the cell $$$(1, 2)$$$ (the cell on the first row and second column) are: $$$(3, 2)$$$, $$$(2, 2)$$$, $$$(1, 3)$$$, $$$(1, 1)$$$. They are shown in gray on the image below:  The gray cells show toroidal neighbors of $$$(1, 2)$$$. Is it possible to color all cells with the pigments provided and create a beautiful picture?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"39"You are given a binary matrix $$$a$$$ of size $$$n \times m$$$. A binary matrix is a matrix where each element is either $$$0$$$ or $$$1$$$.You may perform some (possibly zero) operations with this matrix. During each operation you can inverse the row of this matrix or a column of this matrix. Formally, inverting a row is changing all values in this row to the opposite ($$$0$$$ to $$$1$$$, $$$1$$$ to $$$0$$$). Inverting a column is changing all values in this column to the opposite.Your task is to sort the initial matrix by some sequence of such operations. The matrix is considered sorted if the array $$$[a_{1, 1}, a_{1, 2}, \dots, a_{1, m}, a_{2, 1}, a_{2, 2}, \dots, a_{2, m}, \dots, a_{n, m - 1}, a_{n, m}]$$$ is sorted in non-descending order.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"40"Consider an infinite triangle made up of layers. Let's number the layers, starting from one, from the top of the triangle (from top to bottom). The $$$k$$$-th layer of the triangle contains $$$k$$$ points, numbered from left to right. Each point of an infinite triangle is described by a pair of numbers $$$(r, c)$$$ ($$$1 \le c \le r$$$), where $$$r$$$ is the number of the layer, and $$$c$$$ is the number of the point in the layer. From each point $$$(r, c)$$$ there are two directed edges to the points $$$(r+1, c)$$$ and $$$(r+1, c+1)$$$, but only one of the edges is activated. If $$$r + c$$$ is even, then the edge to the point $$$(r+1, c)$$$ is activated, otherwise the edge to the point $$$(r+1, c+1)$$$ is activated. Look at the picture for a better understanding.  Activated edges are colored in black. Non-activated edges are colored in gray. From the point $$$(r_1, c_1)$$$ it is possible to reach the point $$$(r_2, c_2)$$$, if there is a path between them only from activated edges. For example, in the picture above, there is a path from $$$(1, 1)$$$ to $$$(3, 2)$$$, but there is no path from $$$(2, 1)$$$ to $$$(1, 1)$$$.Initially, you are at the point $$$(1, 1)$$$. For each turn, you can:   Replace activated edge for point $$$(r, c)$$$. That is if the edge to the point $$$(r+1, c)$$$ is activated, then instead of it, the edge to the point $$$(r+1, c+1)$$$ becomes activated, otherwise if the edge to the point $$$(r+1, c+1)$$$, then instead if it, the edge to the point $$$(r+1, c)$$$ becomes activated. This action increases the cost of the path by $$$1$$$;  Move from the current point to another by following the activated edge. This action does not increase the cost of the path. You are given a sequence of $$$n$$$ points of an infinite triangle $$$(r_1, c_1), (r_2, c_2), \ldots, (r_n, c_n)$$$. Find the minimum cost path from $$$(1, 1)$$$, passing through all $$$n$$$ points in arbitrary order.","['graphs', 'math']","[0, 0, 1, 1, 0, 0, 0, 0]"41"There are $$$n$$$ beautiful skyscrapers in New York, the height of the $$$i$$$-th one is $$$h_i$$$. Today some villains have set on fire first $$$n - 1$$$ of them, and now the only safety building is $$$n$$$-th skyscraper.Let's call a jump from $$$i$$$-th skyscraper to $$$j$$$-th ($$$i < j$$$) discrete, if all skyscrapers between are strictly lower or higher than both of them. Formally, jump is discrete, if $$$i < j$$$ and one of the following conditions satisfied:   $$$i + 1 = j$$$  $$$\max(h_{i + 1}, \ldots, h_{j - 1}) < \min(h_i, h_j)$$$  $$$\max(h_i, h_j) < \min(h_{i + 1}, \ldots, h_{j - 1})$$$. At the moment, Vasya is staying on the first skyscraper and wants to live a little longer, so his goal is to reach $$$n$$$-th skyscraper with minimal count of discrete jumps. Help him with calcualting this number.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"42"There are $$$n$$$ cities and $$$m$$$ bidirectional roads in the country. The roads in the country form an undirected weighted graph. The graph is not guaranteed to be connected. Each road has it's own parameter $$$w$$$. You can travel through the roads, but the government made a new law: you can only go through two roads at a time (go from city $$$a$$$ to city $$$b$$$ and then from city $$$b$$$ to city $$$c$$$) and you will have to pay $$$(w_{ab} + w_{bc})^2$$$ money to go through those roads. Find out whether it is possible to travel from city $$$1$$$ to every other city $$$t$$$ and what's the minimum amount of money you need to get from $$$1$$$ to $$$t$$$.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"43"Andrew and Eugene are playing a game. Initially, Andrew has string s, consisting of digits. Eugene sends Andrew multiple queries of type ""di → ti"", that means ""replace all digits di in string s with substrings equal to ti"". For example, if s = 123123, then query ""2 → 00"" transforms s to 10031003, and query ""3 → "" (""replace 3 by an empty string"") transforms it to s = 1212. After all the queries Eugene asks Andrew to find the remainder after division of number with decimal representation equal to s by 1000000007 (109 + 7). When you represent s as a decimal number, please ignore the leading zeroes; also if s is an empty string, then it's assumed that the number equals to zero.Andrew got tired of processing Eugene's requests manually and he asked you to write a program for that. Help him!",[],"[0, 0, 0, 0, 0, 0, 0, 0]"44"Little Dima has two sequences of points with integer coordinates: sequence (a1, 1), (a2, 2), ..., (an, n) and sequence (b1, 1), (b2, 2), ..., (bn, n).Now Dima wants to count the number of distinct sequences of points of length 2·n that can be assembled from these sequences, such that the x-coordinates of points in the assembled sequence will not decrease. Help him with that. Note that each element of the initial sequences should be used exactly once in the assembled sequence.Dima considers two assembled sequences (p1, q1), (p2, q2), ..., (p2·n, q2·n) and (x1, y1), (x2, y2), ..., (x2·n, y2·n) distinct, if there is such i (1 ≤ i ≤ 2·n), that (pi, qi) ≠ (xi, yi).As the answer can be rather large, print the remainder from dividing the answer by number m.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"45"You play a computer game. In this game, you lead a party of $$$m$$$ heroes, and you have to clear a dungeon with $$$n$$$ monsters. Each monster is characterized by its power $$$a_i$$$. Each hero is characterized by his power $$$p_i$$$ and endurance $$$s_i$$$.The heroes clear the dungeon day by day. In the beginning of each day, you choose a hero (exactly one) who is going to enter the dungeon this day.When the hero enters the dungeon, he is challenged by the first monster which was not defeated during the previous days (so, if the heroes have already defeated $$$k$$$ monsters, the hero fights with the monster $$$k + 1$$$). When the hero fights the monster, there are two possible outcomes:  if the monster's power is strictly greater than the hero's power, the hero retreats from the dungeon. The current day ends;  otherwise, the monster is defeated. After defeating a monster, the hero either continues fighting with the next monster or leaves the dungeon. He leaves the dungeon either if he has already defeated the number of monsters equal to his endurance during this day (so, the $$$i$$$-th hero cannot defeat more than $$$s_i$$$ monsters during each day), or if all monsters are defeated — otherwise, he fights with the next monster. When the hero leaves the dungeon, the current day ends.Your goal is to defeat the last monster. What is the minimum number of days that you need to achieve your goal? Each day you have to use exactly one hero; it is possible that some heroes don't fight the monsters at all. Each hero can be used arbitrary number of times.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"46"Little Tanya decided to present her dad a postcard on his Birthday. She has already created a message — string s of length n, consisting of uppercase and lowercase English letters. Tanya can't write yet, so she found a newspaper and decided to cut out the letters and glue them into the postcard to achieve string s. The newspaper contains string t, consisting of uppercase and lowercase English letters. We know that the length of string t greater or equal to the length of the string s.The newspaper may possibly have too few of some letters needed to make the text and too many of some other letters. That's why Tanya wants to cut some n letters out of the newspaper and make a message of length exactly n, so that it looked as much as possible like s. If the letter in some position has correct value and correct letter case (in the string s and in the string that Tanya will make), then she shouts joyfully ""YAY!"", and if the letter in the given position has only the correct value but it is in the wrong case, then the girl says ""WHOOPS"".Tanya wants to make such message that lets her shout ""YAY!"" as much as possible. If there are multiple ways to do this, then her second priority is to maximize the number of times she says ""WHOOPS"". Your task is to help Tanya make the message.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"47"Andryusha is an orderly boy and likes to keep things in their place.Today he faced a problem to put his socks in the wardrobe. He has n distinct pairs of socks which are initially in a bag. The pairs are numbered from 1 to n. Andryusha wants to put paired socks together and put them in the wardrobe. He takes the socks one by one from the bag, and for each sock he looks whether the pair of this sock has been already took out of the bag, or not. If not (that means the pair of this sock is still in the bag), he puts the current socks on the table in front of him. Otherwise, he puts both socks from the pair to the wardrobe.Andryusha remembers the order in which he took the socks from the bag. Can you tell him what is the maximum number of socks that were on the table at the same time? ",[],"[0, 0, 0, 0, 0, 0, 0, 0]"48"You are given a positive integer $$$m$$$ and two integer sequence: $$$a=[a_1, a_2, \ldots, a_n]$$$ and $$$b=[b_1, b_2, \ldots, b_n]$$$. Both of these sequence have a length $$$n$$$.Permutation is a sequence of $$$n$$$ different positive integers from $$$1$$$ to $$$n$$$. For example, these sequences are permutations: $$$[1]$$$, $$$[1,2]$$$, $$$[2,1]$$$, $$$[6,7,3,4,1,2,5]$$$. These are not: $$$[0]$$$, $$$[1,1]$$$, $$$[2,3]$$$.You need to find the non-negative integer $$$x$$$, and increase all elements of $$$a_i$$$ by $$$x$$$, modulo $$$m$$$ (i.e. you want to change $$$a_i$$$ to $$$(a_i + x) \bmod m$$$), so it would be possible to rearrange elements of $$$a$$$ to make it equal $$$b$$$, among them you need to find the smallest possible $$$x$$$.In other words, you need to find the smallest non-negative integer $$$x$$$, for which it is possible to find some permutation $$$p=[p_1, p_2, \ldots, p_n]$$$, such that for all $$$1 \leq i \leq n$$$, $$$(a_i + x) \bmod m = b_{p_i}$$$, where $$$y \bmod m$$$ — remainder of division of $$$y$$$ by $$$m$$$.For example, if $$$m=3$$$, $$$a = [0, 0, 2, 1], b = [2, 0, 1, 1]$$$, you can choose $$$x=1$$$, and $$$a$$$ will be equal to $$$[1, 1, 0, 2]$$$ and you can rearrange it to make it equal $$$[2, 0, 1, 1]$$$, which is equal to $$$b$$$.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"49"  William has two numbers $$$a$$$ and $$$b$$$ initially both equal to zero. William mastered performing three different operations with them quickly. Before performing each operation some positive integer $$$k$$$ is picked, which is then used to perform one of the following operations: (note, that for each operation you can choose a new positive integer $$$k$$$)  add number $$$k$$$ to both $$$a$$$ and $$$b$$$, or  add number $$$k$$$ to $$$a$$$ and subtract $$$k$$$ from $$$b$$$, or  add number $$$k$$$ to $$$b$$$ and subtract $$$k$$$ from $$$a$$$. Note that after performing operations, numbers $$$a$$$ and $$$b$$$ may become negative as well.William wants to find out the minimal number of operations he would have to perform to make $$$a$$$ equal to his favorite number $$$c$$$ and $$$b$$$ equal to his second favorite number $$$d$$$.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"50"You are given an array $$$a$$$, consisting of $$$n$$$ positive integers.Let's call a concatenation of numbers $$$x$$$ and $$$y$$$ the number that is obtained by writing down numbers $$$x$$$ and $$$y$$$ one right after another without changing the order. For example, a concatenation of numbers $$$12$$$ and $$$3456$$$ is a number $$$123456$$$.Count the number of ordered pairs of positions $$$(i, j)$$$ ($$$i \neq j$$$) in array $$$a$$$ such that the concatenation of $$$a_i$$$ and $$$a_j$$$ is divisible by $$$k$$$.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"51"While doing some spring cleaning, Daniel found an old calculator that he loves so much. However, it seems like it is broken. When he tries to compute $$$1 + 3$$$ using the calculator, he gets $$$2$$$ instead of $$$4$$$. But when he tries computing $$$1 + 4$$$, he gets the correct answer, $$$5$$$. Puzzled by this mystery, he opened up his calculator and found the answer to the riddle: the full adders became half adders! So, when he tries to compute the sum $$$a + b$$$ using the calculator, he instead gets the xorsum $$$a \oplus b$$$ (read the definition by the link: https://en.wikipedia.org/wiki/Exclusive_or).As he saw earlier, the calculator sometimes gives the correct answer. And so, he wonders, given integers $$$l$$$ and $$$r$$$, how many pairs of integers $$$(a, b)$$$ satisfy the following conditions: $$$$$$a + b = a \oplus b$$$$$$ $$$$$$l \leq a \leq r$$$$$$ $$$$$$l \leq b \leq r$$$$$$However, Daniel the Barman is going to the bar and will return in two hours. He tells you to solve the problem before he returns, or else you will have to enjoy being blocked.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"52"This is the hard version of the problem. You can find the easy version in the Div. 2 contest. Both versions only differ in the number of times you can ask your friend to taste coffee.This is an interactive problem.You're considering moving to another city, where one of your friends already lives. There are $$$n$$$ cafés in this city, where $$$n$$$ is a power of two. The $$$i$$$-th café produces a single variety of coffee $$$a_i$$$. As you're a coffee-lover, before deciding to move or not, you want to know the number $$$d$$$ of distinct varieties of coffees produced in this city.You don't know the values $$$a_1, \ldots, a_n$$$. Fortunately, your friend has a memory of size $$$k$$$, where $$$k$$$ is a power of two.Once per day, you can ask him to taste a cup of coffee produced by the café $$$c$$$, and he will tell you if he tasted a similar coffee during the last $$$k$$$ days.You can also ask him to take a medication that will reset his memory. He will forget all previous cups of coffee tasted. You can reset his memory at most $$$30\ 000$$$ times.More formally, the memory of your friend is a queue $$$S$$$. Doing a query on café $$$c$$$ will:   Tell you if $$$a_c$$$ is in $$$S$$$;  Add $$$a_c$$$ at the back of $$$S$$$;  If $$$|S| > k$$$, pop the front element of $$$S$$$. Doing a reset request will pop all elements out of $$$S$$$.Your friend can taste at most $$$\dfrac{3n^2}{2k}$$$ cups of coffee in total. Find the diversity $$$d$$$ (number of distinct values in the array $$$a$$$).Note that asking your friend to reset his memory does not count towards the number of times you ask your friend to taste a cup of coffee.In some test cases the behavior of the interactor is adaptive. It means that the array $$$a$$$ may be not fixed before the start of the interaction and may depend on your queries. It is guaranteed that at any moment of the interaction, there is at least one array $$$a$$$ consistent with all the answers given so far.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"53"This problem is interactive.You should guess hidden number $$$x$$$ which is between $$$1$$$ and $$$M = 10004205361450474$$$, inclusive.You could use up to $$$5$$$ queries.In each query, you can output an increasing sequence of $$$k \leq x$$$ integers, each between $$$1$$$ and $$$M$$$, inclusive, and you will obtain one of the following as an answer:  either the hidden number belongs to your query sequence, in this case you immediately win;  or you will be given where the hidden number is located with respect to your query sequence, that is, either it is less than all numbers from the sequence, greater than all numbers from the sequence, or you will be given such an $$$i$$$ that the hidden number $$$x$$$ is between the $$$i$$$-th and the $$$(i+1)$$$-st numbers of your sequence. See the interaction section for clarity.Be aware that the interactor is adaptive, i.e. the hidden number can depend on queries the solution makes. However, it is guaranteed that for any solution the interactor works non-distinguishable from the situation when the hidden number is fixed beforehand.Hacks are allowed only with fixed hidden number. A hack is represented by a single integer between $$$1$$$ and $$$M$$$. In all pretests the hidden number is fixed as well.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"54"You are given a sequence of positive integers a1, a2, ..., an. While possible, you perform the following operation: find a pair of equal consecutive elements. If there are more than one such pair, find the leftmost (with the smallest indices of elements). If the two integers are equal to x, delete both and insert a single integer x + 1 on their place. This way the number of elements in the sequence is decreased by 1 on each step. You stop performing the operation when there is no pair of equal consecutive elements.For example, if the initial sequence is [5, 2, 1, 1, 2, 2], then after the first operation you get [5, 2, 2, 2, 2], after the second — [5, 3, 2, 2], after the third — [5, 3, 3], and finally after the fourth you get [5, 4]. After that there are no equal consecutive elements left in the sequence, so you stop the process.Determine the final sequence after you stop performing the operation.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"55"A group of $$$n$$$ friends decide to go to a restaurant. Each of the friends plans to order meals for $$$x_i$$$ burles and has a total of $$$y_i$$$ burles ($$$1 \le i \le n$$$). The friends decide to split their visit to the restaurant into several days. Each day, some group of at least two friends goes to the restaurant. Each of the friends visits the restaurant no more than once (that is, these groups do not intersect). These groups must satisfy the condition that the total budget of each group must be not less than the amount of burles that the friends in the group are going to spend at the restaurant. In other words, the sum of all $$$x_i$$$ values in the group must not exceed the sum of $$$y_i$$$ values in the group.What is the maximum number of days friends can visit the restaurant?For example, let there be $$$n = 6$$$ friends for whom $$$x$$$ = [$$$8, 3, 9, 2, 4, 5$$$] and $$$y$$$ = [$$$5, 3, 1, 4, 5, 10$$$]. Then:   first and sixth friends can go to the restaurant on the first day. They will spend $$$8+5=13$$$ burles at the restaurant, and their total budget is $$$5+10=15$$$ burles. Since $$$15 \ge 13$$$, they can actually form a group.  friends with indices $$$2, 4, 5$$$ can form a second group. They will spend $$$3+2+4=9$$$ burles at the restaurant, and their total budget will be $$$3+4+5=12$$$ burles ($$$12 \ge 9$$$). It can be shown that they will not be able to form more groups so that each group has at least two friends and each group can pay the bill.So, the maximum number of groups the friends can split into is $$$2$$$. Friends will visit the restaurant for a maximum of two days. Note that the $$$3$$$-rd friend will not visit the restaurant at all.Output the maximum number of days the friends can visit the restaurant for given $$$n$$$, $$$x$$$ and $$$y$$$.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"56"Hyakugoku has just retired from being the resident deity of the South Black Snail Temple in order to pursue her dream of becoming a cartoonist. She spent six months in that temple just playing ""Cat's Cradle"" so now she wants to try a different game — ""Snakes and Ladders"". Unfortunately, she already killed all the snakes, so there are only ladders left now. The game is played on a $$$10 \times 10$$$ board as follows:  At the beginning of the game, the player is at the bottom left square.  The objective of the game is for the player to reach the Goal (the top left square) by following the path and climbing vertical ladders. Once the player reaches the Goal, the game ends.  The path is as follows: if a square is not the end of its row, it leads to the square next to it along the direction of its row; if a square is the end of its row, it leads to the square above it. The direction of a row is determined as follows: the direction of the bottom row is to the right; the direction of any other row is opposite the direction of the row below it. See Notes section for visualization of path.  During each turn, the player rolls a standard six-sided dice. Suppose that the number shown on the dice is $$$r$$$. If the Goal is less than $$$r$$$ squares away on the path, the player doesn't move (but the turn is performed). Otherwise, the player advances exactly $$$r$$$ squares along the path and then stops. If the player stops on a square with the bottom of a ladder, the player chooses whether or not to climb up that ladder. If she chooses not to climb, then she stays in that square for the beginning of the next turn.  Some squares have a ladder in them. Ladders are only placed vertically — each one leads to the same square of some of the upper rows. In order for the player to climb up a ladder, after rolling the dice, she must stop at the square containing the bottom of the ladder. After using the ladder, the player will end up in the square containing the top of the ladder. She cannot leave the ladder in the middle of climbing. And if the square containing the top of the ladder also contains the bottom of another ladder, she is not allowed to use that second ladder.  The numbers on the faces of the dice are 1, 2, 3, 4, 5, and 6, with each number having the same probability of being shown. Please note that:      it is possible for ladders to overlap, but the player cannot switch to the other ladder while in the middle of climbing the first one;     it is possible for ladders to go straight to the top row, but not any higher;     it is possible for two ladders to lead to the same tile;     it is possible for a ladder to lead to a tile that also has a ladder, but the player will not be able to use that second ladder if she uses the first one;     the player can only climb up ladders, not climb down. Hyakugoku wants to finish the game as soon as possible. Thus, on each turn she chooses whether to climb the ladder or not optimally. Help her to determine the minimum expected number of turns the game will take.",['probabilities'],"[0, 0, 0, 0, 0, 1, 0, 0]"57"Polycarpus likes studying at school a lot and he is always diligent about his homework. Polycarpus has never had any problems with natural sciences as his great-great-grandfather was the great physicist Seinstein. On the other hand though, Polycarpus has never had an easy time with history.Everybody knows that the World history encompasses exactly n events: the i-th event had continued from the year ai to the year bi inclusive (ai < bi). Polycarpus easily learned the dates when each of n events started and ended (Polycarpus inherited excellent memory from his great-great-granddad). But the teacher gave him a more complicated task: Polycaprus should know when all events began and ended and he should also find out for each event whether it includes another event. Polycarpus' teacher thinks that an event j includes an event i if aj < ai and bi < bj. Your task is simpler: find the number of events that are included in some other event.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"58"In Walrusland public transport tickets are characterized by two integers: by the number of the series and by the number of the ticket in the series. Let the series number be represented by a and the ticket number — by b, then a ticket is described by the ordered pair of numbers (a, b). The walruses believe that a ticket is lucky if a * b = rev(a) * rev(b). The function rev(x) reverses a number written in the decimal system, at that the leading zeroes disappear. For example, rev(12343) = 34321, rev(1200) = 21.The Public Transport Management Committee wants to release x series, each containing y tickets, so that at least w lucky tickets were released and the total number of released tickets (x * y) were minimum. The series are numbered from 1 to x inclusive. The tickets in each series are numbered from 1 to y inclusive. The Transport Committee cannot release more than maxx series and more than maxy tickets in one series.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"59"Inna and Dima bought a table of size n × m in the shop. Each cell of the table contains a single letter: ""D"", ""I"", ""M"", ""A"".Inna loves Dima, so she wants to go through his name as many times as possible as she moves through the table. For that, Inna acts as follows:  initially, Inna chooses some cell of the table where letter ""D"" is written;  then Inna can move to some side-adjacent table cell that contains letter ""I""; then from this cell she can go to one of the side-adjacent table cells that contains the written letter ""M""; then she can go to a side-adjacent cell that contains letter ""A"". Then Inna assumes that she has gone through her sweetheart's name;  Inna's next move can be going to one of the side-adjacent table cells that contains letter ""D"" and then walk on through name DIMA in the similar manner. Inna never skips a letter. So, from the letter ""D"" she always goes to the letter ""I"", from the letter ""I"" she always goes the to letter ""M"", from the letter ""M"" she always goes to the letter ""A"", and from the letter ""A"" she always goes to the letter ""D"". Depending on the choice of the initial table cell, Inna can go through name DIMA either an infinite number of times or some positive finite number of times or she can't go through his name once. Help Inna find out what maximum number of times she can go through name DIMA.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"60"Reading books is one of Sasha's passions. Once while he was reading one book, he became acquainted with an unusual character. The character told about himself like that: ""Many are my names in many countries. Mithrandir among the Elves, Tharkûn to the Dwarves, Olórin I was in my youth in the West that is forgotten, in the South Incánus, in the North Gandalf; to the East I go not.""And at that moment Sasha thought, how would that character be called in the East? In the East all names are palindromes. A string is a palindrome if it reads the same backward as forward. For example, such strings as ""kazak"", ""oo"" and ""r"" are palindromes, but strings ""abb"" and ""ij"" are not. Sasha believed that the hero would be named after one of the gods of the East. As long as there couldn't be two equal names, so in the East people did the following: they wrote the original name as a string on a piece of paper, then cut the paper minimum number of times $$$k$$$, so they got $$$k+1$$$ pieces of paper with substrings of the initial string, and then unite those pieces together to get a new string. Pieces couldn't be turned over, they could be shuffled.In this way, it's possible to achive a string abcdefg from the string f|de|abc|g using $$$3$$$ cuts (by swapping papers with substrings f and abc). The string cbadefg can't be received using the same cuts.More formally, Sasha wants for the given palindrome $$$s$$$ find such minimum $$$k$$$, that you can cut this string into $$$k + 1$$$ parts, and then unite them in such a way that the final string will be a palindrome and it won't be equal to the initial string $$$s$$$. It there is no answer, then print ""Impossible"" (without quotes).",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"61"Santa has $$$n$$$ candies and he wants to gift them to $$$k$$$ kids. He wants to divide as many candies as possible between all $$$k$$$ kids. Santa can't divide one candy into parts but he is allowed to not use some candies at all.Suppose the kid who recieves the minimum number of candies has $$$a$$$ candies and the kid who recieves the maximum number of candies has $$$b$$$ candies. Then Santa will be satisfied, if the both conditions are met at the same time:  $$$b - a \le 1$$$ (it means $$$b = a$$$ or $$$b = a + 1$$$);  the number of kids who has $$$a+1$$$ candies (note that $$$a+1$$$ not necessarily equals $$$b$$$) does not exceed $$$\lfloor\frac{k}{2}\rfloor$$$ (less than or equal to $$$\lfloor\frac{k}{2}\rfloor$$$). $$$\lfloor\frac{k}{2}\rfloor$$$ is $$$k$$$ divided by $$$2$$$ and rounded down to the nearest integer. For example, if $$$k=5$$$ then $$$\lfloor\frac{k}{2}\rfloor=\lfloor\frac{5}{2}\rfloor=2$$$.Your task is to find the maximum number of candies Santa can give to kids so that he will be satisfied.You have to answer $$$t$$$ independent test cases.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"62"Let's denote the size of the maximum matching in a graph $$$G$$$ as $$$\mathit{MM}(G)$$$.You are given a bipartite graph. The vertices of the first part are numbered from $$$1$$$ to $$$n$$$, the vertices of the second part are numbered from $$$n+1$$$ to $$$2n$$$. Each vertex's degree is $$$2$$$.For a tuple of four integers $$$(l, r, L, R)$$$, where $$$1 \le l \le r \le n$$$ and $$$n+1 \le L \le R \le 2n$$$, let's define $$$G'(l, r, L, R)$$$ as the graph which consists of all vertices of the given graph that are included in the segment $$$[l, r]$$$ or in the segment $$$[L, R]$$$, and all edges of the given graph such that each of their endpoints belongs to one of these segments. In other words, to obtain $$$G'(l, r, L, R)$$$ from the original graph, you have to remove all vertices $$$i$$$ such that $$$i \notin [l, r]$$$ and $$$i \notin [L, R]$$$, and all edges incident to these vertices.Calculate the sum of $$$\mathit{MM}(G(l, r, L, R))$$$ over all tuples of integers $$$(l, r, L, R)$$$ having $$$1 \le l \le r \le n$$$ and $$$n+1 \le L \le R \le 2n$$$.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"63"Given an array $$$a$$$ of $$$n$$$ integers and an integer $$$k$$$ ($$$2 \le k \le n$$$), where each element of the array is denoted by $$$a_i$$$ ($$$0 \le i < n$$$). Perform the operation $$$z$$$ given below on $$$a$$$ and print the value of $$$z(a,k)$$$ modulo $$$10^{9}+7$$$.function z(array a, integer k):    if length(a) < k:        return 0    else:        b = empty array        ans = 0        for i = 0 .. (length(a) - k):            temp = a[i]            for j = i .. (i + k - 1):                temp = max(temp, a[j])            append temp to the end of b            ans = ans + temp        return ans + z(b, k)",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"64"Seryozha has a very changeable character. This time he refused to leave the room to Dima and his girlfriend (her hame is Inna, by the way). However, the two lovebirds can always find a way to communicate. Today they are writing text messages to each other.Dima and Inna are using a secret code in their text messages. When Dima wants to send Inna some sentence, he writes out all words, inserting a heart before each word and after the last word. A heart is a sequence of two characters: the ""less"" characters (<) and the digit three (3). After applying the code, a test message looks like that: <3word1<3word2<3 ... wordn<3.Encoding doesn't end here. Then Dima inserts a random number of small English characters, digits, signs ""more"" and ""less"" into any places of the message.Inna knows Dima perfectly well, so she knows what phrase Dima is going to send her beforehand. Inna has just got a text message. Help her find out if Dima encoded the message correctly. In other words, find out if a text message could have been received by encoding in the manner that is described above.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"65"Nikolay has only recently started in competitive programming, but already qualified to the finals of one prestigious olympiad. There going to be $$$n$$$ participants, one of whom is Nikolay. Like any good olympiad, it consists of two rounds. Tired of the traditional rules, in which the participant who solved the largest number of problems wins, the organizers came up with different rules.Suppose in the first round participant A took $$$x$$$-th place and in the second round — $$$y$$$-th place. Then the total score of the participant A is sum $$$x + y$$$. The overall place of the participant A is the number of participants (including A) having their total score less than or equal to the total score of A. Note, that some participants may end up having a common overall place. It is also important to note, that in both the first and the second round there were no two participants tying at a common place. In other words, for every $$$i$$$ from $$$1$$$ to $$$n$$$ exactly one participant took $$$i$$$-th place in first round and exactly one participant took $$$i$$$-th place in second round.Right after the end of the Olympiad, Nikolay was informed that he got $$$x$$$-th place in first round and $$$y$$$-th place in the second round. Nikolay doesn't know the results of other participants, yet he wonders what is the minimum and maximum place he can take, if we consider the most favorable and unfavorable outcome for him. Please help Nikolay to find the answer to this question.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"66"You are given n strings s1, s2, ..., sn consisting of characters 0 and 1. m operations are performed, on each of them you concatenate two existing strings into a new one. On the i-th operation the concatenation saisbi is saved into a new string sn + i (the operations are numbered starting from 1). After each operation you need to find the maximum positive integer k such that all possible strings consisting of 0 and 1 of length k (there are 2k such strings) are substrings of the new string. If there is no such k, print 0.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"67"ZS the Coder and Chris the Baboon has arrived at Udayland! They walked in the park where n trees grow. They decided to be naughty and color the trees in the park. The trees are numbered with integers from 1 to n from left to right.Initially, tree i has color ci. ZS the Coder and Chris the Baboon recognizes only m different colors, so 0 ≤ ci ≤ m, where ci = 0 means that tree i is uncolored.ZS the Coder and Chris the Baboon decides to color only the uncolored trees, i.e. the trees with ci = 0. They can color each of them them in any of the m colors from 1 to m. Coloring the i-th tree with color j requires exactly pi, j litres of paint.The two friends define the beauty of a coloring of the trees as the minimum number of contiguous groups (each group contains some subsegment of trees) you can split all the n trees into so that each group contains trees of the same color. For example, if the colors of the trees from left to right are 2, 1, 1, 1, 3, 2, 2, 3, 1, 3, the beauty of the coloring is 7, since we can partition the trees into 7 contiguous groups of the same color : {2}, {1, 1, 1}, {3}, {2, 2}, {3}, {1}, {3}. ZS the Coder and Chris the Baboon wants to color all uncolored trees so that the beauty of the coloring is exactly k. They need your help to determine the minimum amount of paint (in litres) needed to finish the job.Please note that the friends can't color the trees that are already colored.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"68"Vasya has an array of integers of length n.Vasya performs the following operations on the array: on each step he finds the longest segment of consecutive equal integers (the leftmost, if there are several such segments) and removes it. For example, if Vasya's array is [13, 13, 7, 7, 7, 2, 2, 2], then after one operation it becomes [13, 13, 2, 2, 2].Compute the number of operations Vasya should make until the array becomes empty, i.e. Vasya removes all elements from it.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"69"Arnie the Worm has finished eating an apple house yet again and decided to move. He made up his mind on the plan, the way the rooms are located and how they are joined by corridors. He numbered all the rooms from 1 to n. All the corridors are bidirectional.Arnie wants the new house to look just like the previous one. That is, it should have exactly n rooms and, if a corridor from room i to room j existed in the old house, it should be built in the new one. We know that during the house constructing process Arnie starts to eat an apple starting from some room and only stops when he eats his way through all the corridors and returns to the starting room. It is also known that Arnie eats without stopping. That is, until Arnie finishes constructing the house, he is busy every moment of his time gnawing a new corridor. Arnie doesn't move along the already built corridors.However, gnawing out corridors in one and the same order any time you change a house is a very difficult activity. That's why Arnie, knowing the order in which the corridors were located in the previous house, wants to gnaw corridors in another order. It is represented as a list of rooms in the order in which they should be visited. The new list should be lexicographically smallest, but it also should be strictly lexicographically greater than the previous one. Help the worm. ",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"70"Peter wrote on the board a strictly increasing sequence of positive integers a1, a2, ..., an. Then Vasil replaced some digits in the numbers of this sequence by question marks. Thus, each question mark corresponds to exactly one lost digit.Restore the the original sequence knowing digits remaining on the board.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"71"Ivan plays an old action game called Heretic. He's stuck on one of the final levels of this game, so he needs some help with killing the monsters.The main part of the level is a large corridor (so large and narrow that it can be represented as an infinite coordinate line). The corridor is divided into two parts; let's assume that the point $$$x = 0$$$ is where these parts meet.The right part of the corridor is filled with $$$n$$$ monsters — for each monster, its initial coordinate $$$x_i$$$ is given (and since all monsters are in the right part, every $$$x_i$$$ is positive).The left part of the corridor is filled with crusher traps. If some monster enters the left part of the corridor or the origin (so, its current coordinate becomes less than or equal to $$$0$$$), it gets instantly killed by a trap.The main weapon Ivan uses to kill the monsters is the Phoenix Rod. It can launch a missile that explodes upon impact, obliterating every monster caught in the explosion and throwing all other monsters away from the epicenter. Formally, suppose that Ivan launches a missile so that it explodes in the point $$$c$$$. Then every monster is either killed by explosion or pushed away. Let some monster's current coordinate be $$$y$$$, then:  if $$$c = y$$$, then the monster is killed;  if $$$y < c$$$, then the monster is pushed $$$r$$$ units to the left, so its current coordinate becomes $$$y - r$$$;  if $$$y > c$$$, then the monster is pushed $$$r$$$ units to the right, so its current coordinate becomes $$$y + r$$$. Ivan is going to kill the monsters as follows: choose some integer point $$$d$$$ and launch a missile into that point, then wait until it explodes and all the monsters which are pushed to the left part of the corridor are killed by crusher traps, then, if at least one monster is still alive, choose another integer point (probably the one that was already used) and launch a missile there, and so on.What is the minimum number of missiles Ivan has to launch in order to kill all of the monsters? You may assume that every time Ivan fires the Phoenix Rod, he chooses the impact point optimally.You have to answer $$$q$$$ independent queries.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"72"A bracket sequence is called regular if it is possible to obtain correct arithmetic expression by inserting characters «+» and «1» into this sequence. For example, sequences «(())()», «()» and «(()(()))» are regular, while «)(», «(()» and «(()))(» are not.One day Johnny got bracket sequence. He decided to remove some of the brackets from it in order to obtain a regular bracket sequence. What is the maximum length of a regular bracket sequence which can be obtained?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"73"The weight of a sequence is defined as the number of unordered pairs of indexes $$$(i,j)$$$ (here $$$i \lt j$$$) with same value ($$$a_{i} = a_{j}$$$). For example, the weight of sequence $$$a = [1, 1, 2, 2, 1]$$$ is $$$4$$$. The set of unordered pairs of indexes with same value are $$$(1, 2)$$$, $$$(1, 5)$$$, $$$(2, 5)$$$, and $$$(3, 4)$$$.You are given a sequence $$$a$$$ of $$$n$$$ integers. Print the sum of the weight of all subsegments of $$$a$$$. A sequence $$$b$$$ is a subsegment of a sequence $$$a$$$ if $$$b$$$ can be obtained from $$$a$$$ by deletion of several (possibly, zero or all) elements from the beginning and several (possibly, zero or all) elements from the end.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"74"Luis has a sequence of $$$n+1$$$ integers $$$a_1, a_2, \ldots, a_{n+1}$$$. For each $$$i = 1, 2, \ldots, n+1$$$ it is guaranteed that $$$0\leq a_i < n$$$, or $$$a_i=n^2$$$. He has calculated the sum of all the elements of the sequence, and called this value $$$s$$$. Luis has lost his sequence, but he remembers the values of $$$n$$$ and $$$s$$$. Can you find the number of elements in the sequence that are equal to $$$n^2$$$?We can show that the answer is unique under the given constraints.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"75"There is a binary string $$$a$$$ of length $$$n$$$. In one operation, you can select any prefix of $$$a$$$ with an equal number of $$$0$$$ and $$$1$$$ symbols. Then all symbols in the prefix are inverted: each $$$0$$$ becomes $$$1$$$ and each $$$1$$$ becomes $$$0$$$.For example, suppose $$$a=0111010000$$$.   In the first operation, we can select the prefix of length $$$8$$$ since it has four $$$0$$$'s and four $$$1$$$'s: $$$[01110100]00\to [10001011]00$$$.  In the second operation, we can select the prefix of length $$$2$$$ since it has one $$$0$$$ and one $$$1$$$: $$$[10]00101100\to [01]00101100$$$.  It is illegal to select the prefix of length $$$4$$$ for the third operation, because it has three $$$0$$$'s and one $$$1$$$. Can you transform the string $$$a$$$ into the string $$$b$$$ using some finite number of operations (possibly, none)?",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"76"This is a harder version of the problem. In this version $$$n \le 500\,000$$$The outskirts of the capital are being actively built up in Berland. The company ""Kernel Panic"" manages the construction of a residential complex of skyscrapers in New Berlskva. All skyscrapers are built along the highway. It is known that the company has already bought $$$n$$$ plots along the highway and is preparing to build $$$n$$$ skyscrapers, one skyscraper per plot.Architects must consider several requirements when planning a skyscraper. Firstly, since the land on each plot has different properties, each skyscraper has a limit on the largest number of floors it can have. Secondly, according to the design code of the city, it is unacceptable for a skyscraper to simultaneously have higher skyscrapers both to the left and to the right of it.Formally, let's number the plots from $$$1$$$ to $$$n$$$. Then if the skyscraper on the $$$i$$$-th plot has $$$a_i$$$ floors, it must hold that $$$a_i$$$ is at most $$$m_i$$$ ($$$1 \le a_i \le m_i$$$). Also there mustn't be integers $$$j$$$ and $$$k$$$ such that $$$j < i < k$$$ and $$$a_j > a_i < a_k$$$. Plots $$$j$$$ and $$$k$$$ are not required to be adjacent to $$$i$$$.The company wants the total number of floors in the built skyscrapers to be as large as possible. Help it to choose the number of floors for each skyscraper in an optimal way, i.e. in such a way that all requirements are fulfilled, and among all such construction plans choose any plan with the maximum possible total number of floors.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"77"You are given two arrays of length $$$n$$$: $$$a_1, a_2, \dots, a_n$$$ and $$$b_1, b_2, \dots, b_n$$$.You can perform the following operation any number of times:   Choose integer index $$$i$$$ ($$$1 \le i \le n$$$);  Swap $$$a_i$$$ and $$$b_i$$$. What is the minimum possible sum $$$|a_1 - a_2| + |a_2 - a_3| + \dots + |a_{n-1} - a_n|$$$ $$$+$$$ $$$|b_1 - b_2| + |b_2 - b_3| + \dots + |b_{n-1} - b_n|$$$ (in other words, $$$\sum\limits_{i=1}^{n - 1}{\left(|a_i - a_{i+1}| + |b_i - b_{i+1}|\right)}$$$) you can achieve after performing several (possibly, zero) operations?",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"78"Several days ago you bought a new house and now you are planning to start a renovation. Since winters in your region can be very cold you need to decide how to heat rooms in your house.Your house has $$$n$$$ rooms. In the $$$i$$$-th room you can install at most $$$c_i$$$ heating radiators. Each radiator can have several sections, but the cost of the radiator with $$$k$$$ sections is equal to $$$k^2$$$ burles.Since rooms can have different sizes, you calculated that you need at least $$$sum_i$$$ sections in total in the $$$i$$$-th room. For each room calculate the minimum cost to install at most $$$c_i$$$ radiators with total number of sections not less than $$$sum_i$$$.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"79"Now it's time of Olympiads. Vanya and Egor decided to make his own team to take part in a programming Olympiad. They've been best friends ever since primary school and hopefully, that can somehow help them in teamwork.For each team Olympiad, Vanya takes his play cards with numbers. He takes only the cards containing numbers 1 and 0. The boys are very superstitious. They think that they can do well at the Olympiad if they begin with laying all the cards in a row so that:  there wouldn't be a pair of any side-adjacent cards with zeroes in a row;  there wouldn't be a group of three consecutive cards containing numbers one. Today Vanya brought n cards with zeroes and m cards with numbers one. The number of cards was so much that the friends do not know how to put all those cards in the described way. Help them find the required arrangement of the cards or else tell the guys that it is impossible to arrange cards in such a way.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"80"Egor came up with a new chips puzzle and suggests you to play.The puzzle has the form of a table with $$$n$$$ rows and $$$m$$$ columns, each cell can contain several black or white chips placed in a row. Thus, the state of the cell can be described by a string consisting of characters '0' (a white chip) and '1' (a black chip), possibly empty, and the whole puzzle can be described as a table, where each cell is a string of zeros and ones. The task is to get from one state of the puzzle some other state.To do this, you can use the following operation.  select 2 different cells $$$(x_1, y_1)$$$ and $$$(x_2, y_2)$$$: the cells must be in the same row or in the same column of the table, and the string in the cell $$$(x_1, y_1)$$$ must be non-empty;  in one operation you can move the last character of the string at the cell $$$(x_1, y_1)$$$ to the beginning of the string at the cell $$$(x_2, y_2)$$$. Egor came up with two states of the table for you: the initial state and the final one. It is guaranteed that the number of zeros and ones in the tables are the same. Your goal is with several operations get the final state from the initial state. Of course, Egor does not want the number of operations to be very large. Let's denote as $$$s$$$ the number of characters in each of the tables (which are the same). Then you should use no more than $$$4 \cdot s$$$ operations.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"81"An important meeting is to be held and there are exactly $$$n$$$ people invited. At any moment, any two people can step back and talk in private. The same two people can talk several (as many as they want) times per meeting.Each person has limited sociability. The sociability of the $$$i$$$-th person is a non-negative integer $$$a_i$$$. This means that after exactly $$$a_i$$$ talks this person leaves the meeting (and does not talk to anyone else anymore). If $$$a_i = 0$$$, the $$$i$$$-th person leaves the meeting immediately after it starts.A meeting is considered most productive if the maximum possible number of talks took place during it.You are given an array of sociability $$$a$$$, determine which people should talk to each other so that the total number of talks is as large as possible.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"82"You have an array of positive integers a[1], a[2], ..., a[n] and a set of bad prime numbers b1, b2, ..., bm. The prime numbers that do not occur in the set b are considered good. The beauty of array a is the sum , where function f(s) is determined as follows:  f(1) = 0;  Let's assume that p is the minimum prime divisor of s. If p is a good prime, then , otherwise . You are allowed to perform an arbitrary (probably zero) number of operations to improve array a. The operation of improvement is the following sequence of actions:  Choose some number r (1 ≤ r ≤ n) and calculate the value g = GCD(a[1], a[2], ..., a[r]).  Apply the assignments: , , ..., . What is the maximum beauty of the array you can get? ","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"83"My orzlers, we can optimize this problem from $$$O(S^3)$$$ to $$$O\left(T^\frac{5}{9}\right)$$$!— Spyofgame, founder of Orzlim religionA long time ago, Spyofgame invented the famous array $$$a$$$ ($$$1$$$-indexed) of length $$$n$$$ that contains information about the world and life. After that, he decided to convert it into the matrix $$$b$$$ ($$$0$$$-indexed) of size $$$(n + 1) \times (n + 1)$$$ which contains information about the world, life and beyond.Spyofgame converted $$$a$$$ into $$$b$$$ with the following rules.  $$$b_{i,0} = 0$$$ if $$$0 \leq i \leq n$$$;  $$$b_{0,i} = a_{i}$$$ if $$$1 \leq i \leq n$$$;  $$$b_{i,j} = b_{i,j-1} \oplus b_{i-1,j}$$$ if $$$1 \leq i, j \leq n$$$. Here $$$\oplus$$$ denotes the bitwise XOR operation.Today, archaeologists have discovered the famous matrix $$$b$$$. However, many elements of the matrix has been lost. They only know the values of $$$b_{i,n}$$$ for $$$1 \leq i \leq n$$$ (note that these are some elements of the last column, not the last row).The archaeologists want to know what a possible array of $$$a$$$ is. Can you help them reconstruct any array that could be $$$a$$$?",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"84"Everyone knows that agents in Valorant decide, who will play as attackers, and who will play as defenders. To do that Raze and Breach decided to play $$$t$$$ matches of a digit game...In each of $$$t$$$ matches of the digit game, a positive integer is generated. It consists of $$$n$$$ digits. The digits of this integer are numerated from $$$1$$$ to $$$n$$$ from the highest-order digit to the lowest-order digit. After this integer is announced, the match starts.Agents play in turns. Raze starts. In one turn an agent can choose any unmarked digit and mark it. Raze can choose digits on odd positions, but can not choose digits on even positions. Breach can choose digits on even positions, but can not choose digits on odd positions. The match ends, when there is only one unmarked digit left. If the single last digit is odd, then Raze wins, else Breach wins.It can be proved, that before the end of the match (for every initial integer with $$$n$$$ digits) each agent has an ability to make a turn, i.e. there is at least one unmarked digit, that stands on a position of required parity.For each of $$$t$$$ matches find out, which agent wins, if both of them want to win and play optimally.",['games'],"[1, 0, 0, 0, 0, 0, 0, 0]"85"Немногие знают, что сотрудники ВКонтакте могут менять цвет подсветки в куполе знаменитого Дома Зингера, где расположена штаб-квартира ВКонтакте. Для этого нужно всего лишь отправить сообщение с цветом в специальный чат «Зингер | color», а бот его распознает и сменит подсветку. При этом на время городских мероприятий смена цвета блокируется.Формально, бот обрабатывает три типа сообщений:   lock: заблокировать изменение цвета. Если оно и так заблокировано на данный момент, сообщение игнорируется.  unlock: разблокировать изменение цвета. Если оно и так разблокировано на данный момент, сообщение игнорируется.  red / orange / yellow / green / blue / indigo / violet: изменить цвет купола на заданный, если изменение цвета на данный момент не заблокировано. Вам дана история сообщений, полученных ботом, в хронологическом порядке. Считайте, что перед получением первого сообщения купол подсвечивается голубым (blue), а изменение цвета не заблокировано.Определите, какой цвет будет у купола Дома Зингера после обработки этих сообщений.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"86"Let's call a set of positive integers $$$a_1, a_2, \dots, a_k$$$ quadratic if the product of the factorials of its elements is a square of an integer, i. e. $$$\prod\limits_{i=1}^{k} a_i! = m^2$$$, for some integer $$$m$$$.You are given a positive integer $$$n$$$.Your task is to find a quadratic subset of a set $$$1, 2, \dots, n$$$ of maximum size. If there are multiple answers, print any of them.","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"87"You are given a rooted tree with n vertices. In each leaf vertex there's a single integer — the number of apples in this vertex. The weight of a subtree is the sum of all numbers in this subtree leaves. For instance, the weight of a subtree that corresponds to some leaf is the number written in the leaf.A tree is balanced if for every vertex v of the tree all its subtrees, corresponding to the children of vertex v, are of equal weight. Count the minimum number of apples that you need to remove from the tree (specifically, from some of its leaves) in order to make the tree balanced. Notice that you can always achieve the goal by just removing all apples.","['number theory', 'trees']","[0, 0, 0, 0, 1, 0, 0, 1]"88"Pavel made a photo of his favourite stars in the sky. His camera takes a photo of all points of the sky that belong to some rectangle with sides parallel to the coordinate axes.Strictly speaking, it makes a photo of all points with coordinates $$$(x, y)$$$, such that $$$x_1 \leq x \leq x_2$$$ and $$$y_1 \leq y \leq y_2$$$, where $$$(x_1, y_1)$$$ and $$$(x_2, y_2)$$$ are coordinates of the left bottom and the right top corners of the rectangle being photographed. The area of this rectangle can be zero.After taking the photo, Pavel wrote down coordinates of $$$n$$$ of his favourite stars which appeared in the photo. These points are not necessarily distinct, there can be multiple stars in the same point of the sky.Pavel has lost his camera recently and wants to buy a similar one. Specifically, he wants to know the dimensions of the photo he took earlier. Unfortunately, the photo is also lost. His notes are also of not much help; numbers are written in random order all over his notepad, so it's impossible to tell which numbers specify coordinates of which points.Pavel asked you to help him to determine what are the possible dimensions of the photo according to his notes. As there are multiple possible answers, find the dimensions with the minimal possible area of the rectangle.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"89"You are given array $$$a_1, a_2, \dots, a_n$$$. Find the subsegment $$$a_l, a_{l+1}, \dots, a_r$$$ ($$$1 \le l \le r \le n$$$) with maximum arithmetic mean $$$\frac{1}{r - l + 1}\sum\limits_{i=l}^{r}{a_i}$$$ (in floating-point numbers, i.e. without any rounding).If there are many such subsegments find the longest one.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"90"You are given an array $$$a$$$ consisting of $$$n$$$ positive integers. You have to choose a positive integer $$$d$$$ and paint all elements into two colors. All elements which are divisible by $$$d$$$ will be painted red, and all other elements will be painted blue.The coloring is called beautiful if there are no pairs of adjacent elements with the same color in the array. Your task is to find any value of $$$d$$$ which yields a beautiful coloring, or report that it is impossible.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"91"Allen wants to enter a fan zone that occupies a round square and has $$$n$$$ entrances.There already is a queue of $$$a_i$$$ people in front of the $$$i$$$-th entrance. Each entrance allows one person from its queue to enter the fan zone in one minute.Allen uses the following strategy to enter the fan zone:   Initially he stands in the end of the queue in front of the first entrance.  Each minute, if he is not allowed into the fan zone during the minute (meaning he is not the first in the queue), he leaves the current queue and stands in the end of the queue of the next entrance (or the first entrance if he leaves the last entrance). Determine the entrance through which Allen will finally enter the fan zone.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"92"The legendary Farmer John is throwing a huge party, and animals from all over the world are hanging out at his house. His guests are hungry, so he instructs his cow Bessie to bring out the snacks! Moo!There are $$$n$$$ snacks flavors, numbered with integers $$$1, 2, \ldots, n$$$. Bessie has $$$n$$$ snacks, one snack of each flavor. Every guest has exactly two favorite flavors. The procedure for eating snacks will go as follows:  First, Bessie will line up the guests in some way.  Then in this order, guests will approach the snacks one by one.  Each guest in their turn will eat all remaining snacks of their favorite flavor. In case no favorite flavors are present when a guest goes up, they become very sad.  Help Bessie to minimize the number of sad guests by lining the guests in an optimal way.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"93"There are n knights sitting at the Round Table at an equal distance from each other. Each of them is either in a good or in a bad mood.Merlin, the wizard predicted to King Arthur that the next month will turn out to be particularly fortunate if the regular polygon can be found. On all vertices of the polygon knights in a good mood should be located. Otherwise, the next month will bring misfortunes.A convex polygon is regular if all its sides have same length and all his angles are equal. In this problem we consider only regular polygons with at least 3 vertices, i. e. only nondegenerated.On a picture below some examples of such polygons are present. Green points mean knights in a good mood. Red points mean ones in a bad mood.  King Arthur knows the knights' moods. Help him find out if the next month will be fortunate or not.","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"94Vasya decided to write an anonymous letter cutting the letters out of a newspaper heading. He knows heading s1 and text s2 that he wants to send. Vasya can use every single heading letter no more than once. Vasya doesn't have to cut the spaces out of the heading — he just leaves some blank space to mark them. Help him; find out if he will manage to compose the needed text.,['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"95"You are given a tube which is reflective inside represented as two non-coinciding, but parallel to $$$Ox$$$ lines. Each line has some special integer points — positions of sensors on sides of the tube.You are going to emit a laser ray in the tube. To do so, you have to choose two integer points $$$A$$$ and $$$B$$$ on the first and the second line respectively (coordinates can be negative): the point $$$A$$$ is responsible for the position of the laser, and the point $$$B$$$ — for the direction of the laser ray. The laser ray is a ray starting at $$$A$$$ and directed at $$$B$$$ which will reflect from the sides of the tube (it doesn't matter if there are any sensors at a reflection point or not). A sensor will only register the ray if the ray hits exactly at the position of the sensor.   Examples of laser rays. Note that image contains two examples. The $$$3$$$ sensors (denoted by black bold points on the tube sides) will register the blue ray but only $$$2$$$ will register the red. Calculate the maximum number of sensors which can register your ray if you choose points $$$A$$$ and $$$B$$$ on the first and the second lines respectively.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"96"Little penguin Polo adores integer segments, that is, pairs of integers [l; r] (l ≤ r). He has a set that consists of n integer segments: [l1; r1], [l2; r2], ..., [ln; rn]. We know that no two segments of this set intersect. In one move Polo can either widen any segment of the set 1 unit to the left or 1 unit to the right, that is transform [l; r] to either segment [l - 1; r], or to segment [l; r + 1].The value of a set of segments that consists of n segments [l1; r1], [l2; r2], ..., [ln; rn] is the number of integers x, such that there is integer j, for which the following inequality holds, lj ≤ x ≤ rj.Find the minimum number of moves needed to make the value of the set of Polo's segments divisible by k.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"97"There are $$$n$$$ coins labeled from $$$1$$$ to $$$n$$$. Initially, coin $$$c_i$$$ is on position $$$i$$$ and is facing upwards (($$$c_1, c_2, \dots, c_n)$$$ is a permutation of numbers from $$$1$$$ to $$$n$$$). You can do some operations on these coins. In one operation, you can do the following:Choose $$$2$$$ distinct indices $$$i$$$ and $$$j$$$.Then, swap the coins on positions $$$i$$$ and $$$j$$$.Then, flip both coins on positions $$$i$$$ and $$$j$$$. (If they are initially faced up, they will be faced down after the operation and vice versa)Construct a sequence of at most $$$n+1$$$ operations such that after performing all these operations the coin $$$i$$$ will be on position $$$i$$$ at the end, facing up.Note that you do not need to minimize the number of operations.","['graphs', 'math']","[0, 0, 1, 1, 0, 0, 0, 0]"98"Do you like summer? Residents of Berland do. They especially love eating ice cream in the hot summer. So this summer day a large queue of n Berland residents lined up in front of the ice cream stall. We know that each of them has a certain amount of berland dollars with them. The residents of Berland are nice people, so each person agrees to swap places with the person right behind him for just 1 dollar. More formally, if person a stands just behind person b, then person a can pay person b 1 dollar, then a and b get swapped. Of course, if person a has zero dollars, he can not swap places with person b.Residents of Berland are strange people. In particular, they get upset when there is someone with a strictly smaller sum of money in the line in front of them.Can you help the residents of Berland form such order in the line so that they were all happy? A happy resident is the one who stands first in the line or the one in front of who another resident stands with not less number of dollars. Note that the people of Berland are people of honor and they agree to swap places only in the manner described above.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"99You are given two integers $$$n$$$ and $$$d$$$. You need to construct a rooted binary tree consisting of $$$n$$$ vertices with a root at the vertex $$$1$$$ and the sum of depths of all vertices equals to $$$d$$$.A tree is a connected graph without cycles. A rooted tree has a special vertex called the root. A parent of a vertex $$$v$$$ is the last different from $$$v$$$ vertex on the path from the root to the vertex $$$v$$$. The depth of the vertex $$$v$$$ is the length of the path from the root to the vertex $$$v$$$. Children of vertex $$$v$$$ are all vertices for which $$$v$$$ is the parent. The binary tree is such a tree that no vertex has more than $$$2$$$ children.You have to answer $$$t$$$ independent test cases.,['trees'],"[0, 0, 0, 0, 0, 0, 0, 1]"100"Let's call beauty of an array $$$b_1, b_2, \ldots, b_n$$$ ($$$n > 1$$$)  — $$$\min\limits_{1 \leq i < j \leq n} |b_i - b_j|$$$.You're given an array $$$a_1, a_2, \ldots a_n$$$ and a number $$$k$$$. Calculate the sum of beauty over all subsequences of the array of length exactly $$$k$$$. As this number can be very large, output it modulo $$$998244353$$$.A sequence $$$a$$$ is a subsequence of an array $$$b$$$ if $$$a$$$ can be obtained from $$$b$$$ by deletion of several (possibly, zero or all) elements.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"101"There are $$$n + 1$$$ cities, numbered from $$$0$$$ to $$$n$$$. $$$n$$$ roads connect these cities, the $$$i$$$-th road connects cities $$$i - 1$$$ and $$$i$$$ ($$$i \in [1, n]$$$).Each road has a direction. The directions are given by a string of $$$n$$$ characters such that each character is either L or R. If the $$$i$$$-th character is L, it means that the $$$i$$$-th road initially goes from the city $$$i$$$ to the city $$$i - 1$$$; otherwise it goes from the city $$$i - 1$$$ to the city $$$i$$$.A traveler would like to visit as many cities of this country as possible. Initially, they will choose some city to start their journey from. Each day, the traveler must go from the city where they currently are to a neighboring city using one of the roads, and they can go along a road only if it is directed in the same direction they are going; i. e., if a road is directed from city $$$i$$$ to the city $$$i + 1$$$, it is possible to travel from $$$i$$$ to $$$i + 1$$$, but not from $$$i + 1$$$ to $$$i$$$. After the traveler moves to a neighboring city, all roads change their directions to the opposite ones. If the traveler cannot go from their current city to a neighboring city, their journey ends; it is also possible to end the journey whenever the traveler wants to.The goal of the traveler is to visit as many different cities as possible (they can visit a city multiple times, but only the first visit is counted). For each city $$$i$$$, calculate the maximum number of different cities the traveler can visit during exactly one journey if they start in the city $$$i$$$. ",[],"[0, 0, 0, 0, 0, 0, 0, 0]"102"The construction of subway in Bertown is almost finished! The President of Berland will visit this city soon to look at the new subway himself.There are n stations in the subway. It was built according to the Bertown Transport Law:  For each station i there exists exactly one train that goes from this station. Its destination station is pi, possibly pi = i;  For each station i there exists exactly one station j such that pj = i. The President will consider the convenience of subway after visiting it. The convenience is the number of ordered pairs (x, y) such that person can start at station x and, after taking some subway trains (possibly zero), arrive at station y (1 ≤ x, y ≤ n).The mayor of Bertown thinks that if the subway is not convenient enough, then the President might consider installing a new mayor (and, of course, the current mayor doesn't want it to happen). Before President visits the city mayor has enough time to rebuild some paths of subway, thus changing the values of pi for not more than two subway stations. Of course, breaking the Bertown Transport Law is really bad, so the subway must be built according to the Law even after changes.The mayor wants to do these changes in such a way that the convenience of the subway is maximized. Help him to calculate the maximum possible convenience he can get! ",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"103"You are given $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$.For each $$$a_i$$$ find its two divisors $$$d_1 > 1$$$ and $$$d_2 > 1$$$ such that $$$\gcd(d_1 + d_2, a_i) = 1$$$ (where $$$\gcd(a, b)$$$ is the greatest common divisor of $$$a$$$ and $$$b$$$) or say that there is no such pair.","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"104"Vasya likes taking part in Codeforces contests. When a round is over, Vasya follows all submissions in the system testing tab.There are $$$n$$$ solutions, the $$$i$$$-th of them should be tested on $$$a_i$$$ tests, testing one solution on one test takes $$$1$$$ second. The solutions are judged in the order from $$$1$$$ to $$$n$$$. There are $$$k$$$ testing processes which test solutions simultaneously. Each of them can test at most one solution at a time.At any time moment $$$t$$$ when some testing process is not judging any solution, it takes the first solution from the queue and tests it on each test in increasing order of the test ids. Let this solution have id $$$i$$$, then it is being tested on the first test from time moment $$$t$$$ till time moment $$$t + 1$$$, then on the second test till time moment $$$t + 2$$$ and so on. This solution is fully tested at time moment $$$t + a_i$$$, and after that the testing process immediately starts testing another solution.Consider some time moment, let there be exactly $$$m$$$ fully tested solutions by this moment. There is a caption ""System testing: $$$d$$$%"" on the page with solutions, where $$$d$$$ is calculated as$$$$$$d = round\left(100\cdot\frac{m}{n}\right),$$$$$$where $$$round(x) = \lfloor{x + 0.5}\rfloor$$$ is a function which maps every real to the nearest integer.Vasya calls a submission interesting if there is a time moment (possibly, non-integer) when the solution is being tested on some test $$$q$$$, and the caption says ""System testing: $$$q$$$%"". Find the number of interesting solutions.Please note that in case when multiple processes attempt to take the first submission from the queue at the same moment (for instance, at the initial moment), the order they take the solutions does not matter.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"105"You are given a string $$$s$$$. Each character is either 0 or 1.You want all 1's in the string to form a contiguous subsegment. For example, if the string is 0, 1, 00111 or 01111100, then all 1's form a contiguous subsegment, and if the string is 0101, 100001 or 11111111111101, then this condition is not met.You may erase some (possibly none) 0's from the string. What is the minimum number of 0's that you have to erase?",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"106"An infinitely long railway has a train consisting of n cars, numbered from 1 to n (the numbers of all the cars are distinct) and positioned in arbitrary order. David Blaine wants to sort the railway cars in the order of increasing numbers. In one move he can make one of the cars disappear from its place and teleport it either to the beginning of the train, or to the end of the train, at his desire. What is the minimum number of actions David Blaine needs to perform in order to sort the train?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"107"Polycarp plays a well-known computer game (we won't mention its name). In this game, he can craft tools of two types — shovels and swords. To craft a shovel, Polycarp spends two sticks and one diamond; to craft a sword, Polycarp spends two diamonds and one stick.Each tool can be sold for exactly one emerald. How many emeralds can Polycarp earn, if he has $$$a$$$ sticks and $$$b$$$ diamonds?",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"108"Alice and Bob are playing yet another card game. This time the rules are the following. There are $$$n$$$ cards lying in a row in front of them. The $$$i$$$-th card has value $$$a_i$$$. First, Alice chooses a non-empty consecutive segment of cards $$$[l; r]$$$ ($$$l \le r$$$). After that Bob removes a single card $$$j$$$ from that segment $$$(l \le j \le r)$$$. The score of the game is the total value of the remaining cards on the segment $$$(a_l + a_{l + 1} + \dots + a_{j - 1} + a_{j + 1} + \dots + a_{r - 1} + a_r)$$$. In particular, if Alice chooses a segment with just one element, then the score after Bob removes the only card is $$$0$$$.Alice wants to make the score as big as possible. Bob takes such a card that the score is as small as possible.What segment should Alice choose so that the score is maximum possible? Output the maximum score.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"109"A tree is a connected undirected graph consisting of n vertices and n  -  1 edges. Vertices are numbered 1 through n.Limak is a little polar bear. He once had a tree with n vertices but he lost it. He still remembers something about the lost tree though.You are given m pairs of vertices (a1, b1), (a2, b2), ..., (am, bm). Limak remembers that for each i there was no edge between ai and bi. He also remembers that vertex 1 was incident to exactly k edges (its degree was equal to k).Is it possible that Limak remembers everything correctly? Check whether there exists a tree satisfying the given conditions.","['graphs', 'trees']","[0, 0, 1, 0, 0, 0, 0, 1]"110"Today, Yasser and Adel are at the shop buying cupcakes. There are $$$n$$$ cupcake types, arranged from $$$1$$$ to $$$n$$$ on the shelf, and there are infinitely many of each type. The tastiness of a cupcake of type $$$i$$$ is an integer $$$a_i$$$. There are both tasty and nasty cupcakes, so the tastiness can be positive, zero or negative.Yasser, of course, wants to try them all, so he will buy exactly one cupcake of each type.On the other hand, Adel will choose some segment $$$[l, r]$$$ $$$(1 \le l \le r \le n)$$$ that does not include all of cupcakes (he can't choose $$$[l, r] = [1, n]$$$) and buy exactly one cupcake of each of types $$$l, l + 1, \dots, r$$$.After that they will compare the total tastiness of the cupcakes each of them have bought. Yasser will be happy if the total tastiness of cupcakes he buys is strictly greater than the total tastiness of cupcakes Adel buys regardless of Adel's choice.For example, let the tastinesses of the cupcakes be $$$[7, 4, -1]$$$. Yasser will buy all of them, the total tastiness will be $$$7 + 4 - 1 = 10$$$. Adel can choose segments $$$[7], [4], [-1], [7, 4]$$$ or $$$[4, -1]$$$, their total tastinesses are $$$7, 4, -1, 11$$$ and $$$3$$$, respectively. Adel can choose segment with tastiness $$$11$$$, and as $$$10$$$ is not strictly greater than $$$11$$$, Yasser won't be happy :(Find out if Yasser will be happy after visiting the shop.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"111"This is the hard version of the problem. The difference between versions is the constraints on $$$n$$$ and $$$a_i$$$. You can make hacks only if all versions of the problem are solved.First, Aoi came up with the following idea for the competitive programming problem:Yuzu is a girl who collecting candies. Originally, she has $$$x$$$ candies. There are also $$$n$$$ enemies numbered with integers from $$$1$$$ to $$$n$$$. Enemy $$$i$$$ has $$$a_i$$$ candies.Yuzu is going to determine a permutation $$$P$$$. A permutation is an array consisting of $$$n$$$ distinct integers from $$$1$$$ to $$$n$$$ in arbitrary order. For example, $$$\{2,3,1,5,4\}$$$ is a permutation, but $$$\{1,2,2\}$$$ is not a permutation ($$$2$$$ appears twice in the array) and $$$\{1,3,4\}$$$ is also not a permutation (because $$$n=3$$$ but there is the number $$$4$$$ in the array).After that, she will do $$$n$$$ duels with the enemies with the following rules:  If Yuzu has equal or more number of candies than enemy $$$P_i$$$, she wins the duel and gets $$$1$$$ candy. Otherwise, she loses the duel and gets nothing.  The candy which Yuzu gets will be used in the next duels. Yuzu wants to win all duels. How many valid permutations $$$P$$$ exist?This problem was easy and wasn't interesting for Akari, who is a friend of Aoi. And Akari made the following problem from the above idea:Let's define $$$f(x)$$$ as the number of valid permutations for the integer $$$x$$$.You are given $$$n$$$, $$$a$$$ and a prime number $$$p \le n$$$. Let's call a positive integer $$$x$$$ good, if the value $$$f(x)$$$ is not divisible by $$$p$$$. Find all good integers $$$x$$$.Your task is to solve this problem made by Akari.","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"112"Given a sequence of integers a1, ..., an and q queries x1, ..., xq on it. For each query xi you have to count the number of pairs (l, r) such that 1 ≤ l ≤ r ≤ n and gcd(al, al + 1, ..., ar) = xi. is a greatest common divisor of v1, v2, ..., vn, that is equal to a largest positive integer that divides all vi.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"113"You have a knapsack with the capacity of $$$W$$$. There are also $$$n$$$ items, the $$$i$$$-th one has weight $$$w_i$$$. You want to put some of these items into the knapsack in such a way that their total weight $$$C$$$ is at least half of its size, but (obviously) does not exceed it. Formally, $$$C$$$ should satisfy: $$$\lceil \frac{W}{2}\rceil \le C \le W$$$. Output the list of items you will put into the knapsack or determine that fulfilling the conditions is impossible. If there are several possible lists of items satisfying the conditions, you can output any. Note that you don't have to maximize the sum of weights of items in the knapsack.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"114"Zookeeper is playing a game. In this game, Zookeeper must use bombs to bomb a string that consists of letters 'A' and 'B'. He can use bombs to bomb a substring which is either ""AB"" or ""BB"". When he bombs such a substring, the substring gets deleted from the string and the remaining parts of the string get concatenated.For example, Zookeeper can use two such operations: AABABBA $$$\to$$$ AABBA $$$\to$$$ AAA.Zookeeper wonders what the shortest string he can make is. Can you help him find the length of the shortest string?",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"115"Giant chess is quite common in Geraldion. We will not delve into the rules of the game, we'll just say that the game takes place on an h × w field, and it is painted in two colors, but not like in chess. Almost all cells of the field are white and only some of them are black. Currently Gerald is finishing a game of giant chess against his friend Pollard. Gerald has almost won, and the only thing he needs to win is to bring the pawn from the upper left corner of the board, where it is now standing, to the lower right corner. Gerald is so confident of victory that he became interested, in how many ways can he win?The pawn, which Gerald has got left can go in two ways: one cell down or one cell to the right. In addition, it can not go to the black cells, otherwise the Gerald still loses. There are no other pawns or pieces left on the field, so that, according to the rules of giant chess Gerald moves his pawn until the game is over, and Pollard is just watching this process.","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"116"Let us remind you part of the rules of Codeforces. The given rules slightly simplified, use the problem statement as a formal document.In the beginning of the round the contestants are divided into rooms. Each room contains exactly n participants. During the contest the participants are suggested to solve five problems, A, B, C, D and E. For each of these problem, depending on when the given problem was solved and whether it was solved at all, the participants receive some points. Besides, a contestant can perform hacks on other contestants. For each successful hack a contestant earns 100 points, for each unsuccessful hack a contestant loses 50 points. The number of points for every contestant is represented by the sum of points he has received from all his problems, including hacks.You are suggested to determine the leader for some room; the leader is a participant who has maximum points.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"117"Petya loves lucky numbers. We all know that lucky numbers are the positive integers whose decimal representations contain only the lucky digits 4 and 7. For example, numbers 47, 744, 4 are lucky and 5, 17, 467 are not.Petya recently learned to determine whether a string of lowercase Latin letters is lucky. For each individual letter all its positions in the string are written out in the increasing order. This results in 26 lists of numbers; some of them can be empty. A string is considered lucky if and only if in each list the absolute difference of any two adjacent numbers is a lucky number. For example, let's consider string ""zbcdzefdzc"". The lists of positions of equal letters are: b: 2 c: 3, 10 d: 4, 8 e: 6 f: 7 z: 1, 5, 9 Lists of positions of letters a, g, h, ..., y are empty.This string is lucky as all differences are lucky numbers. For letters z: 5 - 1 = 4, 9 - 5 = 4, for letters c: 10 - 3 = 7, for letters d: 8 - 4 = 4. Note that if some letter occurs only once in a string, it doesn't influence the string's luckiness after building the lists of positions of equal letters. The string where all the letters are distinct is considered lucky.Find the lexicographically minimal lucky string whose length equals n.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"118"Throughout Igor K.'s life he has had many situations worthy of attention. We remember the story with the virus, the story of his mathematical career and of course, his famous programming achievements. However, one does not always adopt new hobbies, one can quit something as well.This time Igor K. got disappointed in one of his hobbies: editing and voicing videos. Moreover, he got disappointed in it so much, that he decided to destroy his secret archive for good. Igor K. use Pindows XR operation system which represents files and folders by small icons. At that, m icons can fit in a horizontal row in any window.Igor K.'s computer contains n folders in the D: disk's root catalog. The folders are numbered from 1 to n in the order from the left to the right and from top to bottom (see the images). At that the folders with secret videos have numbers from a to b inclusive. Igor K. wants to delete them forever, at that making as few frame selections as possible, and then pressing Shift+Delete exactly once. What is the minimum number of times Igor K. will have to select the folder in order to select folders from a to b and only them? Let us note that if some selected folder is selected repeatedly, then it is deselected. Each selection possesses the shape of some rectangle with sides parallel to the screen's borders.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"119"Polycarp has $$$26$$$ tasks. Each task is designated by a capital letter of the Latin alphabet.The teacher asked Polycarp to solve tasks in the following way: if Polycarp began to solve some task, then he must solve it to the end, without being distracted by another task. After switching to another task, Polycarp cannot return to the previous task.Polycarp can only solve one task during the day. Every day he wrote down what task he solved. Now the teacher wants to know if Polycarp followed his advice.For example, if Polycarp solved tasks in the following order: ""DDBBCCCBBEZ"", then the teacher will see that on the third day Polycarp began to solve the task 'B', then on the fifth day he got distracted and began to solve the task 'C', on the eighth day Polycarp returned to the task 'B'. Other examples of when the teacher is suspicious: ""BAB"", ""AABBCCDDEEBZZ"" and ""AAAAZAAAAA"".If Polycarp solved the tasks as follows: ""FFGZZZY"", then the teacher cannot have any suspicions. Please note that Polycarp is not obligated to solve all tasks. Other examples of when the teacher doesn't have any suspicious: ""BA"", ""AFFFCC"" and ""YYYYY"".Help Polycarp find out if his teacher might be suspicious.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"120"You have a stripe of checkered paper of length $$$n$$$. Each cell is either white or black.What is the minimum number of cells that must be recolored from white to black in order to have a segment of $$$k$$$ consecutive black cells on the stripe?If the input data is such that a segment of $$$k$$$ consecutive black cells already exists, then print 0. ",[],"[0, 0, 0, 0, 0, 0, 0, 0]"121"Homer has two friends Alice and Bob. Both of them are string fans. One day, Alice and Bob decide to play a game on a string $$$s = s_1 s_2 \dots s_n$$$ of length $$$n$$$ consisting of lowercase English letters. They move in turns alternatively and Alice makes the first move.In a move, a player must choose an index $$$i$$$ ($$$1 \leq i \leq n$$$) that has not been chosen before, and change $$$s_i$$$ to any other lowercase English letter $$$c$$$ that $$$c \neq s_i$$$.When all indices have been chosen, the game ends. The goal of Alice is to make the final string lexicographically as small as possible, while the goal of Bob is to make the final string lexicographically as large as possible. Both of them are game experts, so they always play games optimally. Homer is not a game expert, so he wonders what the final string will be.A string $$$a$$$ is lexicographically smaller than a string $$$b$$$ if and only if one of the following holds:   $$$a$$$ is a prefix of $$$b$$$, but $$$a \ne b$$$;  in the first position where $$$a$$$ and $$$b$$$ differ, the string $$$a$$$ has a letter that appears earlier in the alphabet than the corresponding letter in $$$b$$$. ","['games', 'strings']","[1, 0, 0, 0, 0, 0, 1, 0]"122"You are given a tree consisting of $$$n$$$ vertices. A tree is an undirected connected acyclic graph.    Example of a tree. You have to paint each vertex into one of three colors. For each vertex, you know the cost of painting it in every color.You have to paint the vertices so that any path consisting of exactly three distinct vertices does not contain any vertices with equal colors. In other words, let's consider all triples $$$(x, y, z)$$$ such that $$$x \neq y, y \neq z, x \neq z$$$, $$$x$$$ is connected by an edge with $$$y$$$, and $$$y$$$ is connected by an edge with $$$z$$$. The colours of $$$x$$$, $$$y$$$ and $$$z$$$ should be pairwise distinct. Let's call a painting which meets this condition good.You have to calculate the minimum cost of a good painting and find one of the optimal paintings. If there is no good painting, report about it.","['graphs', 'trees']","[0, 0, 1, 0, 0, 0, 0, 1]"123"Since Sonya has just learned the basics of matrices, she decided to play with them a little bit.Sonya imagined a new type of matrices that she called rhombic matrices. These matrices have exactly one zero, while all other cells have the Manhattan distance to the cell containing the zero. The cells with equal numbers have the form of a rhombus, that is why Sonya called this type so.The Manhattan distance between two cells ($$$x_1$$$, $$$y_1$$$) and ($$$x_2$$$, $$$y_2$$$) is defined as $$$|x_1 - x_2| + |y_1 - y_2|$$$. For example, the Manhattan distance between the cells $$$(5, 2)$$$ and $$$(7, 1)$$$ equals to $$$|5-7|+|2-1|=3$$$.    Example of a rhombic matrix. Note that rhombic matrices are uniquely defined by $$$n$$$, $$$m$$$, and the coordinates of the cell containing the zero.She drew a $$$n\times m$$$ rhombic matrix. She believes that you can not recreate the matrix if she gives you only the elements of this matrix in some arbitrary order (i.e., the sequence of $$$n\cdot m$$$ numbers). Note that Sonya will not give you $$$n$$$ and $$$m$$$, so only the sequence of numbers in this matrix will be at your disposal.Write a program that finds such an $$$n\times m$$$ rhombic matrix whose elements are the same as the elements in the sequence in some order.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"124"During the research on properties of the greatest common divisor (GCD) of a set of numbers, Ildar, a famous mathematician, introduced a brand new concept of the weakened common divisor (WCD) of a list of pairs of integers.For a given list of pairs of integers $$$(a_1, b_1)$$$, $$$(a_2, b_2)$$$, ..., $$$(a_n, b_n)$$$ their WCD is arbitrary integer greater than $$$1$$$, such that it divides at least one element in each pair. WCD may not exist for some lists.For example, if the list looks like $$$[(12, 15), (25, 18), (10, 24)]$$$, then their WCD can be equal to $$$2$$$, $$$3$$$, $$$5$$$ or $$$6$$$ (each of these numbers is strictly greater than $$$1$$$ and divides at least one number in each pair).You're currently pursuing your PhD degree under Ildar's mentorship, and that's why this problem was delegated to you. Your task is to calculate WCD efficiently.",['number theory'],"[0, 0, 0, 0, 1, 0, 0, 0]"125"You are storing an integer array of length $$$m$$$ in a database. To maintain internal integrity and protect data, the database stores $$$n$$$ copies of this array.Unfortunately, the recent incident may have altered the stored information in every copy in the database.It's believed, that the incident altered at most two elements in every copy. You need to recover the original array based on the current state of the database.In case there are multiple ways to restore the array, report any. If there is no array that differs from every copy in no more than two positions, report that as well.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"126"You are given an array $$$a[0 \ldots n - 1] = [a_0, a_1, \ldots, a_{n - 1}]$$$ of zeroes and ones only. Note that in this problem, unlike the others, the array indexes are numbered from zero, not from one.In one step, the array $$$a$$$ is replaced by another array of length $$$n$$$ according to the following rules:   First, a new array $$$a^{\rightarrow d}$$$ is defined as a cyclic shift of the array $$$a$$$ to the right by $$$d$$$ cells. The elements of this array can be defined as $$$a^{\rightarrow d}_i = a_{(i + n - d) \bmod n}$$$, where $$$(i + n - d) \bmod n$$$ is the remainder of integer division of $$$i + n - d$$$ by $$$n$$$.  It means that the whole array $$$a^{\rightarrow d}$$$ can be represented as a sequence $$$$$$a^{\rightarrow d} = [a_{n - d}, a_{n - d + 1}, \ldots, a_{n - 1}, a_0, a_1, \ldots, a_{n - d - 1}]$$$$$$  Then each element of the array $$$a_i$$$ is replaced by $$$a_i \,\&\, a^{\rightarrow d}_i$$$, where $$$\&$$$ is a logical ""AND"" operator. For example, if $$$a = [0, 0, 1, 1]$$$ and $$$d = 1$$$, then $$$a^{\rightarrow d} = [1, 0, 0, 1]$$$ and the value of $$$a$$$ after the first step will be $$$[0 \,\&\, 1, 0 \,\&\, 0, 1 \,\&\, 0, 1 \,\&\, 1]$$$, that is $$$[0, 0, 0, 1]$$$.The process ends when the array stops changing. For a given array $$$a$$$, determine whether it will consist of only zeros at the end of the process. If yes, also find the number of steps the process will take before it finishes.","['graphs', 'math', 'number theory']","[0, 0, 1, 1, 1, 0, 0, 0]"127"Nick's company employed n people. Now Nick needs to build a tree hierarchy of «supervisor-surbodinate» relations in the company (this is to say that each employee, except one, has exactly one supervisor). There are m applications written in the following form: «employee ai is ready to become a supervisor of employee bi at extra cost ci». The qualification qj of each employee is known, and for each application the following is true: qai > qbi. Would you help Nick calculate the minimum cost of such a hierarchy, or find out that it is impossible to build it.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"128"You finally woke up after this crazy dream and decided to walk around to clear your head. Outside you saw your house's fence — so plain and boring, that you'd like to repaint it.  You have a fence consisting of $$$n$$$ planks, where the $$$i$$$-th plank has the color $$$a_i$$$. You want to repaint the fence in such a way that the $$$i$$$-th plank has the color $$$b_i$$$.You've invited $$$m$$$ painters for this purpose. The $$$j$$$-th painter will arrive at the moment $$$j$$$ and will recolor exactly one plank to color $$$c_j$$$. For each painter you can choose which plank to recolor, but you can't turn them down, i. e. each painter has to color exactly one plank.Can you get the coloring $$$b$$$ you want? If it's possible, print for each painter which plank he must paint.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"129"You are fed up with your messy room, so you decided to clean it up.Your room is a bracket sequence $$$s=s_{1}s_{2}\dots s_{n}$$$ of length $$$n$$$. Each character of this string is either an opening bracket '(' or a closing bracket ')'.In one operation you can choose any consecutive substring of $$$s$$$ and reverse it. In other words, you can choose any substring $$$s[l \dots r]=s_l, s_{l+1}, \dots, s_r$$$ and change the order of elements in it into $$$s_r, s_{r-1}, \dots, s_{l}$$$.For example, if you will decide to reverse substring $$$s[2 \dots 4]$$$ of string $$$s=$$$""((()))"" it will be equal to $$$s=$$$""()(())"".A regular (aka balanced) bracket sequence is a bracket sequence that can be transformed into a correct arithmetic expression by inserting characters '1' and '+' between the original characters of the sequence. For example, bracket sequences ""()()"", ""(())"" are regular (the resulting expressions are: ""(1)+(1)"", ""((1+1)+1)""), and "")("" and ""("" are not.A prefix of a string $$$s$$$ is a substring that starts at position $$$1$$$. For example, for $$$s=$$$""(())()"" there are $$$6$$$ prefixes: ""("", ""(("", ""(()"", ""(())"", ""(())("" and ""(())()"".In your opinion, a neat and clean room $$$s$$$ is a bracket sequence that:  the whole string $$$s$$$ is a regular bracket sequence;  and there are exactly $$$k$$$ prefixes of this sequence which are regular (including whole $$$s$$$ itself). For example, if $$$k = 2$$$, then ""(())()"" is a neat and clean room.You want to use at most $$$n$$$ operations to make your room neat and clean. Operations are applied one after another sequentially.It is guaranteed that the answer exists. Note that you do not need to minimize the number of operations: find any way to achieve the desired configuration in $$$n$$$ or less operations.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"130"You came to the exhibition and one exhibit has drawn your attention. It consists of $$$n$$$ stacks of blocks, where the $$$i$$$-th stack consists of $$$a_i$$$ blocks resting on the surface.The height of the exhibit is equal to $$$m$$$. Consequently, the number of blocks in each stack is less than or equal to $$$m$$$.There is a camera on the ceiling that sees the top view of the blocks and a camera on the right wall that sees the side view of the blocks. Find the maximum number of blocks you can remove such that the views for both the cameras would not change.Note, that while originally all blocks are stacked on the floor, it is not required for them to stay connected to the floor after some blocks are removed. There is no gravity in the whole exhibition, so no block would fall down, even if the block underneath is removed. It is not allowed to move blocks by hand either.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"131"Gena loves sequences of numbers. Recently, he has discovered a new type of sequences which he called an almost arithmetical progression. A sequence is an almost arithmetical progression, if its elements can be represented as:  a1 = p, where p is some integer;  ai = ai - 1 + ( - 1)i + 1·q (i > 1), where q is some integer. Right now Gena has a piece of paper with sequence b, consisting of n integers. Help Gena, find there the longest subsequence of integers that is an almost arithmetical progression.Sequence s1,  s2,  ...,  sk is a subsequence of sequence b1,  b2,  ...,  bn, if there is such increasing sequence of indexes i1, i2, ..., ik (1  ≤  i1  <  i2  < ...   <  ik  ≤  n), that bij  =  sj. In other words, sequence s can be obtained from b by crossing out some elements.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"132"Katya studies in a fifth grade. Recently her class studied right triangles and the Pythagorean theorem. It appeared, that there are triples of positive integers such that you can construct a right triangle with segments of lengths corresponding to triple. Such triples are called Pythagorean triples.For example, triples (3, 4, 5), (5, 12, 13) and (6, 8, 10) are Pythagorean triples.Here Katya wondered if she can specify the length of some side of right triangle and find any Pythagorean triple corresponding to such length? Note that the side which length is specified can be a cathetus as well as hypotenuse.Katya had no problems with completing this task. Will you do the same?","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"133"You are the owner of a harvesting field which can be modeled as an infinite line, whose positions are identified by integers.It will rain for the next $$$n$$$ days. On the $$$i$$$-th day, the rain will be centered at position $$$x_i$$$ and it will have intensity $$$p_i$$$. Due to these rains, some rainfall will accumulate; let $$$a_j$$$ be the amount of rainfall accumulated at integer position $$$j$$$. Initially $$$a_j$$$ is $$$0$$$, and it will increase by $$$\max(0,p_i-|x_i-j|)$$$ after the $$$i$$$-th day's rain.A flood will hit your field if, at any moment, there is a position $$$j$$$ with accumulated rainfall $$$a_j>m$$$.You can use a magical spell to erase exactly one day's rain, i.e., setting $$$p_i=0$$$. For each $$$i$$$ from $$$1$$$ to $$$n$$$, check whether in case of erasing the $$$i$$$-th day's rain there is no flood.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"134"An array $$$b$$$ of length $$$k$$$ is called good if its arithmetic mean is equal to $$$1$$$. More formally, if $$$$$$\frac{b_1 + \cdots + b_k}{k}=1.$$$$$$Note that the value $$$\frac{b_1+\cdots+b_k}{k}$$$ is not rounded up or down. For example, the array $$$[1,1,1,2]$$$ has an arithmetic mean of $$$1.25$$$, which is not equal to $$$1$$$.You are given an integer array $$$a$$$ of length $$$n$$$. In an operation, you can append a non-negative integer to the end of the array. What's the minimum number of operations required to make the array good?We have a proof that it is always possible with finitely many operations.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"135"An array of integers $$$p_1, p_2, \dots, p_n$$$ is called a permutation if it contains each number from $$$1$$$ to $$$n$$$ exactly once. For example, the following arrays are permutations: $$$[3, 1, 2]$$$, $$$[1]$$$, $$$[1, 2, 3, 4, 5]$$$ and $$$[4, 3, 1, 2]$$$. The following arrays are not permutations: $$$[2]$$$, $$$[1, 1]$$$, $$$[2, 3, 4]$$$.Polycarp invented a really cool permutation $$$p_1, p_2, \dots, p_n$$$ of length $$$n$$$. It is very disappointing, but he forgot this permutation. He only remembers the array $$$q_1, q_2, \dots, q_{n-1}$$$ of length $$$n-1$$$, where $$$q_i=p_{i+1}-p_i$$$.Given $$$n$$$ and $$$q=q_1, q_2, \dots, q_{n-1}$$$, help Polycarp restore the invented permutation.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"136"Alice is visiting New York City. To make the trip fun, Alice will take photos of the city skyline and give the set of photos as a present to Bob. However, she wants to find the set of photos with maximum beauty and she needs your help. There are $$$n$$$ buildings in the city, the $$$i$$$-th of them has positive height $$$h_i$$$. All $$$n$$$ building heights in the city are different. In addition, each building has a beauty value $$$b_i$$$. Note that beauty can be positive or negative, as there are ugly buildings in the city too. A set of photos consists of one or more photos of the buildings in the skyline. Each photo includes one or more buildings in the skyline that form a contiguous segment of indices. Each building needs to be in exactly one photo. This means that if a building does not appear in any photo, or if a building appears in more than one photo, the set of pictures is not valid. The beauty of a photo is equivalent to the beauty $$$b_i$$$ of the shortest building in it. The total beauty of a set of photos is the sum of the beauty of all photos in it. Help Alice to find the maximum beauty a valid set of photos can have. ",[],"[0, 0, 0, 0, 0, 0, 0, 0]"137"You are given an array $$$a$$$ of $$$2n$$$ distinct integers. You want to arrange the elements of the array in a circle such that no element is equal to the the arithmetic mean of its $$$2$$$ neighbours.More formally, find an array $$$b$$$, such that: $$$b$$$ is a permutation of $$$a$$$.For every $$$i$$$ from $$$1$$$ to $$$2n$$$, $$$b_i \neq \frac{b_{i-1}+b_{i+1}}{2}$$$, where $$$b_0 = b_{2n}$$$ and $$$b_{2n+1} = b_1$$$. It can be proved that under the constraints of this problem, such array $$$b$$$ always exists.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"138"Ancient Egyptians are known to have used a large set of symbols  to write on the walls of the temples. Fafa and Fifa went to one of the temples and found two non-empty words S1 and S2 of equal lengths on the wall of temple written one below the other. Since this temple is very ancient, some symbols from the words were erased. The symbols in the set  have equal probability for being in the position of any erased symbol.Fifa challenged Fafa to calculate the probability that S1 is lexicographically greater than S2. Can you help Fafa with this task?You know that , i. e. there were m distinct characters in Egyptians' alphabet, in this problem these characters are denoted by integers from 1 to m in alphabet order. A word x is lexicographically greater than a word y of the same length, if the words are same up to some position, and then the word x has a larger character, than the word y.We can prove that the probability equals to some fraction , where P and Q are coprime integers, and . Print as the answer the value , i. e. such a non-negative integer less than 109 + 7, such that , where  means that a and b give the same remainders when divided by m.","['math', 'probabilities']","[0, 0, 0, 1, 0, 1, 0, 0]"139"Vasiliy lives at point (a, b) of the coordinate plane. He is hurrying up to work so he wants to get out of his house as soon as possible. New app suggested n available Beru-taxi nearby. The i-th taxi is located at point (xi, yi) and moves with a speed vi. Consider that each of n drivers will move directly to Vasiliy and with a maximum possible speed. Compute the minimum time when Vasiliy will get in any of Beru-taxi cars.",['geometry'],"[0, 1, 0, 0, 0, 0, 0, 0]"140"Volodya likes listening to heavy metal and (occasionally) reading. No wonder Volodya is especially interested in texts concerning his favourite music style.Volodya calls a string powerful if it starts with ""heavy"" and ends with ""metal"". Finding all powerful substrings (by substring Volodya means a subsequence of consecutive characters in a string) in a given text makes our hero especially joyful. Recently he felt an enormous fit of energy while reading a certain text. So Volodya decided to count all powerful substrings in this text and brag about it all day long. Help him in this difficult task. Two substrings are considered different if they appear at the different positions in the text.For simplicity, let us assume that Volodya's text can be represented as a single string.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"141"Let's call a non-empty sequence of positive integers a1, a2... ak coprime if the greatest common divisor of all elements of this sequence is equal to 1.Given an array a consisting of n positive integers, find the number of its coprime subsequences. Since the answer may be very large, print it modulo 109 + 7.Note that two subsequences are considered different if chosen indices are different. For example, in the array [1, 1] there are 3 different subsequences: [1], [1] and [1, 1].",['number theory'],"[0, 0, 0, 0, 1, 0, 0, 0]"142"The only difference between easy and hard versions is the length of the string.You are given a string $$$s$$$ and a string $$$t$$$, both consisting only of lowercase Latin letters. It is guaranteed that $$$t$$$ can be obtained from $$$s$$$ by removing some (possibly, zero) number of characters (not necessary contiguous) from $$$s$$$ without changing order of remaining characters (in other words, it is guaranteed that $$$t$$$ is a subsequence of $$$s$$$).For example, the strings ""test"", ""tst"", ""tt"", ""et"" and """" are subsequences of the string ""test"". But the strings ""tset"", ""se"", ""contest"" are not subsequences of the string ""test"".You want to remove some substring (contiguous subsequence) from $$$s$$$ of maximum possible length such that after removing this substring $$$t$$$ will remain a subsequence of $$$s$$$.If you want to remove the substring $$$s[l;r]$$$ then the string $$$s$$$ will be transformed to $$$s_1 s_2 \dots s_{l-1} s_{r+1} s_{r+2} \dots s_{|s|-1} s_{|s|}$$$ (where $$$|s|$$$ is the length of $$$s$$$).Your task is to find the maximum possible length of the substring you can remove so that $$$t$$$ is still a subsequence of $$$s$$$.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"143"Monocarp is playing a computer game once again. He is a wizard apprentice, who only knows a single spell. Luckily, this spell can damage the monsters.The level he's currently on contains $$$n$$$ monsters. The $$$i$$$-th of them appears $$$k_i$$$ seconds after the start of the level and has $$$h_i$$$ health points. As an additional constraint, $$$h_i \le k_i$$$ for all $$$1 \le i \le n$$$. All $$$k_i$$$ are different.Monocarp can cast the spell at moments which are positive integer amounts of second after the start of the level: $$$1, 2, 3, \dots$$$ The damage of the spell is calculated as follows. If he didn't cast the spell at the previous second, the damage is $$$1$$$. Otherwise, let the damage at the previous second be $$$x$$$. Then he can choose the damage to be either $$$x + 1$$$ or $$$1$$$. A spell uses mana: casting a spell with damage $$$x$$$ uses $$$x$$$ mana. Mana doesn't regenerate.To kill the $$$i$$$-th monster, Monocarp has to cast a spell with damage at least $$$h_i$$$ at the exact moment the monster appears, which is $$$k_i$$$.Note that Monocarp can cast the spell even when there is no monster at the current second.The mana amount required to cast the spells is the sum of mana usages for all cast spells. Calculate the least amount of mana required for Monocarp to kill all monsters.It can be shown that it's always possible to kill all monsters under the constraints of the problem.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"144"While playing yet another strategy game, Mans has recruited $$$n$$$ Swedish heroes, whose powers which can be represented as an array $$$a$$$.Unfortunately, not all of those mighty heroes were created as capable as he wanted, so that he decided to do something about it. In order to accomplish his goal, he can pick two consecutive heroes, with powers $$$a_i$$$ and $$$a_{i+1}$$$, remove them and insert a hero with power $$$-(a_i+a_{i+1})$$$ back in the same position. For example if the array contains the elements $$$[5, 6, 7, 8]$$$, he can pick $$$6$$$ and $$$7$$$ and get $$$[5, -(6+7), 8] = [5, -13, 8]$$$.After he will perform this operation $$$n-1$$$ times, Mans will end up having only one hero. He wants his power to be as big as possible. What's the largest possible power he can achieve?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"145"The Olympic Games in Bercouver are in full swing now. Here everyone has their own objectives: sportsmen compete for medals, and sport commentators compete for more convenient positions to give a running commentary. Today the main sport events take place at three round stadiums, and the commentator's objective is to choose the best point of observation, that is to say the point from where all the three stadiums can be observed. As all the sport competitions are of the same importance, the stadiums should be observed at the same angle. If the number of points meeting the conditions is more than one, the point with the maximum angle of observation is prefered. Would you, please, help the famous Berland commentator G. Berniev to find the best point of observation. It should be noted, that the stadiums do not hide each other, the commentator can easily see one stadium through the other.",['geometry'],"[0, 1, 0, 0, 0, 0, 0, 0]"146"It's hard times now. Today Petya needs to score 100 points on Informatics exam. The tasks seem easy to Petya, but he thinks he lacks time to finish them all, so he asks you to help with one..There is a glob pattern in the statements (a string consisting of lowercase English letters, characters ""?"" and ""*""). It is known that character ""*"" occurs no more than once in the pattern.Also, n query strings are given, it is required to determine for each of them if the pattern matches it or not.Everything seemed easy to Petya, but then he discovered that the special pattern characters differ from their usual meaning.A pattern matches a string if it is possible to replace each character ""?"" with one good lowercase English letter, and the character ""*"" (if there is one) with any, including empty, string of bad lowercase English letters, so that the resulting string is the same as the given string.The good letters are given to Petya. All the others are bad.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"147"Polycarp loves geometric progressions very much. Since he was only three years old, he loves only the progressions of length three. He also has a favorite integer k and a sequence a, consisting of n integers.He wants to know how many subsequences of length three can be selected from a, so that they form a geometric progression with common ratio k.A subsequence of length three is a combination of three such indexes i1, i2, i3, that 1 ≤ i1 < i2 < i3 ≤ n. That is, a subsequence of length three are such groups of three elements that are not necessarily consecutive in the sequence, but their indexes are strictly increasing.A geometric progression with common ratio k is a sequence of numbers of the form b·k0, b·k1, ..., b·kr - 1.Polycarp is only three years old, so he can not calculate this number himself. Help him to do it.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"148"Karen has just arrived at school, and she has a math test today!  The test is about basic addition and subtraction. Unfortunately, the teachers were too busy writing tasks for Codeforces rounds, and had no time to make an actual test. So, they just put one question in the test that is worth all the points.There are n integers written on a row. Karen must alternately add and subtract each pair of adjacent integers, and write down the sums or differences on the next row. She must repeat this process on the values on the next row, and so on, until only one integer remains. The first operation should be addition.Note that, if she ended the previous row by adding the integers, she should start the next row by subtracting, and vice versa.The teachers will simply look at the last integer, and then if it is correct, Karen gets a perfect score, otherwise, she gets a zero for the test.Karen has studied well for this test, but she is scared that she might make a mistake somewhere and it will cause her final answer to be wrong. If the process is followed, what number can she expect to be written on the last row?Since this number can be quite large, output only the non-negative remainder after dividing it by 109 + 7.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"149"Jack decides to invite Emma out for a dinner. Jack is a modest student, he doesn't want to go to an expensive restaurant. Emma is a girl with high taste, she prefers elite places.Munhattan consists of n streets and m avenues. There is exactly one restaurant on the intersection of each street and avenue. The streets are numbered with integers from 1 to n and the avenues are numbered with integers from 1 to m. The cost of dinner in the restaurant at the intersection of the i-th street and the j-th avenue is cij.Jack and Emma decide to choose the restaurant in the following way. Firstly Emma chooses the street to dinner and then Jack chooses the avenue. Emma and Jack makes their choice optimally: Emma wants to maximize the cost of the dinner, Jack wants to minimize it. Emma takes into account that Jack wants to minimize the cost of the dinner. Find the cost of the dinner for the couple in love.",['games'],"[1, 0, 0, 0, 0, 0, 0, 0]"150"The pandemic is upon us, and the world is in shortage of the most important resource: toilet paper. As one of the best prepared nations for this crisis, BubbleLand promised to help all other world nations with this valuable resource. To do that, the country will send airplanes to other countries carrying toilet paper.In BubbleLand, there are $$$N$$$ toilet paper factories, and $$$N$$$ airports. Because of how much it takes to build a road, and of course legal issues, every factory must send paper to only one airport, and every airport can only take toilet paper from one factory.Also, a road can't be built between all airport-factory pairs, again because of legal issues. Every possible road has number $$$d$$$ given, number of days it takes to build that road.Your job is to choose $$$N$$$ factory-airport pairs, such that if the country starts building all roads at the same time, it takes the least amount of days to complete them.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"151"Two semifinals have just been in the running tournament. Each semifinal had n participants. There are n participants advancing to the finals, they are chosen as follows: from each semifinal, we choose k people (0 ≤ 2k ≤ n) who showed the best result in their semifinals and all other places in the finals go to the people who haven't ranked in the top k in their semifinal but got to the n - 2k of the best among the others.The tournament organizers hasn't yet determined the k value, so the participants want to know who else has any chance to get to the finals and who can go home.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"152"One fine October day a mathematics teacher Vasily Petrov went to a class and saw there n pupils who sat at the  desks, two people at each desk. Vasily quickly realized that number n is even. Like all true mathematicians, Vasily has all students numbered from 1 to n.But Vasily Petrov did not like the way the children were seated at the desks. According to him, the students whose numbers differ by 1, can not sit together, as they talk to each other all the time, distract others and misbehave.On the other hand, if a righthanded student sits at the left end of the desk and a lefthanded student sits at the right end of the desk, they hit elbows all the time and distract each other. In other cases, the students who sit at the same desk, do not interfere with each other.Vasily knows very well which students are lefthanders and which ones are righthanders, and he asks you to come up with any order that meets these two uncomplicated conditions (students do not talk to each other and do not bump their elbows). It is guaranteed that the input is such that at least one way to seat the students always exists.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"153"This is yet another problem dealing with regular bracket sequences.We should remind you that a bracket sequence is called regular, if by inserting «+» and «1» into it we can get a correct mathematical expression. For example, sequences «(())()», «()» and «(()(()))» are regular, while «)(», «(()» and «(()))(» are not. You are given a string of «(» and «)» characters. You are to find its longest substring that is a regular bracket sequence. You are to find the number of such substrings as well.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"154"Two integer sequences existed initially, one of them was strictly increasing, and another one — strictly decreasing.Strictly increasing sequence is a sequence of integers $$$[x_1 < x_2 < \dots < x_k]$$$. And strictly decreasing sequence is a sequence of integers $$$[y_1 > y_2 > \dots > y_l]$$$. Note that the empty sequence and the sequence consisting of one element can be considered as increasing or decreasing.Elements of increasing sequence were inserted between elements of the decreasing one (and, possibly, before its first element and after its last element) without changing the order. For example, sequences $$$[1, 3, 4]$$$ and $$$[10, 4, 2]$$$ can produce the following resulting sequences: $$$[10, \textbf{1}, \textbf{3}, 4, 2, \textbf{4}]$$$, $$$[\textbf{1}, \textbf{3}, \textbf{4}, 10, 4, 2]$$$. The following sequence cannot be the result of these insertions: $$$[\textbf{1}, 10, \textbf{4}, 4, \textbf{3}, 2]$$$ because the order of elements in the increasing sequence was changed.Let the obtained sequence be $$$a$$$. This sequence $$$a$$$ is given in the input. Your task is to find any two suitable initial sequences. One of them should be strictly increasing, and another one — strictly decreasing. Note that the empty sequence and the sequence consisting of one element can be considered as increasing or decreasing.If there is a contradiction in the input and it is impossible to split the given sequence $$$a$$$ into one increasing sequence and one decreasing sequence, print ""NO"".",[],"[0, 0, 0, 0, 0, 0, 0, 0]"155"Once Bob decided to lay a parquet floor in his living room. The living room is of size n × m metres. Bob had planks of three types: a planks 1 × 2 meters, b planks 2 × 1 meters, and c planks 2 × 2 meters. Help Bob find out, if it is possible to parquet the living room with such a set of planks, and if it is possible, find one of the possible ways to do so. Bob doesn't have to use all the planks.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"156"Vladik and Chloe decided to determine who of them is better at math. Vladik claimed that for any positive integer n he can represent fraction  as a sum of three distinct positive fractions in form .Help Vladik with that, i.e for a given n find three distinct positive integers x, y and z such that . Because Chloe can't check Vladik's answer if the numbers are large, he asks you to print numbers not exceeding 109.If there is no such answer, print -1.","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"157"Do you know a story about the three musketeers? Anyway, you will learn about its origins now.Richelimakieu is a cardinal in the city of Bearis. He is tired of dealing with crime by himself. He needs three brave warriors to help him to fight against bad guys.There are n warriors. Richelimakieu wants to choose three of them to become musketeers but it's not that easy. The most important condition is that musketeers must know each other to cooperate efficiently. And they shouldn't be too well known because they could be betrayed by old friends. For each musketeer his recognition is the number of warriors he knows, excluding other two musketeers.Help Richelimakieu! Find if it is possible to choose three musketeers knowing each other, and what is minimum possible sum of their recognitions.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"158"Профиль горного хребта схематично задан в виде прямоугольной таблицы из символов «.» (пустое пространство) и «*» (часть горы). Каждый столбец таблицы содержит хотя бы одну «звёздочку». Гарантируется, что любой из символов «*» либо находится в нижней строке матрицы, либо непосредственно под ним находится другой символ «*».  ....................*..*.......*.**.......*.**..*...**.***********   Пример изображения горного хребта.  Маршрут туриста проходит через весь горный хребет слева направо. Каждый день турист перемещается вправо — в соседний столбец в схематичном изображении. Конечно, каждый раз он поднимается (или опускается) в самую верхнюю точку горы, которая находится в соответствующем столбце.Считая, что изначально турист находится в самой верхней точке в первом столбце, а закончит свой маршрут в самой верхней точке в последнем столбце, найдите две величины:  наибольший подъём за день (равен 0, если в профиле горного хребта нет ни одного подъёма),  наибольший спуск за день (равен 0, если в профиле горного хребта нет ни одного спуска). ",[],"[0, 0, 0, 0, 0, 0, 0, 0]"159"Sagheer is working at a kindergarten. There are n children and m different toys. These children use well-defined protocols for playing with the toys:  Each child has a lovely set of toys that he loves to play with. He requests the toys one after another at distinct moments of time. A child starts playing if and only if he is granted all the toys in his lovely set. If a child starts playing, then sooner or later he gives the toys back. No child keeps the toys forever. Children request toys at distinct moments of time. No two children request a toy at the same time. If a child is granted a toy, he never gives it back until he finishes playing with his lovely set. If a child is not granted a toy, he waits until he is granted this toy. He can't request another toy while waiting. If two children are waiting for the same toy, then the child who requested it first will take the toy first.Children don't like to play with each other. That's why they never share toys. When a child requests a toy, then granting the toy to this child depends on whether the toy is free or not. If the toy is free, Sagheer will give it to the child. Otherwise, the child has to wait for it and can't request another toy.Children are smart and can detect if they have to wait forever before they get the toys they want. In such case they start crying. In other words, a crying set is a set of children in which each child is waiting for a toy that is kept by another child in the set.Now, we have reached a scenario where all the children made all the requests for their lovely sets, except for one child x that still has one last request for his lovely set. Some children are playing while others are waiting for a toy, but no child is crying, and no one has yet finished playing. If the child x is currently waiting for some toy, he makes his last request just after getting that toy. Otherwise, he makes the request right away. When child x will make his last request, how many children will start crying?You will be given the scenario and q independent queries. Each query will be of the form x y meaning that the last request of the child x is for the toy y. Your task is to help Sagheer find the size of the maximal crying set when child x makes his last request.","['graphs', 'trees']","[0, 0, 1, 0, 0, 0, 0, 1]"160"Shubham has an array $$$a$$$ of size $$$n$$$, and wants to select exactly $$$x$$$ elements from it, such that their sum is odd. These elements do not have to be consecutive. The elements of the array are not guaranteed to be distinct.Tell him whether he can do so.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"161"After many unsuccessful tries, Mashtali decided to copy modify an AtCoder problem. So here is his copied new problem:There is a tree with $$$n$$$ vertices and some non-empty set of the vertices are pinned to the ground.Two players play a game against each other on the tree. They alternately perform the following action:  Remove an edge from the tree, then remove every connected component that has no pinned vertex.The player who cannot move loses(every edge has been deleted already). You are given the tree, but not the set of the pinned vertices. Your task is to determine, for each $$$k$$$, the winner of the game, if only the vertices $$$1, 2, 3, \ldots, k$$$ are pinned and both players play optimally.","['games', 'trees']","[1, 0, 0, 0, 0, 0, 0, 1]"162"This is the easy version of the problem. The difference between the versions is the constraints on $$$a_i$$$. You can make hacks only if all versions of the problem are solved.Little Dormi has recently received a puzzle from his friend and needs your help to solve it. The puzzle consists of an upright board with $$$n$$$ rows and $$$m$$$ columns of cells, some empty and some filled with blocks of sand, and $$$m$$$ non-negative integers $$$a_1,a_2,\ldots,a_m$$$ ($$$0 \leq a_i \leq n$$$). In this version of the problem, $$$a_i$$$ will be equal to the number of blocks of sand in column $$$i$$$.When a cell filled with a block of sand is disturbed, the block of sand will fall from its cell to the sand counter at the bottom of the column (each column has a sand counter). While a block of sand is falling, other blocks of sand that are adjacent at any point to the falling block of sand will also be disturbed and start to fall. Specifically, a block of sand disturbed at a cell $$$(i,j)$$$ will pass through all cells below and including the cell $$$(i,j)$$$ within the column, disturbing all adjacent cells along the way. Here, the cells adjacent to a cell $$$(i,j)$$$ are defined as $$$(i-1,j)$$$, $$$(i,j-1)$$$, $$$(i+1,j)$$$, and $$$(i,j+1)$$$ (if they are within the grid). Note that the newly falling blocks can disturb other blocks.In one operation you are able to disturb any piece of sand. The puzzle is solved when there are at least $$$a_i$$$ blocks of sand counted in the $$$i$$$-th sand counter for each column from $$$1$$$ to $$$m$$$.You are now tasked with finding the minimum amount of operations in order to solve the puzzle. Note that Little Dormi will never give you a puzzle that is impossible to solve.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"163"The girl named Masha was walking in the forest and found a complete binary tree of height $$$n$$$ and a permutation $$$p$$$ of length $$$m=2^n$$$.A complete binary tree of height $$$n$$$ is a rooted tree such that every vertex except the leaves has exactly two sons, and the length of the path from the root to any of the leaves is $$$n$$$. The picture below shows the complete binary tree for $$$n=2$$$.A permutation is an array consisting of $$$n$$$ different integers from $$$1$$$ to $$$n$$$. For example, [$$$2,3,1,5,4$$$] is a permutation, but [$$$1,2,2$$$] is not ($$$2$$$ occurs twice), and [$$$1,3,4$$$] is also not a permutation ($$$n=3$$$, but there is $$$4$$$ in the array).Let's enumerate $$$m$$$ leaves of this tree from left to right. The leaf with the number $$$i$$$ contains the value $$$p_i$$$ ($$$1 \le i \le m$$$).For example, if $$$n = 2$$$, $$$p = [3, 1, 4, 2]$$$, the tree will look like this:  Masha considers a tree beautiful if the values in its leaves are ordered from left to right in increasing order.In one operation, Masha can choose any non-leaf vertex of the tree and swap its left and right sons (along with their subtrees).For example, if Masha applies this operation to the root of the tree discussed above, it will take the following form:  Help Masha understand if she can make a tree beautiful in a certain number of operations. If she can, then output the minimum number of operations to make the tree beautiful.","['graphs', 'trees']","[0, 0, 1, 0, 0, 0, 0, 1]"164"A class of students got bored wearing the same pair of shoes every day, so they decided to shuffle their shoes among themselves. In this problem, a pair of shoes is inseparable and is considered as a single object.There are $$$n$$$ students in the class, and you are given an array $$$s$$$ in non-decreasing order, where $$$s_i$$$ is the shoe size of the $$$i$$$-th student. A shuffling of shoes is valid only if no student gets their own shoes and if every student gets shoes of size greater than or equal to their size. You have to output a permutation $$$p$$$ of $$$\{1,2,\ldots,n\}$$$ denoting a valid shuffling of shoes, where the $$$i$$$-th student gets the shoes of the $$$p_i$$$-th student ($$$p_i \ne i$$$). And output $$$-1$$$ if a valid shuffling does not exist.A permutation is an array consisting of $$$n$$$ distinct integers from $$$1$$$ to $$$n$$$ in arbitrary order. For example, $$$[2,3,1,5,4]$$$ is a permutation, but $$$[1,2,2]$$$ is not a permutation ($$$2$$$ appears twice in the array) and $$$[1,3,4]$$$ is also not a permutation ($$$n=3$$$ but there is $$$4$$$ in the array).",[],"[0, 0, 0, 0, 0, 0, 0, 0]"165"Helen works in Metropolis airport. She is responsible for creating a departure schedule. There are n flights that must depart today, the i-th of them is planned to depart at the i-th minute of the day.Metropolis airport is the main transport hub of Metropolia, so it is difficult to keep the schedule intact. This is exactly the case today: because of technical issues, no flights were able to depart during the first k minutes of the day, so now the new departure schedule must be created.All n scheduled flights must now depart at different minutes between (k + 1)-th and (k + n)-th, inclusive. However, it's not mandatory for the flights to depart in the same order they were initially scheduled to do so — their order in the new schedule can be different. There is only one restriction: no flight is allowed to depart earlier than it was supposed to depart in the initial schedule.Helen knows that each minute of delay of the i-th flight costs airport ci burles. Help her find the order for flights to depart in the new schedule that minimizes the total cost for the airport.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"166"Let's call a number a binary decimal if it's a positive integer and all digits in its decimal notation are either $$$0$$$ or $$$1$$$. For example, $$$1\,010\,111$$$ is a binary decimal, while $$$10\,201$$$ and $$$787\,788$$$ are not.Given a number $$$n$$$, you are asked to represent $$$n$$$ as a sum of some (not necessarily distinct) binary decimals. Compute the smallest number of binary decimals required for that.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"167"Adilbek's house is located on a street which can be represented as the OX axis. This street is really dark, so Adilbek wants to install some post lamps to illuminate it. Street has $$$n$$$ positions to install lamps, they correspond to the integer numbers from $$$0$$$ to $$$n - 1$$$ on the OX axis. However, some positions are blocked and no post lamp can be placed there.There are post lamps of different types which differ only by their power. When placed in position $$$x$$$, post lamp of power $$$l$$$ illuminates the segment $$$[x; x + l]$$$. The power of each post lamp is always a positive integer number.The post lamp shop provides an infinite amount of lamps of each type from power $$$1$$$ to power $$$k$$$. Though each customer is only allowed to order post lamps of exactly one type. Post lamps of power $$$l$$$ cost $$$a_l$$$ each.What is the minimal total cost of the post lamps of exactly one type Adilbek can buy to illuminate the entire segment $$$[0; n]$$$ of the street? If some lamps illuminate any other segment of the street, Adilbek does not care, so, for example, he may place a lamp of power $$$3$$$ in position $$$n - 1$$$ (even though its illumination zone doesn't completely belong to segment $$$[0; n]$$$).",[],"[0, 0, 0, 0, 0, 0, 0, 0]"168"You are the gym teacher in the school.There are $$$n$$$ students in the row. And there are two rivalling students among them. The first one is in position $$$a$$$, the second in position $$$b$$$. Positions are numbered from $$$1$$$ to $$$n$$$ from left to right.Since they are rivals, you want to maximize the distance between them. If students are in positions $$$p$$$ and $$$s$$$ respectively, then distance between them is $$$|p - s|$$$. You can do the following operation at most $$$x$$$ times: choose two adjacent (neighbouring) students and swap them.Calculate the maximum distance between two rivalling students after at most $$$x$$$ swaps.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"169"Miyako came to the flea kingdom with a ukulele. She became good friends with local flea residents and played beautiful music for them every day.In return, the fleas made a bigger ukulele for her: it has $$$n$$$ strings, and each string has $$$(10^{18} + 1)$$$ frets numerated from $$$0$$$ to $$$10^{18}$$$. The fleas use the array $$$s_1, s_2, \ldots, s_n$$$ to describe the ukulele's tuning, that is, the pitch of the $$$j$$$-th fret on the $$$i$$$-th string is the integer $$$s_i + j$$$.Miyako is about to leave the kingdom, but the fleas hope that Miyako will answer some last questions for them.Each question is in the form of: ""How many different pitches are there, if we consider frets between $$$l$$$ and $$$r$$$ (inclusive) on all strings?""Miyako is about to visit the cricket kingdom and has no time to answer all the questions. Please help her with this task!Formally, you are given a matrix with $$$n$$$ rows and $$$(10^{18}+1)$$$ columns, where the cell in the $$$i$$$-th row and $$$j$$$-th column ($$$0 \le j \le 10^{18}$$$) contains the integer $$$s_i + j$$$. You are to answer $$$q$$$ queries, in the $$$k$$$-th query you have to answer the number of distinct integers in the matrix from the $$$l_k$$$-th to the $$$r_k$$$-th columns, inclusive.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"170"Andi and Budi were given an assignment to tidy up their bookshelf of $$$n$$$ books. Each book is represented by the book title — a string $$$s_i$$$ numbered from $$$1$$$ to $$$n$$$, each with length $$$m$$$. Andi really wants to sort the book lexicographically ascending, while Budi wants to sort it lexicographically descending.Settling their fight, they decided to combine their idea and sort it asc-desc-endingly, where the odd-indexed characters will be compared ascendingly, and the even-indexed characters will be compared descendingly.A string $$$a$$$ occurs before a string $$$b$$$ in asc-desc-ending order if and only if in the first position where $$$a$$$ and $$$b$$$ differ, the following holds:  if it is an odd position, the string $$$a$$$ has a letter that appears earlier in the alphabet than the corresponding letter in $$$b$$$;  if it is an even position, the string $$$a$$$ has a letter that appears later in the alphabet than the corresponding letter in $$$b$$$. ",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"171"Dima worked all day and wrote down on a long paper strip his favorite number $$$n$$$ consisting of $$$l$$$ digits. Unfortunately, the strip turned out to be so long that it didn't fit in the Dima's bookshelf.To solve the issue, Dima decided to split the strip into two non-empty parts so that each of them contains a positive integer without leading zeros. After that he will compute the sum of the two integers and write it down on a new strip.Dima wants the resulting integer to be as small as possible, because it increases the chances that the sum will fit it in the bookshelf. Help Dima decide what is the minimum sum he can obtain.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"172"Billy investigates the question of applying greedy algorithm to different spheres of life. At the moment he is studying the application of greedy algorithm to the problem about change. There is an amount of n coins of different face values, and the coins of each value are not limited in number. The task is to collect the sum x with the minimum amount of coins. Greedy algorithm with each its step takes the coin of the highest face value, not exceeding x. Obviously, if among the coins' face values exists the face value 1, any sum x can be collected with the help of greedy algorithm. However, greedy algorithm does not always give the optimal representation of the sum, i.e. the representation with the minimum amount of coins. For example, if there are face values {1, 3, 4} and it is asked to collect the sum 6, greedy algorithm will represent the sum as 4 + 1 + 1, while the optimal representation is 3 + 3, containing one coin less. By the given set of face values find out if there exist such a sum x that greedy algorithm will collect in a non-optimal way. If such a sum exists, find out the smallest of these sums.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"173"Vasya has an array a consisting of positive integer numbers. Vasya wants to divide this array into two non-empty consecutive parts (the prefix and the suffix) so that the sum of all elements in the first part equals to the sum of elements in the second part. It is not always possible, so Vasya will move some element before dividing the array (Vasya will erase some element and insert it into an arbitrary position).Inserting an element in the same position he was erased from is also considered moving.Can Vasya divide the array after choosing the right element to move and its new position?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"174"Berland regional ICPC contest has just ended. There were $$$m$$$ participants numbered from $$$1$$$ to $$$m$$$, who competed on a problemset of $$$n$$$ problems numbered from $$$1$$$ to $$$n$$$.Now the editorial is about to take place. There are two problem authors, each of them is going to tell the tutorial to exactly $$$k$$$ consecutive tasks of the problemset. The authors choose the segment of $$$k$$$ consecutive tasks for themselves independently of each other. The segments can coincide, intersect or not intersect at all.The $$$i$$$-th participant is interested in listening to the tutorial of all consecutive tasks from $$$l_i$$$ to $$$r_i$$$. Each participant always chooses to listen to only the problem author that tells the tutorials to the maximum number of tasks he is interested in. Let this maximum number be $$$a_i$$$. No participant can listen to both of the authors, even if their segments don't intersect.The authors want to choose the segments of $$$k$$$ consecutive tasks for themselves in such a way that the sum of $$$a_i$$$ over all participants is maximized.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"175"Developers often face with regular expression patterns. A pattern is usually defined as a string consisting of characters and metacharacters that sets the rules for your search. These patterns are most often used to check whether a particular string meets the certain rules.In this task, a pattern will be a string consisting of small English letters and question marks ('?'). The question mark in the pattern is a metacharacter that denotes an arbitrary small letter of the English alphabet. We will assume that a string matches the pattern if we can transform the string into the pattern by replacing the question marks by the appropriate characters. For example, string aba matches patterns: ???, ??a, a?a, aba.Programmers that work for the R1 company love puzzling each other (and themselves) with riddles. One of them is as follows: you are given n patterns of the same length, you need to find a pattern that contains as few question marks as possible, and intersects with each of the given patterns. Two patterns intersect if there is a string that matches both the first and the second pattern. Can you solve this riddle?",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"176"After the fourth season Sherlock and Moriary have realized the whole foolishness of the battle between them and decided to continue their competitions in peaceful game of Credit Cards.Rules of this game are simple: each player bring his favourite n-digit credit card. Then both players name the digits written on their cards one by one. If two digits are not equal, then the player, whose digit is smaller gets a flick (knock in the forehead usually made with a forefinger) from the other player. For example, if n = 3, Sherlock's card is 123 and Moriarty's card has number 321, first Sherlock names 1 and Moriarty names 3 so Sherlock gets a flick. Then they both digit 2 so no one gets a flick. Finally, Sherlock names 3, while Moriarty names 1 and gets a flick.Of course, Sherlock will play honestly naming digits one by one in the order they are given, while Moriary, as a true villain, plans to cheat. He is going to name his digits in some other order (however, he is not going to change the overall number of occurences of each digit). For example, in case above Moriarty could name 1, 2, 3 and get no flicks at all, or he can name 2, 3 and 1 to give Sherlock two flicks.Your goal is to find out the minimum possible number of flicks Moriarty will get (no one likes flicks) and the maximum possible number of flicks Sherlock can get from Moriarty. Note, that these two goals are different and the optimal result may be obtained by using different strategies.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"177"The only difference between easy and hard versions is a number of elements in the array.You are given an array $$$a$$$ consisting of $$$n$$$ integers. The value of the $$$i$$$-th element of the array is $$$a_i$$$.You are also given a set of $$$m$$$ segments. The $$$j$$$-th segment is $$$[l_j; r_j]$$$, where $$$1 \le l_j \le r_j \le n$$$.You can choose some subset of the given set of segments and decrease values on each of the chosen segments by one (independently). For example, if the initial array $$$a = [0, 0, 0, 0, 0]$$$ and the given segments are $$$[1; 3]$$$ and $$$[2; 4]$$$ then you can choose both of them and the array will become $$$b = [-1, -2, -2, -1, 0]$$$.You have to choose some subset of the given segments (each segment can be chosen at most once) in such a way that if you apply this subset of segments to the array $$$a$$$ and obtain the array $$$b$$$ then the value $$$\max\limits_{i=1}^{n}b_i - \min\limits_{i=1}^{n}b_i$$$ will be maximum possible.Note that you can choose the empty set.If there are multiple answers, you can print any.If you are Python programmer, consider using PyPy instead of Python when you submit your code.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"178"Gardener Alexey teaches competitive programming to high school students. To congratulate Alexey on the Teacher's Day, the students have gifted him a collection of wooden sticks, where every stick has an integer length. Now Alexey wants to grow a tree from them.The tree looks like a polyline on the plane, consisting of all sticks. The polyline starts at the point $$$(0, 0)$$$. While constructing the polyline, Alexey will attach sticks to it one by one in arbitrary order. Each stick must be either vertical or horizontal (that is, parallel to $$$OX$$$ or $$$OY$$$ axis). It is not allowed for two consecutive sticks to be aligned simultaneously horizontally or simultaneously vertically. See the images below for clarification.Alexey wants to make a polyline in such a way that its end is as far as possible from $$$(0, 0)$$$. Please help him to grow the tree this way.Note that the polyline defining the form of the tree may have self-intersections and self-touches, but it can be proved that the optimal answer does not contain any self-intersections or self-touches.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"179"Levian works as an accountant in a large company. Levian knows how much the company has earned in each of the $$$n$$$ consecutive months — in the $$$i$$$-th month the company had income equal to $$$a_i$$$ (positive income means profit, negative income means loss, zero income means no change). Because of the general self-isolation, the first $$$\lceil \tfrac{n}{2} \rceil$$$ months income might have been completely unstable, but then everything stabilized and for the last $$$\lfloor \tfrac{n}{2} \rfloor$$$ months the income was the same.Levian decided to tell the directors $$$n-k+1$$$ numbers — the total income of the company for each $$$k$$$ consecutive months. In other words, for each $$$i$$$ between $$$1$$$ and $$$n-k+1$$$ he will say the value $$$a_i + a_{i+1} + \ldots + a_{i + k - 1}$$$. For example, if $$$a=[-1, 0, 1, 2, 2]$$$ and $$$k=3$$$ he will say the numbers $$$0, 3, 5$$$.Unfortunately, if at least one total income reported by Levian is not a profit (income $$$\le 0$$$), the directors will get angry and fire the failed accountant.Save Levian's career: find any such $$$k$$$, that for each $$$k$$$ months in a row the company had made a profit, or report that it is impossible.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"180"  William is not only interested in trading but also in betting on sports matches. $$$n$$$ teams participate in each match. Each team is characterized by strength $$$a_i$$$. Each two teams $$$i < j$$$ play with each other exactly once. Team $$$i$$$ wins with probability $$$\frac{a_i}{a_i + a_j}$$$ and team $$$j$$$ wins with probability $$$\frac{a_j}{a_i + a_j}$$$.The team is called a winner if it directly or indirectly defeated all other teams. Team $$$a$$$ defeated (directly or indirectly) team $$$b$$$ if there is a sequence of teams $$$c_1$$$, $$$c_2$$$, ... $$$c_k$$$ such that $$$c_1 = a$$$, $$$c_k = b$$$ and team $$$c_i$$$ defeated team $$$c_{i + 1}$$$ for all $$$i$$$ from $$$1$$$ to $$$k - 1$$$. Note that it is possible that team $$$a$$$ defeated team $$$b$$$ and in the same time team $$$b$$$ defeated team $$$a$$$.William wants you to find the expected value of the number of winners.","['graphs', 'math', 'probabilities']","[0, 0, 1, 1, 0, 1, 0, 0]"181"Xenia is an amateur programmer. Today on the IT lesson she learned about the Hamming distance.The Hamming distance between two strings s = s1s2... sn and t = t1t2... tn of equal length n is value . Record [si ≠ ti] is the Iverson notation and represents the following: if si ≠ ti, it is one, otherwise — zero.Now Xenia wants to calculate the Hamming distance between two long strings a and b. The first string a is the concatenation of n copies of string x, that is, . The second string b is the concatenation of m copies of string y. Help Xenia, calculate the required Hamming distance, given n, x, m, y.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"182"Three sons inherited from their father a rectangular corn fiend divided into n × m squares. For each square we know how many tons of corn grows on it. The father, an old farmer did not love all three sons equally, which is why he bequeathed to divide his field into three parts containing A, B and C tons of corn.The field should be divided by two parallel lines. The lines should be parallel to one side of the field and to each other. The lines should go strictly between the squares of the field. Each resulting part of the field should consist of at least one square. Your task is to find the number of ways to divide the field as is described above, that is, to mark two lines, dividing the field in three parts so that on one of the resulting parts grew A tons of corn, B on another one and C on the remaining one.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"183Let $$$f(x)$$$ be the sum of digits of a decimal number $$$x$$$.Find the smallest non-negative integer $$$x$$$ such that $$$f(x) + f(x + 1) + \dots + f(x + k) = n$$$.,[],"[0, 0, 0, 0, 0, 0, 0, 0]"184"The city Valera lives in is going to hold elections to the city Parliament.The city has n districts and n - 1 bidirectional roads. We know that from any district there is a path along the roads to any other district. Let's enumerate all districts in some way by integers from 1 to n, inclusive. Furthermore, for each road the residents decided if it is the problem road or not. A problem road is a road that needs to be repaired.There are n candidates running the elections. Let's enumerate all candidates in some way by integers from 1 to n, inclusive. If the candidate number i will be elected in the city Parliament, he will perform exactly one promise — to repair all problem roads on the way from the i-th district to the district 1, where the city Parliament is located.Help Valera and determine the subset of candidates such that if all candidates from the subset will be elected to the city Parliament, all problem roads in the city will be repaired. If there are several such subsets, you should choose the subset consisting of the minimum number of candidates.","['graphs', 'trees']","[0, 0, 1, 0, 0, 0, 0, 1]"185"Despite his bad reputation, Captain Flint is a friendly person (at least, friendly to animals). Now Captain Flint is searching worthy sailors to join his new crew (solely for peaceful purposes). A sailor is considered as worthy if he can solve Flint's task.Recently, out of blue Captain Flint has been interested in math and even defined a new class of integers. Let's define a positive integer $$$x$$$ as nearly prime if it can be represented as $$$p \cdot q$$$, where $$$1 < p < q$$$ and $$$p$$$ and $$$q$$$ are prime numbers. For example, integers $$$6$$$ and $$$10$$$ are nearly primes (since $$$2 \cdot 3 = 6$$$ and $$$2 \cdot 5 = 10$$$), but integers $$$1$$$, $$$3$$$, $$$4$$$, $$$16$$$, $$$17$$$ or $$$44$$$ are not.Captain Flint guessed an integer $$$n$$$ and asked you: can you represent it as the sum of $$$4$$$ different positive integers where at least $$$3$$$ of them should be nearly prime.Uncle Bogdan easily solved the task and joined the crew. Can you do the same?","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"186"Do you know what tubular bells are? They are a musical instrument made up of cylindrical metal tubes. In an orchestra, tubular bells are used to mimic the ringing of bells.Mike has tubular bells, too! They consist of $$$n$$$ tubes, and each of the tubes has a length that can be expressed by a integer from $$$l$$$ to $$$r$$$ inclusive. It is clear that the lengths of all the tubes are different (it makes no sense to make the same tubes). It is also known that $$$r-l+1 = n$$$.Formally, we can say that Mike's tubular bells are described by a permutation $$$a$$$ of length $$$n$$$ that contains all numbers from $$$l$$$ to $$$r$$$ inclusive, with $$$a_i$$$ denoting the length of the $$$i$$$-th tube.You are offered an interesting task: to guess what Mike's instrument looks like. Simply, you must guess the permutation.Mike won't tell you $$$l$$$ or $$$r$$$. He will only tell you $$$n$$$, and will allow you to ask no more than $$$n + 5000$$$ queries.In each query, you name two positive integers $$$x$$$, $$$y$$$ such that $$$1 \le x, y \le n, x \neq y$$$. In response to this query, the program written by Mike will give you $$$\mathrm{lcm}(a_x, a_y)$$$, where $$$\mathrm{lcm}(c,d)$$$ denotes the least common multiple of $$$c$$$ and $$$d$$$.Solve Mike's problem!","['math', 'number theory', 'probabilities']","[0, 0, 0, 1, 1, 1, 0, 0]"187"This is the hard version of the problem. The only difference is maximum value of $$$a_i$$$.Once in Kostomuksha Divan found an array $$$a$$$ consisting of positive integers. Now he wants to reorder the elements of $$$a$$$ to maximize the value of the following function: $$$$$$\sum_{i=1}^n \operatorname{gcd}(a_1, \, a_2, \, \dots, \, a_i),$$$$$$ where $$$\operatorname{gcd}(x_1, x_2, \ldots, x_k)$$$ denotes the greatest common divisor of integers $$$x_1, x_2, \ldots, x_k$$$, and $$$\operatorname{gcd}(x) = x$$$ for any integer $$$x$$$.Reordering elements of an array means changing the order of elements in the array arbitrary, or leaving the initial order.Of course, Divan can solve this problem. However, he found it interesting, so he decided to share it with you.",['number theory'],"[0, 0, 0, 0, 1, 0, 0, 0]"188You are given an undirected tree consisting of $$$n$$$ vertices. An undirected tree is a connected undirected graph with $$$n - 1$$$ edges.Your task is to add the minimum number of edges in such a way that the length of the shortest path from the vertex $$$1$$$ to any other vertex is at most $$$2$$$. Note that you are not allowed to add loops and multiple edges.,['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"189"This is the hard version of this problem. The only difference is the limit of $$$n$$$ - the length of the input string. In this version, $$$1 \leq n \leq 10^6$$$.Let's define a correct bracket sequence and its depth as follow:  An empty string is a correct bracket sequence with depth $$$0$$$.  If ""s"" is a correct bracket sequence with depth $$$d$$$ then ""(s)"" is a correct bracket sequence with depth $$$d + 1$$$.  If ""s"" and ""t"" are both correct bracket sequences then their concatenation ""st"" is a correct bracket sequence with depth equal to the maximum depth of $$$s$$$ and $$$t$$$. For a (not necessarily correct) bracket sequence $$$s$$$, we define its depth as the maximum depth of any correct bracket sequence induced by removing some characters from $$$s$$$ (possibly zero). For example: the bracket sequence $$$s = $$$""())(())"" has depth $$$2$$$, because by removing the third character we obtain a correct bracket sequence ""()(())"" with depth $$$2$$$.Given a string $$$a$$$ consists of only characters '(', ')' and '?'. Consider all (not necessarily correct) bracket sequences obtained by replacing all characters '?' in $$$a$$$ by either '(' or ')'. Calculate the sum of all the depths of all these bracket sequences. As this number can be large, find it modulo $$$998244353$$$.Hacks in this problem can be done only if easy and hard versions of this problem was solved.",['probabilities'],"[0, 0, 0, 0, 0, 1, 0, 0]"190"We had a really tough time generating tests for problem D. In order to prepare strong tests, we had to solve the following problem.Given an undirected labeled tree consisting of $$$n$$$ vertices, find a set of segments such that:  both endpoints of each segment are integers from $$$1$$$ to $$$2n$$$, and each integer from $$$1$$$ to $$$2n$$$ should appear as an endpoint of exactly one segment;  all segments are non-degenerate;  for each pair $$$(i, j)$$$ such that $$$i \ne j$$$, $$$i \in [1, n]$$$ and $$$j \in [1, n]$$$, the vertices $$$i$$$ and $$$j$$$ are connected with an edge if and only if the segments $$$i$$$ and $$$j$$$ intersect, but neither segment $$$i$$$ is fully contained in segment $$$j$$$, nor segment $$$j$$$ is fully contained in segment $$$i$$$. Can you solve this problem too?",['trees'],"[0, 0, 0, 0, 0, 0, 0, 1]"191"One day Vasya decided to have a look at the results of Berland 1910 Football Championship’s finals. Unfortunately he didn't find the overall score of the match; however, he got hold of a profound description of the match's process. On the whole there are n lines in that description each of which described one goal. Every goal was marked with the name of the team that had scored it. Help Vasya, learn the name of the team that won the finals. It is guaranteed that the match did not end in a tie.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"192"AquaMoon had $$$n$$$ strings of length $$$m$$$ each. $$$n$$$ is an odd number.When AquaMoon was gone, Cirno tried to pair these $$$n$$$ strings together. After making $$$\frac{n-1}{2}$$$ pairs, she found out that there was exactly one string without the pair!In her rage, she disrupted each pair of strings. For each pair, she selected some positions (at least $$$1$$$ and at most $$$m$$$) and swapped the letters in the two strings of this pair at the selected positions.For example, if $$$m = 6$$$ and two strings ""abcdef"" and ""xyzklm"" are in one pair and Cirno selected positions $$$2$$$, $$$3$$$ and $$$6$$$ she will swap 'b' with 'y', 'c' with 'z' and 'f' with 'm'. The resulting strings will be ""ayzdem"" and ""xbcklf"".Cirno then stole away the string without pair and shuffled all remaining strings in arbitrary order.AquaMoon found the remaining $$$n-1$$$ strings in complete disarray. Also, she remembers the initial $$$n$$$ strings. She wants to know which string was stolen, but she is not good at programming. Can you help her?",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"193"This morning Tolik has understood that while he was sleeping he had invented an incredible problem which will be a perfect fit for Codeforces! But, as a ""Discuss tasks"" project hasn't been born yet (in English, well), he decides to test a problem and asks his uncle.After a long time thinking, Tolik's uncle hasn't any ideas on how to solve it. But, he doesn't want to tell Tolik about his inability to solve it, so he hasn't found anything better than asking you how to solve this task.In this task you are given a cell field $$$n \cdot m$$$, consisting of $$$n$$$ rows and $$$m$$$ columns, where point's coordinates $$$(x, y)$$$ mean it is situated in the $$$x$$$-th row and $$$y$$$-th column, considering numeration from one ($$$1 \leq x \leq n, 1 \leq y \leq m$$$). Initially, you stand in the cell $$$(1, 1)$$$. Every move you can jump from cell $$$(x, y)$$$, which you stand in, by any non-zero vector $$$(dx, dy)$$$, thus you will stand in the $$$(x+dx, y+dy)$$$ cell. Obviously, you can't leave the field, but also there is one more important condition — you're not allowed to use one vector twice. Your task is to visit each cell of the field exactly once (the initial cell is considered as already visited).Tolik's uncle is a very respectful person. Help him to solve this task!",[],"[0, 0, 0, 0, 0, 0, 0, 0]"194"There is a bookshelf which can fit $$$n$$$ books. The $$$i$$$-th position of bookshelf is $$$a_i = 1$$$ if there is a book on this position and $$$a_i = 0$$$ otherwise. It is guaranteed that there is at least one book on the bookshelf.In one move, you can choose some contiguous segment $$$[l; r]$$$ consisting of books (i.e. for each $$$i$$$ from $$$l$$$ to $$$r$$$ the condition $$$a_i = 1$$$ holds) and:  Shift it to the right by $$$1$$$: move the book at index $$$i$$$ to $$$i + 1$$$ for all $$$l \le i \le r$$$. This move can be done only if $$$r+1 \le n$$$ and there is no book at the position $$$r+1$$$.  Shift it to the left by $$$1$$$: move the book at index $$$i$$$ to $$$i-1$$$ for all $$$l \le i \le r$$$. This move can be done only if $$$l-1 \ge 1$$$ and there is no book at the position $$$l-1$$$. Your task is to find the minimum number of moves required to collect all the books on the shelf as a contiguous (consecutive) segment (i.e. the segment without any gaps).For example, for $$$a = [0, 0, 1, 0, 1]$$$ there is a gap between books ($$$a_4 = 0$$$ when $$$a_3 = 1$$$ and $$$a_5 = 1$$$), for $$$a = [1, 1, 0]$$$ there are no gaps between books and for $$$a = [0, 0,0]$$$ there are also no gaps between books.You have to answer $$$t$$$ independent test cases.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"195"Little X has met the following problem recently. Let's define f(x) as the sum of digits in decimal representation of number x (for example, f(1234) = 1 + 2 + 3 + 4). You are to calculate Of course Little X has solved this problem quickly, has locked it, and then has tried to hack others. He has seen the following C++ code:     ans = solve(l, r) % a;    if (ans <= 0)      ans += a; This code will fail only on the test with . You are given number a, help Little X to find a proper test for hack.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"196"You are given an array $$$a$$$ of length $$$2n$$$. Consider a partition of array $$$a$$$ into two subsequences $$$p$$$ and $$$q$$$ of length $$$n$$$ each (each element of array $$$a$$$ should be in exactly one subsequence: either in $$$p$$$ or in $$$q$$$).Let's sort $$$p$$$ in non-decreasing order, and $$$q$$$ in non-increasing order, we can denote the sorted versions by $$$x$$$ and $$$y$$$, respectively. Then the cost of a partition is defined as $$$f(p, q) = \sum_{i = 1}^n |x_i - y_i|$$$.Find the sum of $$$f(p, q)$$$ over all correct partitions of array $$$a$$$. Since the answer might be too big, print its remainder modulo $$$998244353$$$.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"197"Maxim wants to buy some games at the local game shop. There are $$$n$$$ games in the shop, the $$$i$$$-th game costs $$$c_i$$$.Maxim has a wallet which can be represented as an array of integers. His wallet contains $$$m$$$ bills, the $$$j$$$-th bill has value $$$a_j$$$.Games in the shop are ordered from left to right, Maxim tries to buy every game in that order.When Maxim stands at the position $$$i$$$ in the shop, he takes the first bill from his wallet (if his wallet is empty then he proceeds to the next position immediately) and tries to buy the $$$i$$$-th game using this bill. After Maxim tried to buy the $$$n$$$-th game, he leaves the shop.Maxim buys the $$$i$$$-th game if and only if the value of the first bill (which he takes) from his wallet is greater or equal to the cost of the $$$i$$$-th game. If he successfully buys the $$$i$$$-th game, the first bill from his wallet disappears and the next bill becomes first. Otherwise Maxim leaves the first bill in his wallet (this bill still remains the first one) and proceeds to the next game.For example, for array $$$c = [2, 4, 5, 2, 4]$$$ and array $$$a = [5, 3, 4, 6]$$$ the following process takes place: Maxim buys the first game using the first bill (its value is $$$5$$$), the bill disappears, after that the second bill (with value $$$3$$$) becomes the first one in Maxim's wallet, then Maxim doesn't buy the second game because $$$c_2 > a_2$$$, the same with the third game, then he buys the fourth game using the bill of value $$$a_2$$$ (the third bill becomes the first one in Maxim's wallet) and buys the fifth game using the bill of value $$$a_3$$$.Your task is to get the number of games Maxim will buy.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"198"Palindromic characteristics of string s with length |s| is a sequence of |s| integers, where k-th number is the total number of non-empty substrings of s which are k-palindromes.A string is 1-palindrome if and only if it reads the same backward as forward.A string is k-palindrome (k > 1) if and only if:   Its left half equals to its right half.  Its left and right halfs are non-empty (k - 1)-palindromes. The left half of string t is its prefix of length ⌊|t| / 2⌋, and right half — the suffix of the same length. ⌊|t| / 2⌋ denotes the length of string t divided by 2, rounded down.Note that each substring is counted as many times as it appears in the string. For example, in the string ""aaa"" the substring ""a"" appears 3 times.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"199"There was a string $$$s$$$ which was supposed to be encrypted. For this reason, all $$$26$$$ lowercase English letters were arranged in a circle in some order, afterwards, each letter in $$$s$$$ was replaced with the one that follows in clockwise order, in that way the string $$$t$$$ was obtained. You are given a string $$$t$$$. Determine the lexicographically smallest string $$$s$$$ that could be a prototype of the given string $$$t$$$.A string $$$a$$$ is lexicographically smaller than a string $$$b$$$ of the same length if and only if:   in the first position where $$$a$$$ and $$$b$$$ differ, the string $$$a$$$ has a letter, that appears earlier in the alphabet than the corresponding letter in $$$b$$$.  ","['graphs', 'strings']","[0, 0, 1, 0, 0, 0, 1, 0]"200"There are n student groups at the university. During the study day, each group can take no more than 7 classes. Seven time slots numbered from 1 to 7 are allocated for the classes.The schedule on Monday is known for each group, i. e. time slots when group will have classes are known.Your task is to determine the minimum number of rooms needed to hold classes for all groups on Monday. Note that one room can hold at most one group class in a single time slot.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"201"A rooted tree is a non-directed connected graph without any cycles with a distinguished vertex, which is called the tree root. Consider the vertices of a rooted tree, that consists of n vertices, numbered from 1 to n. In this problem the tree root is the vertex number 1.Let's represent the length of the shortest by the number of edges path in the tree between vertices v and u as d(v, u).A parent of vertex v in the rooted tree with the root in vertex r (v ≠ r) is vertex pv, such that d(r, pv) + 1 = d(r, v) and d(pv, v) = 1. For example, on the picture the parent of vertex v = 5 is vertex p5 = 2.One day Polycarpus came across a rooted tree, consisting of n vertices. The tree wasn't exactly ordinary: it had strings written on its edges. Polycarpus positioned the tree on the plane so as to make all edges lead from top to bottom if you go from the vertex parent to the vertex (see the picture). For any edge that lead from vertex pv to vertex v (1 < v ≤ n), he knows string sv that is written on it. All strings are written on the edges from top to bottom. For example, on the picture s7=""ba"". The characters in the strings are numbered starting from 0.    An example of Polycarpus's tree (corresponds to the example from the statement) Polycarpus defines the position in this tree as a specific letter on a specific string. The position is written as a pair of integers (v, x) that means that the position is the x-th letter of the string sv (1 < v ≤ n, 0 ≤ x < |sv|), where |sv| is the length of string sv. For example, the highlighted letters are positions (2, 1) and (3, 1).Let's consider the pair of positions (v, x) and (u, y) in Polycarpus' tree, such that the way from the first position to the second goes down on each step. We will consider that the pair of such positions defines string z. String z consists of all letters on the way from (v, x) to (u, y), written in the order of this path. For example, in the picture the highlighted positions define string ""bacaba"".Polycarpus has a string t, he wants to know the number of pairs of positions that define string t. Note that the way from the first position to the second in the pair must go down everywhere. Help him with this challenging tree-string problem!",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"202"You are given a binary string $$$s$$$ of length $$$n$$$.Let's define $$$d_i$$$ as the number whose decimal representation is $$$s_i s_{i+1}$$$ (possibly, with a leading zero). We define $$$f(s)$$$ to be the sum of all the valid $$$d_i$$$. In other words, $$$f(s) = \sum\limits_{i=1}^{n-1} d_i$$$.For example, for the string $$$s = 1011$$$:   $$$d_1 = 10$$$ (ten);  $$$d_2 = 01$$$ (one)  $$$d_3 = 11$$$ (eleven);  $$$f(s) = 10 + 01 + 11 = 22$$$. In one operation you can swap any two adjacent elements of the string. Find the minimum value of $$$f(s)$$$ that can be achieved if at most $$$k$$$ operations are allowed.","['math', 'strings']","[0, 0, 0, 1, 0, 0, 1, 0]"203"A new e-mail service ""Berlandesk"" is going to be opened in Berland in the near future. The site administration wants to launch their project as soon as possible, that's why they ask you to help. You're suggested to implement the prototype of site registration system. The system should work on the following principle. Each time a new user wants to register, he sends to the system a request with his name. If such a name does not exist in the system database, it is inserted into the database, and the user gets the response OK, confirming the successful registration. If the name already exists in the system database, the system makes up a new user name, sends it to the user as a prompt and also inserts the prompt into the database. The new name is formed by the following rule. Numbers, starting with 1, are appended one after another to name (name1, name2, ...), among these numbers the least i is found so that namei does not yet exist in the database.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"204"Madoka as a child was an extremely capricious girl, and one of her favorite pranks was drawing on her wall. According to Madoka's memories, the wall was a table of $$$n$$$ rows and $$$m$$$ columns, consisting only of zeroes and ones. The coordinate of the cell in the $$$i$$$-th row and the $$$j$$$-th column ($$$1 \le i \le n$$$, $$$1 \le j \le m$$$) is $$$(i, j)$$$.One day she saw a picture ""Mahou Shoujo Madoka Magica"" and decided to draw it on her wall. Initially, the Madoka's table is a table of size $$$n \times m$$$ filled with zeroes. Then she applies the following operation any number of times:Madoka selects any rectangular subtable of the table and paints it in a chess coloring (the upper left corner of the subtable always has the color $$$0$$$). Note that some cells may be colored several times. In this case, the final color of the cell is equal to the color obtained during the last repainting.  White color means $$$0$$$, black means $$$1$$$. So, for example, the table in the first picture is painted in a chess coloring, and the others are not. For better understanding of the statement, we recommend you to read the explanation of the first test.Help Madoka and find some sequence of no more than $$$n \cdot m$$$ operations that allows you to obtain the picture she wants, or determine that this is impossible.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"205"An array is sorted if it has no inversionsA Young BoyYou are given an array of $$$n$$$ positive integers $$$a_1,a_2,\ldots,a_n$$$. In one operation you do the following:   Choose any integer $$$x$$$.  For all $$$i$$$ such that $$$a_i = x$$$, do $$$a_i := 0$$$ (assign $$$0$$$ to $$$a_i$$$). Find the minimum number of operations required to sort the array in non-decreasing order.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"206" As the boat drifts down the river, a wood full of blossoms shows up on the riverfront.""I've been here once,"" Mino exclaims with delight, ""it's breathtakingly amazing.""""What is it like?""""Look, Kanno, you've got your paintbrush, and I've got my words. Have a try, shall we?"" There are four kinds of flowers in the wood, Amaranths, Begonias, Centaureas and Dianthuses.The wood can be represented by a rectangular grid of $$$n$$$ rows and $$$m$$$ columns. In each cell of the grid, there is exactly one type of flowers.According to Mino, the numbers of connected components formed by each kind of flowers are $$$a$$$, $$$b$$$, $$$c$$$ and $$$d$$$ respectively. Two cells are considered in the same connected component if and only if a path exists between them that moves between cells sharing common edges and passes only through cells containing the same flowers.You are to help Kanno depict such a grid of flowers, with $$$n$$$ and $$$m$$$ arbitrarily chosen under the constraints given below. It can be shown that at least one solution exists under the constraints of this problem.Note that you can choose arbitrary $$$n$$$ and $$$m$$$ under the constraints below, they are not given in the input.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"207"""Eat a beaver, save a tree!"" — That will be the motto of ecologists' urgent meeting in Beaverley Hills.And the whole point is that the population of beavers on the Earth has reached incredible sizes! Each day their number increases in several times and they don't even realize how much their unhealthy obsession with trees harms the nature and the humankind. The amount of oxygen in the atmosphere has dropped to 17 per cent and, as the best minds of the world think, that is not the end.In the middle of the 50-s of the previous century a group of soviet scientists succeed in foreseeing the situation with beavers and worked out a secret technology to clean territory. The technology bears a mysterious title ""Beavermuncher-0xFF"". Now the fate of the planet lies on the fragile shoulders of a small group of people who has dedicated their lives to science.The prototype is ready, you now need to urgently carry out its experiments in practice. You are given a tree, completely occupied by beavers. A tree is a connected undirected graph without cycles. The tree consists of n vertices, the i-th vertex contains ki beavers. ""Beavermuncher-0xFF"" works by the following principle: being at some vertex u, it can go to the vertex v, if they are connected by an edge, and eat exactly one beaver located at the vertex v. It is impossible to move to the vertex v if there are no beavers left in v. ""Beavermuncher-0xFF"" cannot just stand at some vertex and eat beavers in it. ""Beavermuncher-0xFF"" must move without stops.Why does the ""Beavermuncher-0xFF"" works like this? Because the developers have not provided place for the battery in it and eating beavers is necessary for converting their mass into pure energy.It is guaranteed that the beavers will be shocked by what is happening, which is why they will not be able to move from a vertex of the tree to another one. As for the ""Beavermuncher-0xFF"", it can move along each edge in both directions while conditions described above are fulfilled.The root of the tree is located at the vertex s. This means that the ""Beavermuncher-0xFF"" begins its mission at the vertex s and it must return there at the end of experiment, because no one is going to take it down from a high place. Determine the maximum number of beavers ""Beavermuncher-0xFF"" can eat and return to the starting vertex.",['trees'],"[0, 0, 0, 0, 0, 0, 0, 1]"208"Let's call an undirected graph $$$G = (V, E)$$$ relatively prime if and only if for each edge $$$(v, u) \in E$$$  $$$GCD(v, u) = 1$$$ (the greatest common divisor of $$$v$$$ and $$$u$$$ is $$$1$$$). If there is no edge between some pair of vertices $$$v$$$ and $$$u$$$ then the value of $$$GCD(v, u)$$$ doesn't matter. The vertices are numbered from $$$1$$$ to $$$|V|$$$.Construct a relatively prime graph with $$$n$$$ vertices and $$$m$$$ edges such that it is connected and it contains neither self-loops nor multiple edges.If there exists no valid graph with the given number of vertices and edges then output ""Impossible"".If there are multiple answers then print any of them.","['graphs', 'math']","[0, 0, 1, 1, 0, 0, 0, 0]"209"Recently, Duff has been practicing weight lifting. As a hard practice, Malek gave her a task. He gave her a sequence of weights. Weight of i-th of them is 2wi pounds. In each step, Duff can lift some of the remaining weights and throw them away. She does this until there's no more weight left. Malek asked her to minimize the number of steps.  Duff is a competitive programming fan. That's why in each step, she can only lift and throw away a sequence of weights 2a1, ..., 2ak if and only if there exists a non-negative integer x such that 2a1 + 2a2 + ... + 2ak = 2x, i. e. the sum of those numbers is a power of two.Duff is a competitive programming fan, but not a programmer. That's why she asked for your help. Help her minimize the number of steps. ",[],"[0, 0, 0, 0, 0, 0, 0, 0]"210"Gildong is playing a video game called Block Adventure. In Block Adventure, there are $$$n$$$ columns of blocks in a row, and the columns are numbered from $$$1$$$ to $$$n$$$. All blocks have equal heights. The height of the $$$i$$$-th column is represented as $$$h_i$$$, which is the number of blocks stacked in the $$$i$$$-th column.Gildong plays the game as a character that can stand only on the top of the columns. At the beginning, the character is standing on the top of the $$$1$$$-st column. The goal of the game is to move the character to the top of the $$$n$$$-th column.The character also has a bag that can hold infinitely many blocks. When the character is on the top of the $$$i$$$-th column, Gildong can take one of the following three actions as many times as he wants:   if there is at least one block on the column, remove one block from the top of the $$$i$$$-th column and put it in the bag;  if there is at least one block in the bag, take one block out of the bag and place it on the top of the $$$i$$$-th column;  if $$$i < n$$$ and $$$|h_i - h_{i+1}| \le k$$$, move the character to the top of the $$$i+1$$$-st column. $$$k$$$ is a non-negative integer given at the beginning of the game. Note that it is only possible to move to the next column. In actions of the first two types the character remains in the $$$i$$$-th column, and the value $$$h_i$$$ changes.The character initially has $$$m$$$ blocks in the bag. Gildong wants to know if it is possible to win the game. Help Gildong find the answer to his question.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"211"You are given $$$n$$$ segments on a coordinate axis $$$OX$$$. The $$$i$$$-th segment has borders $$$[l_i; r_i]$$$. All points $$$x$$$, for which $$$l_i \le x \le r_i$$$ holds, belong to the $$$i$$$-th segment.Your task is to choose the maximum by size (the number of segments) subset of the given set of segments such that each pair of segments in this subset either non-intersecting or one of them lies inside the other one.Two segments $$$[l_i; r_i]$$$ and $$$[l_j; r_j]$$$ are non-intersecting if they have no common points. For example, segments $$$[1; 2]$$$ and $$$[3; 4]$$$, $$$[1; 3]$$$ and $$$[5; 5]$$$ are non-intersecting, while segments $$$[1; 2]$$$ and $$$[2; 3]$$$, $$$[1; 2]$$$ and $$$[2; 2]$$$ are intersecting.The segment $$$[l_i; r_i]$$$ lies inside the segment $$$[l_j; r_j]$$$ if $$$l_j \le l_i$$$ and $$$r_i \le r_j$$$. For example, segments $$$[2; 2]$$$, $$$[2, 3]$$$, $$$[3; 4]$$$ and $$$[2; 4]$$$ lie inside the segment $$$[2; 4]$$$, while $$$[2; 5]$$$ and $$$[1; 4]$$$ are not.You have to answer $$$t$$$ independent test cases.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"212"Email address in Berland is a string of the form A@B, where A and B are arbitrary strings consisting of small Latin letters. Bob is a system administrator in «Bersoft» company. He keeps a list of email addresses of the company's staff. This list is as a large string, where all addresses are written in arbitrary order, separated by commas. The same address can be written more than once.Suddenly, because of unknown reasons, all commas in Bob's list disappeared. Now Bob has a string, where all addresses are written one after another without any separators, and there is impossible to determine, where the boundaries between addresses are. Unfortunately, on the same day his chief asked him to bring the initial list of addresses. Now Bob wants to disjoin addresses in some valid way. Help him to do that.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"213"You are given two binary strings $$$x$$$ and $$$y$$$, which are binary representations of some two integers (let's denote these integers as $$$f(x)$$$ and $$$f(y)$$$). You can choose any integer $$$k \ge 0$$$, calculate the expression $$$s_k = f(x) + f(y) \cdot 2^k$$$ and write the binary representation of $$$s_k$$$ in reverse order (let's denote it as $$$rev_k$$$). For example, let $$$x = 1010$$$ and $$$y = 11$$$; you've chosen $$$k = 1$$$ and, since $$$2^1 = 10_2$$$, so $$$s_k = 1010_2 + 11_2 \cdot 10_2 = 10000_2$$$ and $$$rev_k = 00001$$$.For given $$$x$$$ and $$$y$$$, you need to choose such $$$k$$$ that $$$rev_k$$$ is lexicographically minimal (read notes if you don't know what does ""lexicographically"" means).It's guaranteed that, with given constraints, $$$k$$$ exists and is finite.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"214"You are given a circular maze such as the ones shown in the figures.    Determine if it can be solved, i.e., if there is a path which goes from the center to the outside of the maze which does not touch any wall. The maze is described by $$$n$$$ walls. Each wall can be either circular or straight.   Circular walls are described by a radius $$$r$$$, the distance from the center, and two angles $$$\theta_1, \theta_2$$$ describing the beginning and the end of the wall in the clockwise direction. Notice that swapping the two angles changes the wall.  Straight walls are described by an angle $$$\theta$$$, the direction of the wall, and two radii $$$r_1 < r_2$$$ describing the beginning and the end of the wall. Angles are measured in degrees; the angle $$$0$$$ corresponds to the upward pointing direction; and angles increase clockwise (hence the east direction corresponds to the angle $$$90$$$).",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"215"Recently you have bought a snow walking robot and brought it home. Suppose your home is a cell $$$(0, 0)$$$ on an infinite grid.You also have the sequence of instructions of this robot. It is written as the string $$$s$$$ consisting of characters 'L', 'R', 'U' and 'D'. If the robot is in the cell $$$(x, y)$$$ right now, he can move to one of the adjacent cells (depending on the current instruction).  If the current instruction is 'L', then the robot can move to the left to $$$(x - 1, y)$$$;  if the current instruction is 'R', then the robot can move to the right to $$$(x + 1, y)$$$;  if the current instruction is 'U', then the robot can move to the top to $$$(x, y + 1)$$$;  if the current instruction is 'D', then the robot can move to the bottom to $$$(x, y - 1)$$$. You've noticed the warning on the last page of the manual: if the robot visits some cell (except $$$(0, 0)$$$) twice then it breaks.So the sequence of instructions is valid if the robot starts in the cell $$$(0, 0)$$$, performs the given instructions, visits no cell other than $$$(0, 0)$$$ two or more times and ends the path in the cell $$$(0, 0)$$$. Also cell $$$(0, 0)$$$ should be visited at most two times: at the beginning and at the end (if the path is empty then it is visited only once). For example, the following sequences of instructions are considered valid: ""UD"", ""RL"", ""UUURULLDDDDLDDRRUU"", and the following are considered invalid: ""U"" (the endpoint is not $$$(0, 0)$$$) and ""UUDD"" (the cell $$$(0, 1)$$$ is visited twice).The initial sequence of instructions, however, might be not valid. You don't want your robot to break so you decided to reprogram it in the following way: you will remove some (possibly, all or none) instructions from the initial sequence of instructions, then rearrange the remaining instructions as you wish and turn on your robot to move. Your task is to remove as few instructions from the initial sequence as possible and rearrange the remaining ones so that the sequence is valid. Report the valid sequence of the maximum length you can obtain.Note that you can choose any order of remaining instructions (you don't need to minimize the number of swaps or any other similar metric).You have to answer $$$q$$$ independent test cases.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"216"A k-multiple free set is a set of integers where there is no pair of integers where one is equal to another integer multiplied by k. That is, there are no two integers x and y (x < y) from the set, such that y = x·k.You're given a set of n distinct positive integers. Your task is to find the size of it's largest k-multiple free subset.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"217"The Cybermen solved that first test much quicker than the Daleks. Luckily for us, the Daleks were angry (shocking!) and they destroyed some of the Cybermen.After the fighting stopped, Heidi gave them another task to waste their time on.There are $$$n$$$ points on a plane. Given a radius $$$r$$$, find the maximum number of points that can be covered by an $$$L^1$$$-ball with radius $$$r$$$.An $$$L^1$$$-ball with radius $$$r$$$ and center $$$(x_0, y_0)$$$ in a 2D-plane is defined as the set of points $$$(x, y)$$$ such that the Manhattan distance between $$$(x_0, y_0)$$$ and $$$(x, y)$$$ is at most $$$r$$$.Manhattan distance between $$$(x_0, y_0)$$$ and $$$(x, y)$$$ is defined as $$$|x - x_0| + |y - y_0|$$$.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"218There are n boys and m girls studying in the class. They should stand in a line so that boys and girls alternated there as much as possible. Let's assume that positions in the line are indexed from left to right by numbers from 1 to n + m. Then the number of integers i (1 ≤ i < n + m) such that positions with indexes i and i + 1 contain children of different genders (position i has a girl and position i + 1 has a boy or vice versa) must be as large as possible. Help the children and tell them how to form the line.,[],"[0, 0, 0, 0, 0, 0, 0, 0]"219"Valerian was captured by Shapur. The victory was such a great one that Shapur decided to carve a scene of Valerian's defeat on a mountain. So he had to find the best place to make his victory eternal!He decided to visit all n cities of Persia to find the best available mountain, but after the recent war he was too tired and didn't want to traverse a lot. So he wanted to visit each of these n cities at least once with smallest possible traverse. Persian cities are connected with bidirectional roads. You can go from any city to any other one using these roads and there is a unique path between each two cities.All cities are numbered 1 to n. Shapur is currently in the city 1 and he wants to visit all other cities with minimum possible traverse. He can finish his travels in any city.Help Shapur find how much He should travel.","['graphs', 'trees']","[0, 0, 1, 0, 0, 0, 0, 1]"220"Polycarp has n dice d1, d2, ..., dn. The i-th dice shows numbers from 1 to di. Polycarp rolled all the dice and the sum of numbers they showed is A. Agrippina didn't see which dice showed what number, she knows only the sum A and the values d1, d2, ..., dn. However, she finds it enough to make a series of statements of the following type: dice i couldn't show number r. For example, if Polycarp had two six-faced dice and the total sum is A = 11, then Agrippina can state that each of the two dice couldn't show a value less than five (otherwise, the remaining dice must have a value of at least seven, which is impossible).For each dice find the number of values for which it can be guaranteed that the dice couldn't show these values if the sum of the shown values is A.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"221"Ivan had string s consisting of small English letters. However, his friend Julia decided to make fun of him and hid the string s. Ivan preferred making a new string to finding the old one. Ivan knows some information about the string s. Namely, he remembers, that string ti occurs in string s at least ki times or more, he also remembers exactly ki positions where the string ti occurs in string s: these positions are xi, 1, xi, 2, ..., xi, ki. He remembers n such strings ti.You are to reconstruct lexicographically minimal string s such that it fits all the information Ivan remembers. Strings ti and string s consist of small English letters only.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"222"This is an easy version of the problem. The difference between the easy and hard versions is that in this version, you can output any permutation with the smallest weight.You are given a permutation $$$p_1, p_2, \ldots, p_n$$$ of integers from $$$1$$$ to $$$n$$$.Let's define the weight of the permutation $$$q_1, q_2, \ldots, q_n$$$ of integers from $$$1$$$ to $$$n$$$ as $$$$$$|q_1 - p_{q_{2}}| + |q_2 - p_{q_{3}}| + \ldots + |q_{n-1} - p_{q_{n}}| + |q_n - p_{q_{1}}|$$$$$$You want your permutation to be as lightweight as possible. Find any permutation $$$q$$$ with the smallest possible weight.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"223"Note that this is the first problem of the two similar problems. You can hack this problem only if you solve both problems.You are given a tree with $$$n$$$ nodes. In the beginning, $$$0$$$ is written on all edges. In one operation, you can choose any $$$2$$$ distinct leaves $$$u$$$, $$$v$$$ and any real number $$$x$$$ and add $$$x$$$ to values written on all edges on the simple path between $$$u$$$ and $$$v$$$.For example, on the picture below you can see the result of applying two operations to the graph: adding $$$2$$$ on the path from $$$7$$$ to $$$6$$$, and then adding $$$-0.5$$$ on the path from $$$4$$$ to $$$5$$$.   Is it true that for any configuration of real numbers written on edges, we can achieve it with a finite number of operations?Leaf is a node of a tree of degree $$$1$$$. Simple path is a path that doesn't contain any node twice.",['trees'],"[0, 0, 0, 0, 0, 0, 0, 1]"224"Ezzat has an array of $$$n$$$ integers (maybe negative). He wants to split it into two non-empty subsequences $$$a$$$ and $$$b$$$, such that every element from the array belongs to exactly one subsequence, and the value of $$$f(a) + f(b)$$$ is the maximum possible value, where $$$f(x)$$$ is the average of the subsequence $$$x$$$. A sequence $$$x$$$ is a subsequence of a sequence $$$y$$$ if $$$x$$$ can be obtained from $$$y$$$ by deletion of several (possibly, zero or all) elements.The average of a subsequence is the sum of the numbers of this subsequence divided by the size of the subsequence.For example, the average of $$$[1,5,6]$$$ is $$$(1+5+6)/3 = 12/3 = 4$$$, so $$$f([1,5,6]) = 4$$$.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"225"Roma (a popular Russian name that means 'Roman') loves the Little Lvov Elephant's lucky numbers.Let us remind you that lucky numbers are positive integers whose decimal representation only contains lucky digits 4 and 7. For example, numbers 47, 744, 4 are lucky and 5, 17, 467 are not.Roma's got n positive integers. He wonders, how many of those integers have not more than k lucky digits? Help him, write the program that solves the problem.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"226"In BerSoft $$$n$$$ programmers work, the programmer $$$i$$$ is characterized by a skill $$$r_i$$$.A programmer $$$a$$$ can be a mentor of a programmer $$$b$$$ if and only if the skill of the programmer $$$a$$$ is strictly greater than the skill of the programmer $$$b$$$ $$$(r_a > r_b)$$$ and programmers $$$a$$$ and $$$b$$$ are not in a quarrel.You are given the skills of each programmers and a list of $$$k$$$ pairs of the programmers, which are in a quarrel (pairs are unordered). For each programmer $$$i$$$, find the number of programmers, for which the programmer $$$i$$$ can be a mentor.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"227"This is an interactive problem. Remember to flush your output while communicating with the testing program. You may use fflush(stdout) in C++, system.out.flush() in Java, stdout.flush() in Python or flush(output) in Pascal to flush the output. If you use some other programming language, consult its documentation. You may also refer to the guide on interactive problems: https://codeforces.com/blog/entry/45307.The jury picked an integer $$$x$$$ not less than $$$0$$$ and not greater than $$$2^{14} - 1$$$. You have to guess this integer.To do so, you may ask no more than $$$2$$$ queries. Each query should consist of $$$100$$$ integer numbers $$$a_1$$$, $$$a_2$$$, ..., $$$a_{100}$$$ (each integer should be not less than $$$0$$$ and not greater than $$$2^{14} - 1$$$). In response to your query, the jury will pick one integer $$$i$$$ ($$$1 \le i \le 100$$$) and tell you the value of $$$a_i \oplus x$$$ (the bitwise XOR of $$$a_i$$$ and $$$x$$$). There is an additional constraint on the queries: all $$$200$$$ integers you use in the queries should be distinct.It is guaranteed that the value of $$$x$$$ is fixed beforehand in each test, but the choice of $$$i$$$ in every query may depend on the integers you send.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"228"Analyzing the mistakes people make while typing search queries is a complex and an interesting work. As there is no guaranteed way to determine what the user originally meant by typing some query, we have to use different sorts of heuristics.Polycarp needed to write a code that could, given two words, check whether they could have been obtained from the same word as a result of typos. Polycarpus suggested that the most common typo is skipping exactly one letter as you type a word.Implement a program that can, given two distinct words S and T of the same length n determine how many words W of length n + 1 are there with such property that you can transform W into both S, and T by deleting exactly one character. Words S and T consist of lowercase English letters. Word W also should consist of lowercase English letters.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"229"There is a chip on the coordinate line. Initially, the chip is located at the point $$$0$$$. You can perform any number of moves; each move increases the coordinate of the chip by some positive integer (which is called the length of the move). The length of the first move you make should be divisible by $$$k$$$, the length of the second move — by $$$k+1$$$, the third — by $$$k+2$$$, and so on.For example, if $$$k=2$$$, then the sequence of moves may look like this: $$$0 \rightarrow 4 \rightarrow 7 \rightarrow 19 \rightarrow 44$$$, because $$$4 - 0 = 4$$$ is divisible by $$$2 = k$$$, $$$7 - 4 = 3$$$ is divisible by $$$3 = k + 1$$$, $$$19 - 7 = 12$$$ is divisible by $$$4 = k + 2$$$, $$$44 - 19 = 25$$$ is divisible by $$$5 = k + 3$$$.You are given two positive integers $$$n$$$ and $$$k$$$. Your task is to count the number of ways to reach the point $$$x$$$, starting from $$$0$$$, for every $$$x \in [1, n]$$$. The number of ways can be very large, so print it modulo $$$998244353$$$. Two ways are considered different if they differ as sets of visited positions.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"230"After Vitaly was expelled from the university, he became interested in the graph theory.Vitaly especially liked the cycles of an odd length in which each vertex occurs at most once.Vitaly was wondering how to solve the following problem. You are given an undirected graph consisting of n vertices and m edges, not necessarily connected, without parallel edges and loops. You need to find t — the minimum number of edges that must be added to the given graph in order to form a simple cycle of an odd length, consisting of more than one vertex. Moreover, he must find w — the number of ways to add t edges in order to form a cycle of an odd length (consisting of more than one vertex). It is prohibited to add loops or parallel edges.Two ways to add edges to the graph are considered equal if they have the same sets of added edges.Since Vitaly does not study at the university, he asked you to help him with this task.","['graphs', 'math']","[0, 0, 1, 1, 0, 0, 0, 0]"231"An atom of element X can exist in n distinct states with energies E1 < E2 < ... < En. Arkady wants to build a laser on this element, using a three-level scheme. Here is a simplified description of the scheme. Three distinct states i, j and k are selected, where i < j < k. After that the following process happens:   initially the atom is in the state i, we spend Ek - Ei energy to put the atom in the state k, the atom emits a photon with useful energy Ek - Ej and changes its state to the state j, the atom spontaneously changes its state to the state i, losing energy Ej - Ei, the process repeats from step 1. Let's define the energy conversion efficiency as , i. e. the ration between the useful energy of the photon and spent energy.Due to some limitations, Arkady can only choose such three states that Ek - Ei ≤ U.Help Arkady to find such the maximum possible energy conversion efficiency within the above constraints.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"232"You've got another geometrical task. You are given two non-degenerate polygons A and B as vertex coordinates. Polygon A is strictly convex. Polygon B is an arbitrary polygon without any self-intersections and self-touches. The vertices of both polygons are given in the clockwise order. For each polygon no three consecutively following vertices are located on the same straight line.Your task is to check whether polygon B is positioned strictly inside polygon A. It means that any point of polygon B should be strictly inside polygon A. ""Strictly"" means that the vertex of polygon B cannot lie on the side of the polygon A.",['geometry'],"[0, 1, 0, 0, 0, 0, 0, 0]"233"Coming up with a new problem isn't as easy as many people think. Sometimes it is hard enough to name it. We'll consider a title original if it doesn't occur as a substring in any titles of recent Codeforces problems. You've got the titles of n last problems — the strings, consisting of lowercase English letters. Your task is to find the shortest original title for the new problem. If there are multiple such titles, choose the lexicographically minimum one. Note, that title of the problem can't be an empty string.A substring s[l... r] (1 ≤ l ≤ r ≤ |s|) of string s = s1s2... s|s| (where |s| is the length of string s) is string slsl + 1... sr.String x = x1x2... xp is lexicographically smaller than string y = y1y2... yq, if either p < q and x1 = y1, x2 = y2, ... , xp = yp, or there exists such number r (r < p, r < q), that x1 = y1, x2 = y2, ... , xr = yr and xr + 1 < yr + 1. The string characters are compared by their ASCII codes.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"234"Omkar's most recent follower, Ajit, has entered the Holy Forest. Ajit realizes that Omkar's forest is an $$$n$$$ by $$$m$$$ grid ($$$1 \leq n, m \leq 2000$$$) of some non-negative integers. Since the forest is blessed by Omkar, it satisfies some special conditions:  For any two adjacent (sharing a side) cells, the absolute value of the difference of numbers in them is at most $$$1$$$.  If the number in some cell is strictly larger than $$$0$$$, it should be strictly greater than the number in at least one of the cells adjacent to it. Unfortunately, Ajit is not fully worthy of Omkar's powers yet. He sees each cell as a ""0"" or a ""#"". If a cell is labeled as ""0"", then the number in it must equal $$$0$$$. Otherwise, the number in it can be any nonnegative integer.Determine how many different assignments of elements exist such that these special conditions are satisfied. Two assignments are considered different if there exists at least one cell such that the numbers written in it in these assignments are different. Since the answer may be enormous, find the answer modulo $$$10^9+7$$$.","['graphs', 'math']","[0, 0, 1, 1, 0, 0, 0, 0]"235"The city of D consists of n towers, built consecutively on a straight line. The height of the tower that goes i-th (from left to right) in the sequence equals hi. The city mayor decided to rebuild the city to make it beautiful. In a beautiful city all towers are are arranged in non-descending order of their height from left to right.The rebuilding consists of performing several (perhaps zero) operations. An operation constitutes using a crane to take any tower and put it altogether on the top of some other neighboring tower. In other words, we can take the tower that stands i-th and put it on the top of either the (i - 1)-th tower (if it exists), or the (i + 1)-th tower (of it exists). The height of the resulting tower equals the sum of heights of the two towers that were put together. After that the two towers can't be split by any means, but more similar operations can be performed on the resulting tower. Note that after each operation the total number of towers on the straight line decreases by 1.Help the mayor determine the minimum number of operations required to make the city beautiful.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"236"Everybody seems to think that the Martians are green, but it turns out they are metallic pink and fat. Ajs has two bags of distinct nonnegative integers. The bags are disjoint, and the union of the sets of numbers in the bags is $$$\{0,1,…,M-1\}$$$, for some positive integer $$$M$$$. Ajs draws a number from the first bag and a number from the second bag, and then sums them modulo $$$M$$$.What are the residues modulo $$$M$$$ that Ajs cannot obtain with this action?",['number theory'],"[0, 0, 0, 0, 1, 0, 0, 0]"237"You are given an integer $$$n$$$. Check if $$$n$$$ has an odd divisor, greater than one (does there exist such a number $$$x$$$ ($$$x > 1$$$) that $$$n$$$ is divisible by $$$x$$$ and $$$x$$$ is odd).For example, if $$$n=6$$$, then there is $$$x=3$$$. If $$$n=4$$$, then such a number does not exist.","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"238"Madoka is going to enroll in ""TSUNS PTU"". But she stumbled upon a difficult task during the entrance computer science exam:  A number is called good if it is a multiple of $$$d$$$.  A number is called beatiful if it is good and it cannot be represented as a product of two good numbers. Notice that a beautiful number must be good.Given a good number $$$x$$$, determine whether it can be represented in at least two different ways as a product of several (possibly, one) beautiful numbers. Two ways are different if the sets of numbers used are different.Solve this problem for Madoka and help her to enroll in the best school in Russia!","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"239"You are given an undirected unweighted graph consisting of $$$n$$$ vertices and $$$m$$$ edges.You have to write a number on each vertex of the graph. Each number should be $$$1$$$, $$$2$$$ or $$$3$$$. The graph becomes beautiful if for each edge the sum of numbers on vertices connected by this edge is odd.Calculate the number of possible ways to write numbers $$$1$$$, $$$2$$$ and $$$3$$$ on vertices so the graph becomes beautiful. Since this number may be large, print it modulo $$$998244353$$$.Note that you have to write exactly one number on each vertex.The graph does not have any self-loops or multiple edges.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"240"The Robot is in a rectangular maze of size n × m. Each cell of the maze is either empty or occupied by an obstacle. The Robot can move between neighboring cells on the side left (the symbol ""L""), right (the symbol ""R""), up (the symbol ""U"") or down (the symbol ""D""). The Robot can move to the cell only if it is empty. Initially, the Robot is in the empty cell.Your task is to find lexicographically minimal Robot's cycle with length exactly k, which begins and ends in the cell where the Robot was initially. It is allowed to the Robot to visit any cell many times (including starting).Consider that Robot's way is given as a line which consists of symbols ""L"", ""R"", ""U"" and ""D"". For example, if firstly the Robot goes down, then left, then right and up, it means that his way is written as ""DLRU"".In this task you don't need to minimize the length of the way. Find the minimum lexicographical (in alphabet order as in the dictionary) line which satisfies requirements above.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"241"The grasshopper is located on the numeric axis at the point with coordinate $$$x_0$$$.Having nothing else to do he starts jumping between integer points on the axis. Making a jump from a point with coordinate $$$x$$$ with a distance $$$d$$$ to the left moves the grasshopper to a point with a coordinate $$$x - d$$$, while jumping to the right moves him to a point with a coordinate $$$x + d$$$.The grasshopper is very fond of positive integers, so for each integer $$$i$$$ starting with $$$1$$$ the following holds: exactly $$$i$$$ minutes after the start he makes a jump with a distance of exactly $$$i$$$. So, in the first minutes he jumps by $$$1$$$, then by $$$2$$$, and so on.The direction of a jump is determined as follows: if the point where the grasshopper was before the jump has an even coordinate, the grasshopper jumps to the left, otherwise he jumps to the right.For example, if after $$$18$$$ consecutive jumps he arrives at the point with a coordinate $$$7$$$, he will jump by a distance of $$$19$$$ to the right, since $$$7$$$ is an odd number, and will end up at a point $$$7 + 19 = 26$$$. Since $$$26$$$ is an even number, the next jump the grasshopper will make to the left by a distance of $$$20$$$, and it will move him to the point $$$26 - 20 = 6$$$.Find exactly which point the grasshopper will be at after exactly $$$n$$$ jumps.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"242"This is the hard version of the problem. The difference between the versions is that the hard version does require you to output the numbers of the rods to be removed. You can make hacks only if all versions of the problem are solved.Stitch likes experimenting with different machines with his friend Sparky. Today they built another machine.The main element of this machine are $$$n$$$ rods arranged along one straight line and numbered from $$$1$$$ to $$$n$$$ inclusive. Each of these rods must carry an electric charge quantitatively equal to either $$$1$$$ or $$$-1$$$ (otherwise the machine will not work). Another condition for this machine to work is that the sign-variable sum of the charge on all rods must be zero.More formally, the rods can be represented as an array of $$$n$$$ numbers characterizing the charge: either $$$1$$$ or $$$-1$$$. Then the condition must hold: $$$a_1 - a_2 + a_3 - a_4 + \ldots = 0$$$, or $$$\sum\limits_{i=1}^n (-1)^{i-1} \cdot a_i = 0$$$.Sparky charged all $$$n$$$ rods with an electric current, but unfortunately it happened that the rods were not charged correctly (the sign-variable sum of the charge is not zero). The friends decided to leave only some of the rods in the machine. Sparky has $$$q$$$ questions. In the $$$i$$$th question Sparky asks: if the machine consisted only of rods with numbers $$$l_i$$$ to $$$r_i$$$ inclusive, what minimal number of rods could be removed from the machine so that the sign-variable sum of charges on the remaining ones would be zero? Also Sparky wants to know the numbers of these rods. Perhaps the friends got something wrong, and the sign-variable sum is already zero. In that case, you don't have to remove the rods at all.If the number of rods is zero, we will assume that the sign-variable sum of charges is zero, that is, we can always remove all rods.Help your friends and answer all of Sparky's questions!",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"243"Grandma Capa has decided to knit a scarf and asked Grandpa Sher to make a pattern for it, a pattern is a string consisting of lowercase English letters. Grandpa Sher wrote a string $$$s$$$ of length $$$n$$$.Grandma Capa wants to knit a beautiful scarf, and in her opinion, a beautiful scarf can only be knit from a string that is a palindrome. She wants to change the pattern written by Grandpa Sher, but to avoid offending him, she will choose one lowercase English letter and erase some (at her choice, possibly none or all) occurrences of that letter in string $$$s$$$.She also wants to minimize the number of erased symbols from the pattern. Please help her and find the minimum number of symbols she can erase to make string $$$s$$$ a palindrome, or tell her that it's impossible. Notice that she can only erase symbols equal to the one letter she chose.A string is a palindrome if it is the same from the left to the right and from the right to the left. For example, the strings 'kek', 'abacaba', 'r' and 'papicipap' are palindromes, while the strings 'abb' and 'iq' are not.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"244"You and your friend Ilya are participating in an individual programming contest consisting of multiple stages. A contestant can get between $$$0$$$ and $$$100$$$ points, inclusive, for each stage, independently of other contestants.Points received by contestants in different stages are used for forming overall contest results. Suppose that $$$k$$$ stages of the contest are completed. For each contestant, $$$k - \lfloor \frac{k}{4} \rfloor$$$ stages with the highest scores are selected, and these scores are added up. This sum is the overall result of the contestant. (Here $$$\lfloor t \rfloor$$$ denotes rounding $$$t$$$ down.)For example, suppose $$$9$$$ stages are completed, and your scores are $$$50, 30, 50, 50, 100, 10, 30, 100, 50$$$. First, $$$7$$$ stages with the highest scores are chosen — for example, all stages except for the $$$2$$$-nd and the $$$6$$$-th can be chosen. Then your overall result is equal to $$$50 + 50 + 50 + 100 + 30 + 100 + 50 = 430$$$.As of now, $$$n$$$ stages are completed, and you know the points you and Ilya got for these stages. However, it is unknown how many more stages will be held. You wonder what the smallest number of additional stages is, after which your result might become greater than or equal to Ilya's result, at least in theory. Find this number!",[],"[0, 0, 0, 0, 0, 0, 0, 0]"245"Palo Alto is an unusual city because it is an endless coordinate line. It is also known for the office of Lyft Level 5.Lyft has become so popular so that it is now used by all $$$m$$$ taxi drivers in the city, who every day transport the rest of the city residents — $$$n$$$ riders.Each resident (including taxi drivers) of Palo-Alto lives in its unique location (there is no such pair of residents that their coordinates are the same).The Lyft system is very clever: when a rider calls a taxi, his call does not go to all taxi drivers, but only to the one that is the closest to that person. If there are multiple ones with the same distance, then to taxi driver with a smaller coordinate is selected.But one morning the taxi drivers wondered: how many riders are there that would call the given taxi driver if they were the first to order a taxi on that day? In other words, you need to find for each taxi driver $$$i$$$ the number $$$a_{i}$$$ — the number of riders that would call the $$$i$$$-th taxi driver when all drivers and riders are at their home?The taxi driver can neither transport himself nor other taxi drivers.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"246"Imagine that your city is an infinite 2D plane with Cartesian coordinate system. The only crime-affected road of your city is the x-axis. Currently, there are n criminals along the road. No police station has been built on this road yet, so the mayor wants to build one.As you are going to be in charge of this new police station, the mayor has asked you to choose a suitable position (some integer point) for building it. You should choose the best position for the police station, so that you could minimize the total time of your criminal catching mission. Your mission of catching the criminals will operate only from this station. The new station will have only one patrol car. You will go to the criminals by this car, carry them on the car, bring them back to the police station and put them in prison. The patrol car can carry at most m criminals at a time. Note that, the criminals don't know about your mission. So, they will stay where they are instead of running away.Your task is to find the position for the police station, so that total distance you need to cover to catch all the criminals will be minimum possible. Note that, you also can built the police station on the positions where one or more criminals already exist. In such a case all these criminals are arrested instantly.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"247"Note: The XOR-sum of set $$$\{s_1,s_2,\ldots,s_m\}$$$ is defined as $$$s_1 \oplus s_2 \oplus \ldots \oplus s_m$$$, where $$$\oplus$$$ denotes the bitwise XOR operation.After almost winning IOI, Victor bought himself an $$$n\times n$$$ grid containing integers in each cell. $$$n$$$ is an even integer. The integer in the cell in the $$$i$$$-th row and $$$j$$$-th column is $$$a_{i,j}$$$.Sadly, Mihai stole the grid from Victor and told him he would return it with only one condition: Victor has to tell Mihai the XOR-sum of all the integers in the whole grid.Victor doesn't remember all the elements of the grid, but he remembers some information about it: For each cell, Victor remembers the XOR-sum of all its neighboring cells.Two cells are considered neighbors if they share an edge — in other words, for some integers $$$1 \le i, j, k, l \le n$$$, the cell in the $$$i$$$-th row and $$$j$$$-th column is a neighbor of the cell in the $$$k$$$-th row and $$$l$$$-th column if $$$|i - k| = 1$$$ and $$$j = l$$$, or if $$$i = k$$$ and $$$|j - l| = 1$$$.To get his grid back, Victor is asking you for your help. Can you use the information Victor remembers to find the XOR-sum of the whole grid?It can be proven that the answer is unique.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"248"Ashish has a binary string $$$s$$$ of length $$$n$$$ that he wants to sort in non-decreasing order.He can perform the following operation:   Choose a subsequence of any length such that its elements are in non-increasing order. Formally, choose any $$$k$$$ such that $$$1 \leq k \leq n$$$ and any sequence of $$$k$$$ indices $$$1 \le i_1 \lt i_2 \lt \ldots \lt i_k \le n$$$ such that $$$s_{i_1} \ge s_{i_2} \ge \ldots \ge s_{i_k}$$$.  Reverse this subsequence in-place. Formally, swap $$$s_{i_1}$$$ with $$$s_{i_k}$$$, swap $$$s_{i_2}$$$ with $$$s_{i_{k-1}}$$$, $$$\ldots$$$ and swap $$$s_{i_{\lfloor k/2 \rfloor}}$$$ with $$$s_{i_{\lceil k/2 \rceil + 1}}$$$ (Here $$$\lfloor x \rfloor$$$ denotes the largest integer not exceeding $$$x$$$, and $$$\lceil x \rceil$$$ denotes the smallest integer not less than $$$x$$$) Find the minimum number of operations required to sort the string in non-decreasing order. It can be proven that it is always possible to sort the given binary string in at most $$$n$$$ operations.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"249"You are given a sequence a consisting of n integers. Find the maximum possible value of  (integer remainder of ai divided by aj), where 1 ≤ i, j ≤ n and ai ≥ aj.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"250"Having read half of the book called ""Storm and Calm"" on the IT lesson, Innocentius was absolutely determined to finish the book on the maths lessons. All was fine until the math teacher Ms. Watkins saw Innocentius reading fiction books instead of solving equations of the fifth degree. As during the last maths class Innocentius suggested the algorithm of solving equations of the fifth degree in the general case, Ms. Watkins had no other choice but to give him a new task.The teacher asked to write consecutively (without spaces) all words from the ""Storm and Calm"" in one long string s. She thought that a string is good if the number of vowels in the string is no more than twice more than the number of consonants. That is, the string with v vowels and c consonants is good if and only if v ≤ 2c.The task Innocentius had to solve turned out to be rather simple: he should find the number of the longest good substrings of the string s.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"251"Happy new year! The year 2020 is also known as Year Gyeongja (경자년, gyeongja-nyeon) in Korea. Where did the name come from? Let's briefly look at the Gapja system, which is traditionally used in Korea to name the years.There are two sequences of $$$n$$$ strings $$$s_1, s_2, s_3, \ldots, s_{n}$$$ and $$$m$$$ strings $$$t_1, t_2, t_3, \ldots, t_{m}$$$. These strings contain only lowercase letters. There might be duplicates among these strings.Let's call a concatenation of strings $$$x$$$ and $$$y$$$ as the string that is obtained by writing down strings $$$x$$$ and $$$y$$$ one right after another without changing the order. For example, the concatenation of the strings ""code"" and ""forces"" is the string ""codeforces"".The year 1 has a name which is the concatenation of the two strings $$$s_1$$$ and $$$t_1$$$. When the year increases by one, we concatenate the next two strings in order from each of the respective sequences. If the string that is currently being used is at the end of its sequence, we go back to the first string in that sequence.For example, if $$$n = 3, m = 4, s = $$${""a"", ""b"", ""c""}, $$$t =$$$ {""d"", ""e"", ""f"", ""g""}, the following table denotes the resulting year names. Note that the names of the years may repeat.  You are given two sequences of strings of size $$$n$$$ and $$$m$$$ and also $$$q$$$ queries. For each query, you will be given the current year. Could you find the name corresponding to the given year, according to the Gapja system?",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"252"Recently personal training sessions have finished in the Berland State University Olympiad Programmer Training Centre. By the results of these training sessions teams are composed for the oncoming team contest season. Each team consists of three people. All the students of the Centre possess numbers from 1 to 3n, and all the teams possess numbers from 1 to n. The splitting of students into teams is performed in the following manner: while there are people who are not part of a team, a person with the best total score is chosen among them (the captain of a new team), this person chooses for himself two teammates from those who is left according to his list of priorities. The list of every person's priorities is represented as a permutation from the rest of 3n - 1 students who attend the centre, besides himself.You are given the results of personal training sessions which are a permutation of numbers from 1 to 3n, where the i-th number is the number of student who has won the i-th place. No two students share a place. You are also given the arrangement of the already formed teams in the order in which they has been created. Your task is to determine the list of priorities for the student number k. If there are several priority lists, choose the lexicographically minimal one.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"253"You are given a number $$$n$$$ and an array $$$b_1, b_2, \ldots, b_{n+2}$$$, obtained according to the following algorithm:   some array $$$a_1, a_2, \ldots, a_n$$$ was guessed;  array $$$a$$$ was written to array $$$b$$$, i.e. $$$b_i = a_i$$$ ($$$1 \le i \le n$$$);  The $$$(n+1)$$$-th element of the array $$$b$$$ is the sum of the numbers in the array $$$a$$$, i.e. $$$b_{n+1} = a_1+a_2+\ldots+a_n$$$;  The $$$(n+2)$$$-th element of the array $$$b$$$ was written some number $$$x$$$ ($$$1 \le x \le 10^9$$$), i.e. $$$b_{n+2} = x$$$; The  array $$$b$$$ was shuffled. For example, the array $$$b=[2, 3, 7, 12 ,2]$$$ it could be obtained in the following ways:   $$$a=[2, 2, 3]$$$ and $$$x=12$$$;  $$$a=[3, 2, 7]$$$ and $$$x=2$$$. For the given array $$$b$$$, find any array $$$a$$$ that could have been guessed initially.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"254"There are n psychos standing in a line. Each psycho is assigned a unique integer from 1 to n. At each step every psycho who has an id greater than the psycho to his right (if exists) kills his right neighbor in the line. Note that a psycho might kill and get killed at the same step. You're given the initial arrangement of the psychos in the line. Calculate how many steps are needed to the moment of time such, that nobody kills his neighbor after that moment. Look notes to understand the statement more precise.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"255"Recall that the sequence $$$b$$$ is a a subsequence of the sequence $$$a$$$ if $$$b$$$ can be derived from $$$a$$$ by removing zero or more elements without changing the order of the remaining elements. For example, if $$$a=[1, 2, 1, 3, 1, 2, 1]$$$, then possible subsequences are: $$$[1, 1, 1, 1]$$$, $$$[3]$$$ and $$$[1, 2, 1, 3, 1, 2, 1]$$$, but not $$$[3, 2, 3]$$$ and $$$[1, 1, 1, 1, 2]$$$.You are given a sequence $$$a$$$ consisting of $$$n$$$ positive and negative elements (there is no zeros in the sequence).Your task is to choose maximum by size (length) alternating subsequence of the given sequence (i.e. the sign of each next element is the opposite from the sign of the current element, like positive-negative-positive and so on or negative-positive-negative and so on). Among all such subsequences, you have to choose one which has the maximum sum of elements.In other words, if the maximum length of alternating subsequence is $$$k$$$ then your task is to find the maximum sum of elements of some alternating subsequence of length $$$k$$$.You have to answer $$$t$$$ independent test cases.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"256"Numbers $$$1, 2, 3, \dots n$$$ (each integer from $$$1$$$ to $$$n$$$ once) are written on a board. In one operation you can erase any two numbers $$$a$$$ and $$$b$$$ from the board and write one integer $$$\frac{a + b}{2}$$$ rounded up instead.You should perform the given operation $$$n - 1$$$ times and make the resulting number that will be left on the board as small as possible. For example, if $$$n = 4$$$, the following course of action is optimal:  choose $$$a = 4$$$ and $$$b = 2$$$, so the new number is $$$3$$$, and the whiteboard contains $$$[1, 3, 3]$$$;  choose $$$a = 3$$$ and $$$b = 3$$$, so the new number is $$$3$$$, and the whiteboard contains $$$[1, 3]$$$;  choose $$$a = 1$$$ and $$$b = 3$$$, so the new number is $$$2$$$, and the whiteboard contains $$$[2]$$$. It's easy to see that after $$$n - 1$$$ operations, there will be left only one number. Your goal is to minimize it.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"257"One day mum asked Petya to sort his toys and get rid of some of them. Petya found a whole box of toy spiders. They were quite dear to him and the boy didn't want to throw them away. Petya conjured a cunning plan: he will glue all the spiders together and attach them to the ceiling. Besides, Petya knows that the lower the spiders will hang, the more mum is going to like it and then she won't throw his favourite toys away. Help Petya carry out the plan.A spider consists of k beads tied together by k - 1 threads. Each thread connects two different beads, at that any pair of beads that make up a spider is either directly connected by a thread, or is connected via some chain of threads and beads.Petya may glue spiders together directly gluing their beads. The length of each thread equals 1. The sizes of the beads can be neglected. That's why we can consider that gluing spiders happens by identifying some of the beads (see the picture). Besides, the construction resulting from the gluing process should also represent a spider, that is, it should have the given features. After Petya glues all spiders together, he measures the length of the resulting toy. The distance between a pair of beads is identified as the total length of the threads that connect these two beads. The length of the resulting construction is the largest distance between all pairs of beads. Petya wants to make the spider whose length is as much as possible.    The picture two shows two spiders from the second sample. We can glue to the bead number 2 of the first spider the bead number 1 of the second spider. The threads in the spiders that form the sequence of threads of maximum lengths are highlighted on the picture.",['trees'],"[0, 0, 0, 0, 0, 0, 0, 1]"258"You are given a permutation of integers from 1 to n. Exactly once you apply the following operation to this permutation: pick a random segment and shuffle its elements. Formally:  Pick a random segment (continuous subsequence) from l to r. All  segments are equiprobable.  Let k = r - l + 1, i.e. the length of the chosen segment. Pick a random permutation of integers from 1 to k, p1, p2, ..., pk. All k! permutation are equiprobable.  This permutation is applied to elements of the chosen segment, i.e. permutation a1, a2, ..., al - 1, al, al + 1, ..., ar - 1, ar, ar + 1, ..., an is transformed to a1, a2, ..., al - 1, al - 1 + p1, al - 1 + p2, ..., al - 1 + pk - 1, al - 1 + pk, ar + 1, ..., an. Inversion if a pair of elements (not necessary neighbouring) with the wrong relative order. In other words, the number of inversion is equal to the number of pairs (i, j) such that i < j and ai > aj. Find the expected number of inversions after we apply exactly one operation mentioned above.",['probabilities'],"[0, 0, 0, 0, 0, 1, 0, 0]"259"You are given a sequence of n positive integers d1, d2, ..., dn (d1 < d2 < ... < dn). Your task is to construct an undirected graph such that:  there are exactly dn + 1 vertices;  there are no self-loops;  there are no multiple edges;  there are no more than 106 edges;  its degree set is equal to d. Vertices should be numbered 1 through (dn + 1).Degree sequence is an array a with length equal to the number of vertices in a graph such that ai is the number of vertices adjacent to i-th vertex.Degree set is a sorted in increasing order sequence of all distinct values from the degree sequence.It is guaranteed that there exists such a graph that all the conditions hold, and it contains no more than 106 edges.Print the resulting graph.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"260"CQXYM is counting permutations length of $$$2n$$$.A permutation is an array consisting of $$$n$$$ distinct integers from $$$1$$$ to $$$n$$$ in arbitrary order. For example, $$$[2,3,1,5,4]$$$ is a permutation, but $$$[1,2,2]$$$ is not a permutation ($$$2$$$ appears twice in the array) and $$$[1,3,4]$$$ is also not a permutation ($$$n=3$$$ but there is $$$4$$$ in the array).A permutation $$$p$$$(length of $$$2n$$$) will be counted only if the number of $$$i$$$ satisfying $$$p_i<p_{i+1}$$$ is no less than $$$n$$$. For example:  Permutation $$$[1, 2, 3, 4]$$$ will count, because the number of such $$$i$$$ that $$$p_i<p_{i+1}$$$ equals $$$3$$$ ($$$i = 1$$$, $$$i = 2$$$, $$$i = 3$$$). Permutation $$$[3, 2, 1, 4]$$$ won't count, because the number of such $$$i$$$ that $$$p_i<p_{i+1}$$$ equals $$$1$$$ ($$$i = 3$$$). CQXYM wants you to help him to count the number of such permutations modulo $$$1000000007$$$ ($$$10^9+7$$$).In addition, modulo operation is to get the remainder. For example:  $$$7 \mod 3=1$$$, because $$$7 = 3 \cdot 2 + 1$$$,  $$$15 \mod 4=3$$$, because $$$15 = 4 \cdot 3 + 3$$$. ","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"261"Haha, try to solve this, SelectorUnlimited!— antontrygubO_oYour friends Alice and Bob practice fortune telling.Fortune telling is performed as follows. There is a well-known array $$$a$$$ of $$$n$$$ non-negative integers indexed from $$$1$$$ to $$$n$$$. The tellee starts with some non-negative number $$$d$$$ and performs one of the two operations for each $$$i = 1, 2, \ldots, n$$$, in the increasing order of $$$i$$$. The possible operations are: replace their current number $$$d$$$ with $$$d + a_i$$$ replace their current number $$$d$$$ with $$$d \oplus a_i$$$ (hereinafter $$$\oplus$$$ denotes the bitwise XOR operation)Notice that the chosen operation may be different for different $$$i$$$ and for different tellees.One time, Alice decided to start with $$$d = x$$$ and Bob started with $$$d = x + 3$$$. Each of them performed fortune telling and got a particular number in the end. Notice that the friends chose operations independently of each other, that is, they could apply different operations for the same $$$i$$$.You learnt that either Alice or Bob ended up with number $$$y$$$ in the end, but you don't know whose of the two it was. Given the numbers Alice and Bob started with and $$$y$$$, find out who (Alice or Bob) could get the number $$$y$$$ after performing the operations. It is guaranteed that on the jury tests, exactly one of your friends could have actually gotten that number.HacksYou cannot make hacks in this problem.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"262"This is the easy version of the problem. The only difference is that in this version there are no ""remove"" queries.Initially you have a set containing one element — $$$0$$$. You need to handle $$$q$$$ queries of the following types:+ $$$x$$$ — add the integer $$$x$$$ to the set. It is guaranteed that this integer is not contained in the set; ? $$$k$$$ — find the $$$k\text{-mex}$$$ of the set. In our problem, we define the $$$k\text{-mex}$$$ of a set of integers as the smallest non-negative integer $$$x$$$ that is divisible by $$$k$$$ and which is not contained in the set.",['number theory'],"[0, 0, 0, 0, 1, 0, 0, 0]"263"A flowerbed has many flowers and two fountains.You can adjust the water pressure and set any values r1(r1 ≥ 0) and r2(r2 ≥ 0), giving the distances at which the water is spread from the first and second fountain respectively. You have to set such r1 and r2 that all the flowers are watered, that is, for each flower, the distance between the flower and the first fountain doesn't exceed r1, or the distance to the second fountain doesn't exceed r2. It's OK if some flowers are watered by both fountains.You need to decrease the amount of water you need, that is set such r1 and r2 that all the flowers are watered and the r12 + r22 is minimum possible. Find this minimum value.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"264"Petya is preparing for his birthday. He decided that there would be $$$n$$$ different dishes on the dinner table, numbered from $$$1$$$ to $$$n$$$. Since Petya doesn't like to cook, he wants to order these dishes in restaurants.Unfortunately, all dishes are prepared in different restaurants and therefore Petya needs to pick up his orders from $$$n$$$ different places. To speed up this process, he wants to order courier delivery at some restaurants. Thus, for each dish, there are two options for Petya how he can get it:  the dish will be delivered by a courier from the restaurant $$$i$$$, in this case the courier will arrive in $$$a_i$$$ minutes,  Petya goes to the restaurant $$$i$$$ on his own and picks up the dish, he will spend $$$b_i$$$ minutes on this. Each restaurant has its own couriers and they start delivering the order at the moment Petya leaves the house. In other words, all couriers work in parallel. Petya must visit all restaurants in which he has not chosen delivery, he does this consistently.For example, if Petya wants to order $$$n = 4$$$ dishes and $$$a = [3, 7, 4, 5]$$$, and $$$b = [2, 1, 2, 4]$$$, then he can order delivery from the first and the fourth restaurant, and go to the second and third on your own. Then the courier of the first restaurant will bring the order in $$$3$$$ minutes, the courier of the fourth restaurant will bring the order in $$$5$$$ minutes, and Petya will pick up the remaining dishes in $$$1 + 2 = 3$$$ minutes. Thus, in $$$5$$$ minutes all the dishes will be at Petya's house.Find the minimum time after which all the dishes can be at Petya's home.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"265"You are given two integers $$$l$$$ and $$$r$$$, where $$$l < r$$$. We will add $$$1$$$ to $$$l$$$ until the result is equal to $$$r$$$. Thus, there will be exactly $$$r-l$$$ additions performed. For each such addition, let's look at the number of digits that will be changed after it.For example:   if $$$l=909$$$, then adding one will result in $$$910$$$ and $$$2$$$ digits will be changed;  if you add one to $$$l=9$$$, the result will be $$$10$$$ and $$$2$$$ digits will also be changed;  if you add one to $$$l=489999$$$, the result will be $$$490000$$$ and $$$5$$$ digits will be changed. Changed digits always form a suffix of the result written in the decimal system.Output the total number of changed digits, if you want to get $$$r$$$ from $$$l$$$, adding $$$1$$$ each time.","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"266Little Victor adores the sets theory. Let us remind you that a set is a group of numbers where all numbers are pairwise distinct. Today Victor wants to find a set of integers S that has the following properties:  for all x  the following inequality holds l ≤ x ≤ r;  1 ≤ |S| ≤ k;  lets denote the i-th element of the set S as si; value  must be as small as possible. Help Victor find the described set.,['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"267"You are given an integer $$$n$$$. You have to change the minimum number of digits in it in such a way that the resulting number does not have any leading zeroes and is divisible by $$$7$$$.If there are multiple ways to do it, print any of them. If the given number is already divisible by $$$7$$$, leave it unchanged.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"268"Vasya, like many others, likes to participate in a variety of sweepstakes and lotteries. Now he collects wrappings from a famous chocolate bar ""Jupiter"". According to the sweepstake rules, each wrapping has an integer written on it — the number of points that the participant adds to his score as he buys the bar. After a participant earns a certain number of points, he can come to the prize distribution center and exchange the points for prizes. When somebody takes a prize, the prize's cost is simply subtracted from the number of his points.Vasya didn't only bought the bars, he also kept a record of how many points each wrapping cost. Also, he remembers that he always stucks to the greedy strategy — as soon as he could take at least one prize, he went to the prize distribution centre and exchanged the points for prizes. Moreover, if he could choose between multiple prizes, he chose the most expensive one. If after an exchange Vasya had enough points left to get at least one more prize, then he continued to exchange points.The sweepstake has the following prizes (the prizes are sorted by increasing of their cost):   a mug (costs a points),  a towel (costs b points),  a bag (costs c points),  a bicycle (costs d points),  a car (costs e points). Now Vasya wants to recollect what prizes he has received. You know sequence p1, p2, ..., pn, where pi is the number of points Vasya got for the i-th bar. The sequence of points is given in the chronological order. You also know numbers a, b, c, d, e. Your task is to find, how many prizes Vasya received, what prizes they are and how many points he's got left after all operations are completed.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"269"You are given a sorted array $$$a_1, a_2, \dots, a_n$$$ (for each index $$$i > 1$$$ condition $$$a_i \ge a_{i-1}$$$ holds) and an integer $$$k$$$.You are asked to divide this array into $$$k$$$ non-empty consecutive subarrays. Every element in the array should be included in exactly one subarray. Let $$$max(i)$$$ be equal to the maximum in the $$$i$$$-th subarray, and $$$min(i)$$$ be equal to the minimum in the $$$i$$$-th subarray. The cost of division is equal to $$$\sum\limits_{i=1}^{k} (max(i) - min(i))$$$. For example, if $$$a = [2, 4, 5, 5, 8, 11, 19]$$$ and we divide it into $$$3$$$ subarrays in the following way: $$$[2, 4], [5, 5], [8, 11, 19]$$$, then the cost of division is equal to $$$(4 - 2) + (5 - 5) + (19 - 8) = 13$$$.Calculate the minimum cost you can obtain by dividing the array $$$a$$$ into $$$k$$$ non-empty consecutive subarrays. ",[],"[0, 0, 0, 0, 0, 0, 0, 0]"270"You are given a simple weighted connected undirected graph, consisting of $$$n$$$ vertices and $$$m$$$ edges.A path in the graph of length $$$k$$$ is a sequence of $$$k+1$$$ vertices $$$v_1, v_2, \dots, v_{k+1}$$$ such that for each $$$i$$$ $$$(1 \le i \le k)$$$ the edge $$$(v_i, v_{i+1})$$$ is present in the graph. A path from some vertex $$$v$$$ also has vertex $$$v_1=v$$$. Note that edges and vertices are allowed to be included in the path multiple times.The weight of the path is the total weight of edges in it.For each $$$i$$$ from $$$1$$$ to $$$q$$$ consider a path from vertex $$$1$$$ of length $$$i$$$ of the maximum weight. What is the sum of weights of these $$$q$$$ paths?Answer can be quite large, so print it modulo $$$10^9+7$$$.","['geometry', 'graphs']","[0, 1, 1, 0, 0, 0, 0, 0]"271"A boy named Ayrat lives on planet AMI-1511. Each inhabitant of this planet has a talent. Specifically, Ayrat loves running, moreover, just running is not enough for him. He is dreaming of making running a real art.First, he wants to construct the running track with coating t. On planet AMI-1511 the coating of the track is the sequence of colored blocks, where each block is denoted as the small English letter. Therefore, every coating can be treated as a string.Unfortunately, blocks aren't freely sold to non-business customers, but Ayrat found an infinite number of coatings s. Also, he has scissors and glue. Ayrat is going to buy some coatings s, then cut out from each of them exactly one continuous piece (substring) and glue it to the end of his track coating. Moreover, he may choose to flip this block before glueing it. Ayrat want's to know the minimum number of coating s he needs to buy in order to get the coating t for his running track. Of course, he also want's to know some way to achieve the answer.","['strings', 'trees']","[0, 0, 0, 0, 0, 0, 1, 1]"272"Once Petya read a problem about a bracket sequence. He gave it much thought but didn't find a solution. Today you will face it.You are given string s. It represents a correct bracket sequence. A correct bracket sequence is the sequence of opening (""("") and closing ("")"") brackets, such that it is possible to obtain a correct mathematical expression from it, inserting numbers and operators between the brackets. For example, such sequences as ""(())()"" and ""()"" are correct bracket sequences and such sequences as "")()"" and ""(()"" are not.In a correct bracket sequence each bracket corresponds to the matching bracket (an opening bracket corresponds to the matching closing bracket and vice versa). For example, in a bracket sequence shown of the figure below, the third bracket corresponds to the matching sixth one and the fifth bracket corresponds to the fourth one.   You are allowed to color some brackets in the bracket sequence so as all three conditions are fulfilled:   Each bracket is either not colored any color, or is colored red, or is colored blue.  For any pair of matching brackets exactly one of them is colored. In other words, for any bracket the following is true: either it or the matching bracket that corresponds to it is colored.  No two neighboring colored brackets have the same color. Find the number of different ways to color the bracket sequence. The ways should meet the above-given conditions. Two ways of coloring are considered different if they differ in the color of at least one bracket. As the result can be quite large, print it modulo 1000000007 (109 + 7).",[],"[0, 0, 0, 0, 0, 0, 0, 0]"273"You are given a string $$$s$$$ of lowercase Latin letters. The following operation can be used:   select one character (from 'a' to 'z') that occurs at least once in the string. And replace all such characters in the string with the previous one in alphabetical order on the loop. For example, replace all 'c' with 'b' or replace all 'a' with 'z'. And you are given the integer $$$k$$$ — the maximum number of operations that can be performed. Find the minimum lexicographically possible string that can be obtained by performing no more than $$$k$$$ operations.The string $$$a=a_1a_2 \dots a_n$$$ is lexicographically smaller than the string $$$b = b_1b_2 \dots b_n$$$ if there exists an index $$$k$$$ ($$$1 \le k \le n$$$) such that $$$a_1=b_1$$$, $$$a_2=b_2$$$, ..., $$$a_{k-1}=b_{k-1}$$$, but $$$a_k < b_k$$$.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"274"You have a garland consisting of $$$n$$$ lamps. Each lamp is colored red, green or blue. The color of the $$$i$$$-th lamp is $$$s_i$$$ ('R', 'G' and 'B' — colors of lamps in the garland).You have to recolor some lamps in this garland (recoloring a lamp means changing its initial color to another) in such a way that the obtained garland is diverse.A garland is called diverse if any two adjacent (consecutive) lamps (i. e. such lamps that the distance between their positions is $$$1$$$) have distinct colors.In other words, if the obtained garland is $$$t$$$ then for each $$$i$$$ from $$$1$$$ to $$$n-1$$$ the condition $$$t_i \ne t_{i + 1}$$$ should be satisfied.Among all ways to recolor the initial garland to make it diverse you have to choose one with the minimum number of recolored lamps. If there are multiple optimal solutions, print any of them.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"275"You like numbers, don't you? Nastia has a lot of numbers and she wants to share them with you! Isn't it amazing?Let $$$a_i$$$ be how many numbers $$$i$$$ ($$$1 \le i \le k$$$) you have.An $$$n \times n$$$ matrix is called beautiful if it contains all the numbers you have, and for each $$$2 \times 2$$$ submatrix of the original matrix is satisfied:   The number of occupied cells doesn't exceed $$$3$$$;  The numbers on each diagonal are distinct. Make a beautiful matrix of minimum size.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"276"Vasya and Petya are playing a simple game. Vasya thought of number x between 1 and n, and Petya tries to guess the number.Petya can ask questions like: ""Is the unknown number divisible by number y?"".The game is played by the following rules: first Petya asks all the questions that interest him (also, he can ask no questions), and then Vasya responds to each question with a 'yes' or a 'no'. After receiving all the answers Petya should determine the number that Vasya thought of.Unfortunately, Petya is not familiar with the number theory. Help him find the minimum number of questions he should ask to make a guaranteed guess of Vasya's number, and the numbers yi, he should ask the questions about.","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"277"Our brave travelers reached an island where pirates had buried treasure. However as the ship was about to moor, the captain found out that some rat ate a piece of the treasure map.The treasure map can be represented as a rectangle n × m in size. Each cell stands for an islands' square (the square's side length equals to a mile). Some cells stand for the sea and they are impenetrable. All other cells are penetrable (i.e. available) and some of them contain local sights. For example, the large tree on the hills or the cave in the rocks.Besides, the map also has a set of k instructions. Each instruction is in the following form:""Walk n miles in the y direction""The possible directions are: north, south, east, and west. If you follow these instructions carefully (you should fulfill all of them, one by one) then you should reach exactly the place where treasures are buried. Unfortunately the captain doesn't know the place where to start fulfilling the instructions — as that very piece of the map was lost. But the captain very well remembers that the place contained some local sight. Besides, the captain knows that the whole way goes through the island's penetrable squares.The captain wants to know which sights are worth checking. He asks you to help him with that. ",[],"[0, 0, 0, 0, 0, 0, 0, 0]"278"You have n distinct points on a plane, none of them lie on OY axis. Check that there is a point after removal of which the remaining points are located on one side of the OY axis.",['geometry'],"[0, 1, 0, 0, 0, 0, 0, 0]"279"You are given a set of strings $$$S$$$. Each string consists of lowercase Latin letters.For each string in this set, you want to calculate the minimum number of seconds required to type this string. To type a string, you have to start with an empty string and transform it into the string you want to type using the following actions:  if the current string is $$$t$$$, choose some lowercase Latin letter $$$c$$$ and append it to the back of $$$t$$$, so the current string becomes $$$t + c$$$. This action takes $$$1$$$ second;  use autocompletion. When you try to autocomplete the current string $$$t$$$, a list of all strings $$$s \in S$$$ such that $$$t$$$ is a prefix of $$$s$$$ is shown to you. This list includes $$$t$$$ itself, if $$$t$$$ is a string from $$$S$$$, and the strings are ordered lexicographically. You can transform $$$t$$$ into the $$$i$$$-th string from this list in $$$i$$$ seconds. Note that you may choose any string from this list you want, it is not necessarily the string you are trying to type. What is the minimum number of seconds that you have to spend to type each string from $$$S$$$?Note that the strings from $$$S$$$ are given in an unusual way.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"280"Polycarp has $$$n$$$ different binary words. A word called binary if it contains only characters '0' and '1'. For example, these words are binary: ""0001"", ""11"", ""0"" and ""0011100"".Polycarp wants to offer his set of $$$n$$$ binary words to play a game ""words"". In this game, players name words and each next word (starting from the second) must start with the last character of the previous word. The first word can be any. For example, these sequence of words can be named during the game: ""0101"", ""1"", ""10"", ""00"", ""00001"".Word reversal is the operation of reversing the order of the characters. For example, the word ""0111"" after the reversal becomes ""1110"", the word ""11010"" after the reversal becomes ""01011"".Probably, Polycarp has such a set of words that there is no way to put them in the order correspondent to the game rules. In this situation, he wants to reverse some words from his set so that:  the final set of $$$n$$$ words still contains different words (i.e. all words are unique);  there is a way to put all words of the final set of words in the order so that the final sequence of $$$n$$$ words is consistent with the game rules. Polycarp wants to reverse minimal number of words. Please, help him.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"281"There are two strings s and t, consisting only of letters a and b. You can make the following operation several times: choose a prefix of s, a prefix of t and swap them. Prefixes can be empty, also a prefix can coincide with a whole string. Your task is to find a sequence of operations after which one of the strings consists only of a letters and the other consists only of b letters. The number of operations should be minimized.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"282"Let $$$f_{x} = c^{2x-6} \cdot f_{x-1} \cdot f_{x-2} \cdot f_{x-3}$$$ for $$$x \ge 4$$$.You have given integers $$$n$$$, $$$f_{1}$$$, $$$f_{2}$$$, $$$f_{3}$$$, and $$$c$$$. Find $$$f_{n} \bmod (10^{9}+7)$$$.","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"283"Mike decided to teach programming to children in an elementary school. He knows that it is not an easy task to interest children in that age to code. That is why he decided to give each child two sweets.Mike has $$$n$$$ sweets with sizes $$$a_1, a_2, \ldots, a_n$$$. All his sweets have different sizes. That is, there is no such pair $$$(i, j)$$$ ($$$1 \leq i, j \leq n$$$) such that $$$i \ne j$$$ and $$$a_i = a_j$$$.Since Mike has taught for many years, he knows that if he gives two sweets with sizes $$$a_i$$$ and $$$a_j$$$ to one child and $$$a_k$$$ and $$$a_p$$$ to another, where $$$(a_i + a_j) \neq (a_k + a_p)$$$, then a child who has a smaller sum of sizes will be upset. That is, if there are two children who have different sums of sweets, then one of them will be upset. Apparently, Mike does not want somebody to be upset. Mike wants to invite children giving each of them two sweets. Obviously, he can't give one sweet to two or more children. His goal is to invite as many children as he can. Since Mike is busy preparing to his first lecture in the elementary school, he is asking you to find the maximum number of children he can invite giving each of them two sweets in such way that nobody will be upset.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"284"You are given two arrays: an array $$$a$$$ consisting of $$$n$$$ zeros and an array $$$b$$$ consisting of $$$n$$$ integers.You can apply the following operation to the array $$$a$$$ an arbitrary number of times: choose some subsegment of $$$a$$$ of length $$$k$$$ and add the arithmetic progression $$$1, 2, \ldots, k$$$ to this subsegment — i. e. add $$$1$$$ to the first element of the subsegment, $$$2$$$ to the second element, and so on. The chosen subsegment should be inside the borders of the array $$$a$$$ (i.e., if the left border of the chosen subsegment is $$$l$$$, then the condition $$$1 \le l \le l + k - 1 \le n$$$ should be satisfied). Note that the progression added is always $$$1, 2, \ldots, k$$$ but not the $$$k, k - 1, \ldots, 1$$$ or anything else (i.e., the leftmost element of the subsegment always increases by $$$1$$$, the second element always increases by $$$2$$$ and so on).Your task is to find the minimum possible number of operations required to satisfy the condition $$$a_i \ge b_i$$$ for each $$$i$$$ from $$$1$$$ to $$$n$$$. Note that the condition $$$a_i \ge b_i$$$ should be satisfied for all elements at once.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"285"The Bitlandians are quite weird people. They have very peculiar customs.As is customary, Uncle J. wants to have n eggs painted for Bitruz (an ancient Bitland festival). He has asked G. and A. to do the work.The kids are excited because just as is customary, they're going to be paid for the job! Overall uncle J. has got n eggs. G. named his price for painting each egg. Similarly, A. named his price for painting each egg. It turns out that for each egg the sum of the money both A. and G. want for the painting equals 1000.Uncle J. wants to distribute the eggs between the children so as to give each egg to exactly one child. Also, Uncle J. wants the total money paid to A. to be different from the total money paid to G. by no more than 500.Help Uncle J. Find the required distribution of eggs or otherwise say that distributing the eggs in the required manner is impossible.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"286"You are given $$$n$$$ strings $$$s_1, s_2, \ldots, s_n$$$ consisting of lowercase Latin letters.In one operation you can remove a character from a string $$$s_i$$$ and insert it to an arbitrary position in a string $$$s_j$$$ ($$$j$$$ may be equal to $$$i$$$). You may perform this operation any number of times. Is it possible to make all $$$n$$$ strings equal?",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"287"While Mike was walking in the subway, all the stuff in his back-bag dropped on the ground. There were several fax messages among them. He concatenated these strings in some order and now he has string s.  He is not sure if this is his own back-bag or someone else's. He remembered that there were exactly k messages in his own bag, each was a palindrome string and all those strings had the same length.He asked you to help him and tell him if he has worn his own back-bag. Check if the given string s is a concatenation of k palindromes of the same length.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"288"Vasya has recently developed a new algorithm to optimize the reception of customer flow and he considered the following problem.Let the queue to the cashier contain n people, at that each of them is characterized by a positive integer ai — that is the time needed to work with this customer. What is special about this very cashier is that it can serve two customers simultaneously. However, if two customers need ai and aj of time to be served, the time needed to work with both of them customers is equal to max(ai, aj). Please note that working with customers is an uninterruptable process, and therefore, if two people simultaneously come to the cashier, it means that they begin to be served simultaneously, and will both finish simultaneously (it is possible that one of them will have to wait).Vasya used in his algorithm an ingenious heuristic — as long as the queue has more than one person waiting, then some two people of the first three standing in front of the queue are sent simultaneously. If the queue has only one customer number i, then he goes to the cashier, and is served within ai of time. Note that the total number of phases of serving a customer will always be equal to ⌈n / 2⌉.Vasya thinks that this method will help to cope with the queues we all hate. That's why he asked you to work out a program that will determine the minimum time during which the whole queue will be served using this algorithm.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"289"Okabe and Super Hacker Daru are stacking and removing boxes. There are n boxes numbered from 1 to n. Initially there are no boxes on the stack.Okabe, being a control freak, gives Daru 2n commands: n of which are to add a box to the top of the stack, and n of which are to remove a box from the top of the stack and throw it in the trash. Okabe wants Daru to throw away the boxes in the order from 1 to n. Of course, this means that it might be impossible for Daru to perform some of Okabe's remove commands, because the required box is not on the top of the stack.That's why Daru can decide to wait until Okabe looks away and then reorder the boxes in the stack in any way he wants. He can do it at any point of time between Okabe's commands, but he can't add or remove boxes while he does it.Tell Daru the minimum number of times he needs to reorder the boxes so that he can successfully complete all of Okabe's commands. It is guaranteed that every box is added before it is required to be removed.",['trees'],"[0, 0, 0, 0, 0, 0, 0, 1]"290"Inna likes sweets and a game called the ""Candy Matrix"". Today, she came up with the new game ""Candy Matrix 2: Reload"".The field for the new game is a rectangle table of size n × m. Each line of the table contains one cell with a dwarf figurine, one cell with a candy, the other cells of the line are empty. The game lasts for several moves. During each move the player should choose all lines of the matrix where dwarf is not on the cell with candy and shout ""Let's go!"". After that, all the dwarves from the chosen lines start to simultaneously move to the right. During each second, each dwarf goes to the adjacent cell that is located to the right of its current cell. The movement continues until one of the following events occurs:  some dwarf in one of the chosen lines is located in the rightmost cell of his row;  some dwarf in the chosen lines is located in the cell with the candy. The point of the game is to transport all the dwarves to the candy cells.Inna is fabulous, as she came up with such an interesting game. But what about you? Your task is to play this game optimally well. Specifically, you should say by the given game field what minimum number of moves the player needs to reach the goal of the game.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"291"The robot is placed in the top left corner of a grid, consisting of $$$n$$$ rows and $$$m$$$ columns, in a cell $$$(1, 1)$$$.In one step, it can move into a cell, adjacent by a side to the current one:   $$$(x, y) \rightarrow (x, y + 1)$$$;  $$$(x, y) \rightarrow (x + 1, y)$$$;  $$$(x, y) \rightarrow (x, y - 1)$$$;  $$$(x, y) \rightarrow (x - 1, y)$$$. The robot can't move outside the grid.The cell $$$(s_x, s_y)$$$ contains a deadly laser. If the robot comes into some cell that has distance less than or equal to $$$d$$$ to the laser, it gets evaporated. The distance between two cells $$$(x_1, y_1)$$$ and $$$(x_2, y_2)$$$ is $$$|x_1 - x_2| + |y_1 - y_2|$$$.Print the smallest number of steps that the robot can take to reach the cell $$$(n, m)$$$ without getting evaporated or moving outside the grid. If it's not possible to reach the cell $$$(n, m)$$$, print -1.The laser is neither in the starting cell, nor in the ending cell. The starting cell always has distance greater than $$$d$$$ to the laser.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"292"During a recent research Berland scientists found out that there were n cities in Ancient Berland, joined by two-way paths. Any two cities are joined by no more than one path. No path joins a city with itself. According to a well-known tradition, the road network was built so that it would be impossible to choose three cities from each of which one can get to any other one directly. That is, there was no cycle exactly as long as 3. Unfortunately, the road map has not been preserved till nowadays. Now the scientists are interested how much developed a country Ancient Berland was. Help them - find, what maximal number of roads could be in the country. You also have to restore any of the possible road maps.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"293"A robot cleaner is placed on the floor of a rectangle room, surrounded by walls. The floor consists of $$$n$$$ rows and $$$m$$$ columns. The rows of the floor are numbered from $$$1$$$ to $$$n$$$ from top to bottom, and columns of the floor are numbered from $$$1$$$ to $$$m$$$ from left to right. The cell on the intersection of the $$$r$$$-th row and the $$$c$$$-th column is denoted as $$$(r,c)$$$. The initial position of the robot is $$$(r_b, c_b)$$$.In one second, the robot moves by $$$dr$$$ rows and $$$dc$$$ columns, that is, after one second, the robot moves from the cell $$$(r, c)$$$ to $$$(r + dr, c + dc)$$$. Initially $$$dr = 1$$$, $$$dc = 1$$$. If there is a vertical wall (the left or the right walls) in the movement direction, $$$dc$$$ is reflected before the movement, so the new value of $$$dc$$$ is $$$-dc$$$. And if there is a horizontal wall (the upper or lower walls), $$$dr$$$ is reflected before the movement, so the new value of $$$dr$$$ is $$$-dr$$$.Each second (including the moment before the robot starts moving), the robot cleans every cell lying in the same row or the same column as its position. There is only one dirty cell at $$$(r_d, c_d)$$$. The job of the robot is to clean that dirty cell.  Illustration for the first example. The blue arc is the robot. The red star is the target dirty cell. Each second the robot cleans a row and a column, denoted by yellow stripes. Given the floor size $$$n$$$ and $$$m$$$, the robot's initial position $$$(r_b, c_b)$$$ and the dirty cell's position $$$(r_d, c_d)$$$, find the time for the robot to do its job.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"294"Today the kindergarten has a new group of $$$n$$$ kids who need to be seated at the dinner table. The chairs at the table are numbered from $$$1$$$ to $$$4n$$$. Two kids can't sit on the same chair. It is known that two kids who sit on chairs with numbers $$$a$$$ and $$$b$$$ ($$$a \neq b$$$) will indulge if:   $$$gcd(a, b) = 1$$$ or,  $$$a$$$ divides $$$b$$$ or $$$b$$$ divides $$$a$$$. $$$gcd(a, b)$$$ — the maximum number $$$x$$$ such that $$$a$$$ is divisible by $$$x$$$ and $$$b$$$ is divisible by $$$x$$$.For example, if $$$n=3$$$ and the kids sit on chairs with numbers $$$2$$$, $$$3$$$, $$$4$$$, then they will indulge since $$$4$$$ is divided by $$$2$$$ and $$$gcd(2, 3) = 1$$$. If kids sit on chairs with numbers $$$4$$$, $$$6$$$, $$$10$$$, then they will not indulge.The teacher really doesn't want the mess at the table, so she wants to seat the kids so there are no $$$2$$$ of the kid that can indulge. More formally, she wants no pair of chairs $$$a$$$ and $$$b$$$ that the kids occupy to fulfill the condition above.Since the teacher is very busy with the entertainment of the kids, she asked you to solve this problem.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"295"Alice and Bob are playing a game on a matrix, consisting of $$$2$$$ rows and $$$m$$$ columns. The cell in the $$$i$$$-th row in the $$$j$$$-th column contains $$$a_{i, j}$$$ coins in it.Initially, both Alice and Bob are standing in a cell $$$(1, 1)$$$. They are going to perform a sequence of moves to reach a cell $$$(2, m)$$$.The possible moves are:   Move right — from some cell $$$(x, y)$$$ to $$$(x, y + 1)$$$;  Move down — from some cell $$$(x, y)$$$ to $$$(x + 1, y)$$$. First, Alice makes all her moves until she reaches $$$(2, m)$$$. She collects the coins in all cells she visit (including the starting cell).When Alice finishes, Bob starts his journey. He also performs the moves to reach $$$(2, m)$$$ and collects the coins in all cells that he visited, but Alice didn't.The score of the game is the total number of coins Bob collects.Alice wants to minimize the score. Bob wants to maximize the score. What will the score of the game be if both players play optimally?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"296"A permutation of size n is an array of size n such that each integer from 1 to n occurs exactly once in this array. An inversion in a permutation p is a pair of indices (i, j) such that i > j and ai < aj. For example, a permutation [4, 1, 3, 2] contains 4 inversions: (2, 1), (3, 1), (4, 1), (4, 3).You are given a permutation a of size n and m queries to it. Each query is represented by two indices l and r denoting that you have to reverse the segment [l, r] of the permutation. For example, if a = [1, 2, 3, 4] and a query l = 2, r = 4 is applied, then the resulting permutation is [1, 4, 3, 2].After each query you have to determine whether the number of inversions is odd or even.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"297You are given a graph consisting of $$$n$$$ vertices and $$$m$$$ edges. It is not guaranteed that the given graph is connected. Some edges are already directed and you can't change their direction. Other edges are undirected and you have to choose some direction for all these edges.You have to direct undirected edges in such a way that the resulting graph is directed and acyclic (i.e. the graph with all edges directed and having no directed cycles). Note that you have to direct all undirected edges.You have to answer $$$t$$$ independent test cases.,['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"298"The new generation external memory contains an array of integers $$$a[1 \ldots n] = [a_1, a_2, \ldots, a_n]$$$.This type of memory does not support changing the value of an arbitrary element. Instead, it allows you to cut out any segment of the given array, cyclically shift (rotate) it by any offset and insert it back into the same place.Technically, each cyclic shift consists of two consecutive actions:   You may select arbitrary indices $$$l$$$ and $$$r$$$ ($$$1 \le l < r \le n$$$) as the boundaries of the segment.  Then you replace the segment $$$a[l \ldots r]$$$ with it's cyclic shift to the left by an arbitrary offset $$$d$$$. The concept of a cyclic shift can be also explained by following relations: the sequence $$$[1, 4, 1, 3]$$$ is a cyclic shift of the sequence $$$[3, 1, 4, 1]$$$ to the left by the offset $$$1$$$ and the sequence $$$[4, 1, 3, 1]$$$ is a cyclic shift of the sequence $$$[3, 1, 4, 1]$$$ to the left by the offset $$$2$$$. For example, if $$$a = [1, \color{blue}{3, 2, 8}, 5]$$$, then choosing $$$l = 2$$$, $$$r = 4$$$ and $$$d = 2$$$ yields a segment $$$a[2 \ldots 4] = [3, 2, 8]$$$. This segment is then shifted by the offset $$$d = 2$$$ to the left, and you get a segment $$$[8, 3, 2]$$$ which then takes the place of of the original elements of the segment. In the end you get $$$a = [1, \color{blue}{8, 3, 2}, 5]$$$.Sort the given array $$$a$$$ using no more than $$$n$$$ cyclic shifts of any of its segments. Note that you don't need to minimize the number of cyclic shifts. Any method that requires $$$n$$$ or less cyclic shifts will be accepted.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"299"Alice and Bob play ping-pong with simplified rules.During the game, the player serving the ball commences a play. The server strikes the ball then the receiver makes a return by hitting the ball back. Thereafter, the server and receiver must alternately make a return until one of them doesn't make a return.The one who doesn't make a return loses this play. The winner of the play commences the next play. Alice starts the first play.Alice has $$$x$$$ stamina and Bob has $$$y$$$. To hit the ball (while serving or returning) each player spends $$$1$$$ stamina, so if they don't have any stamina, they can't return the ball (and lose the play) or can't serve the ball (in this case, the other player serves the ball instead). If both players run out of stamina, the game is over.Sometimes, it's strategically optimal not to return the ball, lose the current play, but save the stamina. On the contrary, when the server commences a play, they have to hit the ball, if they have some stamina left.Both Alice and Bob play optimally and want to, firstly, maximize their number of wins and, secondly, minimize the number of wins of their opponent.Calculate the resulting number of Alice's and Bob's wins.","['games', 'math']","[1, 0, 0, 1, 0, 0, 0, 0]"300"As you know, every birthday party has a cake! This time, Babaei is going to prepare the very special birthday party's cake.Simple cake is a cylinder of some radius and height. The volume of the simple cake is equal to the volume of corresponding cylinder. Babaei has n simple cakes and he is going to make a special cake placing some cylinders on each other.However, there are some additional culinary restrictions. The cakes are numbered in such a way that the cake number i can be placed only on the table or on some cake number j where j < i. Moreover, in order to impress friends Babaei will put the cake i on top of the cake j only if the volume of the cake i is strictly greater than the volume of the cake j.Babaei wants to prepare a birthday cake that has a maximum possible total volume. Help him find this value.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"301"Tanya has $$$n$$$ candies numbered from $$$1$$$ to $$$n$$$. The $$$i$$$-th candy has the weight $$$a_i$$$.She plans to eat exactly $$$n-1$$$ candies and give the remaining candy to her dad. Tanya eats candies in order of increasing their numbers, exactly one candy per day.Your task is to find the number of such candies $$$i$$$ (let's call these candies good) that if dad gets the $$$i$$$-th candy then the sum of weights of candies Tanya eats in even days will be equal to the sum of weights of candies Tanya eats in odd days. Note that at first, she will give the candy, after it she will eat the remaining candies one by one.For example, $$$n=4$$$ and weights are $$$[1, 4, 3, 3]$$$. Consider all possible cases to give a candy to dad:  Tanya gives the $$$1$$$-st candy to dad ($$$a_1=1$$$), the remaining candies are $$$[4, 3, 3]$$$. She will eat $$$a_2=4$$$ in the first day, $$$a_3=3$$$ in the second day, $$$a_4=3$$$ in the third day. So in odd days she will eat $$$4+3=7$$$ and in even days she will eat $$$3$$$. Since $$$7 \ne 3$$$ this case shouldn't be counted to the answer (this candy isn't good).  Tanya gives the $$$2$$$-nd candy to dad ($$$a_2=4$$$), the remaining candies are $$$[1, 3, 3]$$$. She will eat $$$a_1=1$$$ in the first day, $$$a_3=3$$$ in the second day, $$$a_4=3$$$ in the third day. So in odd days she will eat $$$1+3=4$$$ and in even days she will eat $$$3$$$. Since $$$4 \ne 3$$$ this case shouldn't be counted to the answer (this candy isn't good).  Tanya gives the $$$3$$$-rd candy to dad ($$$a_3=3$$$), the remaining candies are $$$[1, 4, 3]$$$. She will eat $$$a_1=1$$$ in the first day, $$$a_2=4$$$ in the second day, $$$a_4=3$$$ in the third day. So in odd days she will eat $$$1+3=4$$$ and in even days she will eat $$$4$$$. Since $$$4 = 4$$$ this case should be counted to the answer (this candy is good).  Tanya gives the $$$4$$$-th candy to dad ($$$a_4=3$$$), the remaining candies are $$$[1, 4, 3]$$$. She will eat $$$a_1=1$$$ in the first day, $$$a_2=4$$$ in the second day, $$$a_3=3$$$ in the third day. So in odd days she will eat $$$1+3=4$$$ and in even days she will eat $$$4$$$. Since $$$4 = 4$$$ this case should be counted to the answer (this candy is good). In total there $$$2$$$ cases which should counted (these candies are good), so the answer is $$$2$$$.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"302"You are going to the beach with the idea to build the greatest sand castle ever in your head! The beach is not as three-dimensional as you could have imagined, it can be decribed as a line of spots to pile up sand pillars. Spots are numbered 1 through infinity from left to right. Obviously, there is not enough sand on the beach, so you brought n packs of sand with you. Let height hi of the sand pillar on some spot i be the number of sand packs you spent on it. You can't split a sand pack to multiple pillars, all the sand from it should go to a single one. There is a fence of height equal to the height of pillar with H sand packs to the left of the first spot and you should prevent sand from going over it. Finally you ended up with the following conditions to building the castle:  h1 ≤ H: no sand from the leftmost spot should go over the fence;  For any  |hi - hi + 1| ≤ 1: large difference in heights of two neighboring pillars can lead sand to fall down from the higher one to the lower, you really don't want this to happen;  : you want to spend all the sand you brought with you. As you have infinite spots to build, it is always possible to come up with some valid castle structure. Though you want the castle to be as compact as possible. Your task is to calculate the minimum number of spots you can occupy so that all the aforementioned conditions hold.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"303"Gerald has a friend, Pollard. Pollard is interested in lucky tickets (ticket is a sequence of digits). At first he thought that a ticket is lucky if between some its digits we can add arithmetic signs and brackets so that the result obtained by the arithmetic expression was number 100. But he quickly analyzed all such tickets and moved on to a more general question. Now he explores k-lucky tickets.Pollard sais that a ticket is k-lucky if we can add arithmetic operation signs between its digits to the left or right of them (i.e., ""+"", ""-"", "" × "") and brackets so as to obtain the correct arithmetic expression whose value would equal k. For example, ticket ""224201016"" is 1000-lucky as ( - 2 - (2 + 4)) × (2 + 0) + 1016 = 1000.Pollard was so carried away by the lucky tickets that he signed up for a seminar on lucky tickets and, as far as Gerald knows, Pollard will attend it daily at 7 pm in some famous institute and will commute to it in the same tram for m days. In this tram tickets have eight digits. And Gerald wants to make a surprise for Pollard: each day Pollard will receive a tram k-lucky ticket. The conductor has already agreed to give Pollard certain tickets during all these m days and he only wants Gerald to tell him what kind of tickets to give out. In this regard, help Gerald pick exactly m distinct k-lucky tickets.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"304"A bracket sequence is a string containing only characters ""("" and "")"". A regular bracket sequence is a bracket sequence that can be transformed into a correct arithmetic expression by inserting characters ""1"" and ""+"" between the original characters of the sequence. For example, bracket sequences ""()()"" and ""(())"" are regular (the resulting expressions are: ""(1)+(1)"" and ""((1+1)+1)""), and "")("", ""("" and "")"" are not.You are given an integer $$$n$$$. Your goal is to construct and print exactly $$$n$$$ different regular bracket sequences of length $$$2n$$$.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"305"Ivan wants to have a good dinner. A good dinner should consist of a first course, a second course, a drink, and a dessert.There are $$$n_1$$$ different types of first courses Ivan can buy (the $$$i$$$-th of them costs $$$a_i$$$ coins), $$$n_2$$$ different types of second courses (the $$$i$$$-th of them costs $$$b_i$$$ coins), $$$n_3$$$ different types of drinks (the $$$i$$$-th of them costs $$$c_i$$$ coins) and $$$n_4$$$ different types of desserts (the $$$i$$$-th of them costs $$$d_i$$$ coins).Some dishes don't go well with each other. There are $$$m_1$$$ pairs of first courses and second courses that don't go well with each other, $$$m_2$$$ pairs of second courses and drinks, and $$$m_3$$$ pairs of drinks and desserts that don't go well with each other.Ivan wants to buy exactly one first course, one second course, one drink, and one dessert so that they go well with each other, and the total cost of the dinner is the minimum possible. Help him to find the cheapest dinner option!",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"306"The average miner Vaganych took refresher courses. As soon as a miner completes the courses, he should take exams. The hardest one is a computer test called ""Testing Pants for Sadness"".The test consists of n questions; the questions are to be answered strictly in the order in which they are given, from question 1 to question n. Question i contains ai answer variants, exactly one of them is correct. A click is regarded as selecting any answer in any question. The goal is to select the correct answer for each of the n questions. If Vaganych selects a wrong answer for some question, then all selected answers become unselected and the test starts from the very beginning, from question 1 again. But Vaganych remembers everything. The order of answers for each question and the order of questions remain unchanged, as well as the question and answers themselves.Vaganych is very smart and his memory is superb, yet he is unbelievably unlucky and knows nothing whatsoever about the test's theme. How many clicks will he have to perform in the worst case?",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"307"Vasya came up with a password to register for EatForces — a string $$$s$$$. The password in EatForces should be a string, consisting of lowercase and uppercase Latin letters and digits.But since EatForces takes care of the security of its users, user passwords must contain at least one digit, at least one uppercase Latin letter and at least one lowercase Latin letter. For example, the passwords ""abaCABA12"", ""Z7q"" and ""3R24m"" are valid, and the passwords ""qwerty"", ""qwerty12345"" and ""Password"" are not. A substring of string $$$s$$$ is a string $$$x = s_l s_{l + 1} \dots s_{l + len - 1} (1 \le l \le |s|, 0 \le len \le |s| - l + 1)$$$. $$$len$$$ is the length of the substring. Note that the empty string is also considered a substring of $$$s$$$, it has the length $$$0$$$.Vasya's password, however, may come too weak for the security settings of EatForces. He likes his password, so he wants to replace some its substring with another string of the same length in order to satisfy the above conditions. This operation should be performed exactly once, and the chosen string should have the minimal possible length.Note that the length of $$$s$$$ should not change after the replacement of the substring, and the string itself should contain only lowercase and uppercase Latin letters and digits.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"308"Now Serval is a junior high school student in Japari Middle School, and he is still thrilled on math as before. As a talented boy in mathematics, he likes to play with numbers. This time, he wants to play with numbers on a rooted tree.A tree is a connected graph without cycles. A rooted tree has a special vertex called the root. A parent of a node $$$v$$$ is the last different from $$$v$$$ vertex on the path from the root to the vertex $$$v$$$. Children of vertex $$$v$$$ are all nodes for which $$$v$$$ is the parent. A vertex is a leaf if it has no children.The rooted tree Serval owns has $$$n$$$ nodes, node $$$1$$$ is the root. Serval will write some numbers into all nodes of the tree. However, there are some restrictions. Each of the nodes except leaves has an operation $$$\max$$$ or $$$\min$$$ written in it, indicating that the number in this node should be equal to the maximum or minimum of all the numbers in its sons, respectively. Assume that there are $$$k$$$ leaves in the tree. Serval wants to put integers $$$1, 2, \ldots, k$$$ to the $$$k$$$ leaves (each number should be used exactly once). He loves large numbers, so he wants to maximize the number in the root. As his best friend, can you help him?",['trees'],"[0, 0, 0, 0, 0, 0, 0, 1]"309"Demiurges Shambambukli and Mazukta love to watch the games of ordinary people. Today, they noticed two men who play the following game.There is a rooted tree on n nodes, m of which are leaves (a leaf is a nodes that does not have any children), edges of the tree are directed from parent to children. In the leaves of the tree integers from 1 to m are placed in such a way that each number appears exactly in one leaf.Initially, the root of the tree contains a piece. Two players move this piece in turns, during a move a player moves the piece from its current nodes to one of its children; if the player can not make a move, the game ends immediately. The result of the game is the number placed in the leaf where a piece has completed its movement. The player who makes the first move tries to maximize the result of the game and the second player, on the contrary, tries to minimize the result. We can assume that both players move optimally well.Demiurges are omnipotent, so before the game they can arbitrarily rearrange the numbers placed in the leaves. Shambambukli wants to rearrange numbers so that the result of the game when both players play optimally well is as large as possible, and Mazukta wants the result to be as small as possible. What will be the outcome of the game, if the numbers are rearranged by Shambambukli, and what will it be if the numbers are rearranged by Mazukta? Of course, the Demiurges choose the best possible option of arranging numbers.","['math', 'trees']","[0, 0, 0, 1, 0, 0, 0, 1]"310"Calculate the value of the sum: n mod 1 + n mod 2 + n mod 3 + ... + n mod m. As the result can be very large, you should print the value modulo 109 + 7 (the remainder when divided by 109 + 7).The modulo operator a mod b stands for the remainder after dividing a by b. For example 10 mod 3 = 1.","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"311"We say that a positive integer $$$n$$$ is $$$k$$$-good for some positive integer $$$k$$$ if $$$n$$$ can be expressed as a sum of $$$k$$$ positive integers which give $$$k$$$ distinct remainders when divided by $$$k$$$.Given a positive integer $$$n$$$, find some $$$k \geq 2$$$ so that $$$n$$$ is $$$k$$$-good or tell that such a $$$k$$$ does not exist.","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"312"As you know, Bob's brother lives in Flatland. In Flatland there are n cities, connected by n - 1 two-way roads. The cities are numbered from 1 to n. You can get from one city to another moving along the roads.The «Two Paths» company, where Bob's brother works, has won a tender to repair two paths in Flatland. A path is a sequence of different cities, connected sequentially by roads. The company is allowed to choose by itself the paths to repair. The only condition they have to meet is that the two paths shouldn't cross (i.e. shouldn't have common cities).It is known that the profit, the «Two Paths» company will get, equals the product of the lengths of the two paths. Let's consider the length of each road equals 1, and the length of a path equals the amount of roads in it. Find the maximum possible profit for the company.","['graphs', 'trees']","[0, 0, 1, 0, 0, 0, 0, 1]"313"After watching a certain anime before going to sleep, Mark dreams of standing in an old classroom with a blackboard that has a sequence of $$$n$$$ positive integers $$$a_1, a_2,\dots,a_n$$$ on it.Then, professor Koro comes in. He can perform the following operation:  select an integer $$$x$$$ that appears at least $$$2$$$ times on the board,  erase those $$$2$$$ appearances, and  write $$$x+1$$$ on the board. Professor Koro then asks Mark the question, ""what is the maximum possible number that could appear on the board after some operations?""Mark quickly solves this question, but he is still slower than professor Koro. Thus, professor Koro decides to give Mark additional challenges. He will update the initial sequence of integers $$$q$$$ times. Each time, he will choose positive integers $$$k$$$ and $$$l$$$, then change $$$a_k$$$ to $$$l$$$. After each update, he will ask Mark the same question again.Help Mark answer these questions faster than Professor Koro!Note that the updates are persistent. Changes made to the sequence $$$a$$$ will apply when processing future updates.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"314"You are given a rooted tree consisting of $$$n$$$ vertices. The vertices are numbered from $$$1$$$ to $$$n$$$, and the root is the vertex $$$1$$$. You are also given a score array $$$s_1, s_2, \ldots, s_n$$$.A multiset of $$$k$$$ simple paths is called valid if the following two conditions are both true.   Each path starts from $$$1$$$.  Let $$$c_i$$$ be the number of paths covering vertex $$$i$$$. For each pair of vertices $$$(u,v)$$$ ($$$2\le u,v\le n$$$) that have the same parent, $$$|c_u-c_v|\le 1$$$ holds.  The value of the path multiset is defined as $$$\sum\limits_{i=1}^n c_i s_i$$$.It can be shown that it is always possible to find at least one valid multiset. Find the maximum value among all valid multisets.",['trees'],"[0, 0, 0, 0, 0, 0, 0, 1]"315"The only difference between the two versions is that in this version $$$n \leq 2 \cdot 10^5$$$ and the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.A terminal is a row of $$$n$$$ equal segments numbered $$$1$$$ to $$$n$$$ in order. There are two terminals, one above the other. You are given an array $$$a$$$ of length $$$n$$$. For all $$$i = 1, 2, \dots, n$$$, there should be a straight wire from some point on segment $$$i$$$ of the top terminal to some point on segment $$$a_i$$$ of the bottom terminal. You can't select the endpoints of a segment. For example, the following pictures show two possible wirings if $$$n=7$$$ and $$$a=[4,1,4,6,7,7,5]$$$.  A crossing occurs when two wires share a point in common. In the picture above, crossings are circled in red.What is the maximum number of crossings there can be if you place the wires optimally?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"316"Iahub and Sorin are the best competitive programmers in their town. However, they can't both qualify to an important contest. The selection will be made with the help of a single problem. Blatnatalag, a friend of Iahub, managed to get hold of the problem before the contest. Because he wants to make sure Iahub will be the one qualified, he tells Iahub the following task.You're given an (1-based) array a with n elements. Let's define function f(i, j) (1 ≤ i, j ≤ n) as (i - j)2 + g(i, j)2. Function g is calculated by the following pseudo-code:int g(int i, int j) {    int sum = 0;    for (int k = min(i, j) + 1; k <= max(i, j); k = k + 1)        sum = sum + a[k];    return sum;}Find a value mini ≠ j  f(i, j).Probably by now Iahub already figured out the solution to this problem. Can you?",['geometry'],"[0, 1, 0, 0, 0, 0, 0, 0]"317"Your friend is developing a computer game. He has already decided how the game world should look like — it should consist of $$$n$$$ locations connected by $$$m$$$ two-way passages. The passages are designed in such a way that it should be possible to get from any location to any other location.Of course, some passages should be guarded by the monsters (if you just can go everywhere without any difficulties, then it's not fun, right?). Some crucial passages will be guarded by really fearsome monsters, requiring the hero to prepare for battle and designing his own tactics of defeating them (commonly these kinds of monsters are called bosses). And your friend wants you to help him place these bosses.The game will start in location $$$s$$$ and end in location $$$t$$$, but these locations are not chosen yet. After choosing these locations, your friend will place a boss in each passage such that it is impossible to get from $$$s$$$ to $$$t$$$ without using this passage. Your friend wants to place as much bosses as possible (because more challenges means more fun, right?), so he asks you to help him determine the maximum possible number of bosses, considering that any location can be chosen as $$$s$$$ or as $$$t$$$.","['graphs', 'trees']","[0, 0, 1, 0, 0, 0, 0, 1]"318"In a far away kingdom young pages help to set the table for the King. As they are terribly mischievous, one needs to keep an eye on the control whether they have set everything correctly. This time the royal chef Gerasim had the impression that the pages have played a prank again: they had poured the juice from one cup to another. Now Gerasim wants to check his hypothesis. The good thing is that chef Gerasim always pour the same number of milliliters of juice to all cups in the royal kitchen. Having thoroughly measured the juice in each cup, Gerasim asked you to write a program that will determine from which cup juice was poured to which one; otherwise, the program should determine that this time the pages set the table diligently.To simplify your task we shall consider the cups to be bottomless so that the juice never overfills a cup and pours out, however much it can be. Besides, by some strange reason in a far away kingdom one can only pour to a cup or from one cup to another an integer number of milliliters of juice.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"319"You have an array $$$a_1, a_2, \dots, a_n$$$. Let's call some subarray $$$a_l, a_{l + 1}, \dots , a_r$$$ of this array a subpermutation if it contains all integers from $$$1$$$ to $$$r-l+1$$$ exactly once. For example, array $$$a = [2, 2, 1, 3, 2, 3, 1]$$$ contains $$$6$$$ subarrays which are subpermutations: $$$[a_2 \dots a_3]$$$, $$$[a_2 \dots a_4]$$$, $$$[a_3 \dots a_3]$$$, $$$[a_3 \dots a_5]$$$, $$$[a_5 \dots a_7]$$$, $$$[a_7 \dots a_7]$$$.You are asked to calculate the number of subpermutations.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"320"Berland, 2016. The exchange rate of currency you all know against the burle has increased so much that to simplify the calculations, its fractional part was neglected and the exchange rate is now assumed to be an integer.Reliable sources have informed the financier Anton of some information about the exchange rate of currency you all know against the burle for tomorrow. Now Anton knows that tomorrow the exchange rate will be an even number, which can be obtained from the present rate by swapping exactly two distinct digits in it. Of all the possible values that meet these conditions, the exchange rate for tomorrow will be the maximum possible. It is guaranteed that today the exchange rate is an odd positive integer n. Help Anton to determine the exchange rate of currency you all know for tomorrow!","['math', 'strings']","[0, 0, 0, 1, 0, 0, 1, 0]"321"Polycarp found under the Christmas tree an array $$$a$$$ of $$$n$$$ elements and instructions for playing with it:   At first, choose index $$$i$$$ ($$$1 \leq i \leq n$$$) — starting position in the array. Put the chip at the index $$$i$$$ (on the value $$$a_i$$$).  While $$$i \leq n$$$, add $$$a_i$$$ to your score and move the chip $$$a_i$$$ positions to the right (i.e. replace $$$i$$$ with $$$i + a_i$$$).  If $$$i > n$$$, then Polycarp ends the game. For example, if $$$n = 5$$$ and $$$a = [7, 3, 1, 2, 3]$$$, then the following game options are possible:   Polycarp chooses $$$i = 1$$$. Game process: $$$i = 1 \overset{+7}{\longrightarrow} 8$$$. The score of the game is: $$$a_1 = 7$$$.  Polycarp chooses $$$i = 2$$$. Game process: $$$i = 2 \overset{+3}{\longrightarrow} 5 \overset{+3}{\longrightarrow} 8$$$. The score of the game is: $$$a_2 + a_5 = 6$$$.  Polycarp chooses $$$i = 3$$$. Game process: $$$i = 3 \overset{+1}{\longrightarrow} 4 \overset{+2}{\longrightarrow} 6$$$. The score of the game is: $$$a_3 + a_4 = 3$$$.  Polycarp chooses $$$i = 4$$$. Game process: $$$i = 4 \overset{+2}{\longrightarrow} 6$$$. The score of the game is: $$$a_4 = 2$$$.  Polycarp chooses $$$i = 5$$$. Game process: $$$i = 5 \overset{+3}{\longrightarrow} 8$$$. The score of the game is: $$$a_5 = 3$$$. Help Polycarp to find out the maximum score he can get if he chooses the starting index in an optimal way.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"322"Nauuo is a girl who loves playing games related to portals.One day she was playing a game as follows.In an $$$n\times n$$$ grid, the rows are numbered from $$$1$$$ to $$$n$$$ from top to bottom, the columns are numbered from $$$1$$$ to $$$n$$$ from left to right. We denote a cell on the intersection of the $$$r$$$-th row and $$$c$$$-th column as $$$(r,c)$$$.A portal is a pair of doors. You can travel from one of them to another without changing your direction. More formally, if you walk into a cell with a door, you will teleport to the cell with the other door of the same portal and then walk into the next cell facing the original direction. There can not be more than one doors in a single cell.The ""next cell"" is the nearest cell in the direction you are facing. For example, if you are facing bottom, the next cell of $$$(2,5)$$$ is $$$(3,5)$$$.If you walk into a cell without a door, you must walk into the next cell after that without changing the direction. If the next cell does not exist, you must exit the grid.You have to set some (possibly zero) portals in the grid, so that if you walk into $$$(i,1)$$$ facing right, you will eventually exit the grid from $$$(r_i,n)$$$, if you walk into $$$(1, i)$$$ facing bottom, you will exit the grid from $$$(n,c_i)$$$.It is guaranteed that both $$$r_{1..n}$$$ and $$$c_{1..n}$$$ are permutations of $$$n$$$ elements. A permutation of $$$n$$$ elements is a sequence of numbers $$$p_1,p_2,\ldots,p_n$$$ in which every integer from $$$1$$$ to $$$n$$$ appears exactly once.She got confused while playing the game, can you help her to find a solution?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"323"Johnny's younger sister Megan had a birthday recently. Her brother has bought her a box signed as ""Your beautiful necklace — do it yourself!"". It contains many necklace parts and some magic glue. The necklace part is a chain connecting two pearls. Color of each pearl can be defined by a non-negative integer. The magic glue allows Megan to merge two pearls (possibly from the same necklace part) into one. The beauty of a connection of pearls in colors $$$u$$$ and $$$v$$$ is defined as follows: let $$$2^k$$$ be the greatest power of two dividing $$$u \oplus v$$$ — exclusive or of $$$u$$$ and $$$v$$$. Then the beauty equals $$$k$$$. If $$$u = v$$$, you may assume that beauty is equal to $$$20$$$.Each pearl can be combined with another at most once. Merging two parts of a necklace connects them. Using the glue multiple times, Megan can finally build the necklace, which is a cycle made from connected necklace parts (so every pearl in the necklace is combined with precisely one other pearl in it). The beauty of such a necklace is the minimum beauty of a single connection in it. The girl wants to use all available necklace parts to build exactly one necklace consisting of all of them with the largest possible beauty. Help her!",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"324"Tokitsukaze has two colorful tapes. There are $$$n$$$ distinct colors, numbered $$$1$$$ through $$$n$$$, and each color appears exactly once on each of the two tapes. Denote the color of the $$$i$$$-th position of the first tape as $$$ca_i$$$, and the color of the $$$i$$$-th position of the second tape as $$$cb_i$$$.Now Tokitsukaze wants to select each color an integer value from $$$1$$$ to $$$n$$$, distinct for all the colors. After that she will put down the color values in each colored position on the tapes. Denote the number of the $$$i$$$-th position of the first tape as $$$numa_i$$$, and the number of the $$$i$$$-th position of the second tape as $$$numb_i$$$.  For example, for the above picture, assuming that the color red has value $$$x$$$ ($$$1 \leq x \leq n$$$), it appears at the $$$1$$$-st position of the first tape and the $$$3$$$-rd position of the second tape, so $$$numa_1=numb_3=x$$$.Note that each color $$$i$$$ from $$$1$$$ to $$$n$$$ should have a distinct value, and the same color which appears in both tapes has the same value. After labeling each color, the beauty of the two tapes is calculated as $$$$$$\sum_{i=1}^{n}|numa_i-numb_i|.$$$$$$Please help Tokitsukaze to find the highest possible beauty. ",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"325"You are given two integers $$$n$$$ and $$$m$$$. You have to construct the array $$$a$$$ of length $$$n$$$ consisting of non-negative integers (i.e. integers greater than or equal to zero) such that the sum of elements of this array is exactly $$$m$$$ and the value $$$\sum\limits_{i=1}^{n-1} |a_i - a_{i+1}|$$$ is the maximum possible. Recall that $$$|x|$$$ is the absolute value of $$$x$$$.In other words, you have to maximize the sum of absolute differences between adjacent (consecutive) elements. For example, if the array $$$a=[1, 3, 2, 5, 5, 0]$$$ then the value above for this array is $$$|1-3| + |3-2| + |2-5| + |5-5| + |5-0| = 2 + 1 + 3 + 0 + 5 = 11$$$. Note that this example doesn't show the optimal answer but it shows how the required value for some array is calculated.You have to answer $$$t$$$ independent test cases.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"326"After observing the results of Spy Syndrome, Yash realised the errors of his ways. He now believes that a super spy such as Siddhant can't use a cipher as basic and ancient as Caesar cipher. After many weeks of observation of Siddhant’s sentences, Yash determined a new cipher technique.For a given sentence, the cipher is processed as:   Convert all letters of the sentence to lowercase.  Reverse each of the words of the sentence individually.  Remove all the spaces in the sentence. For example, when this cipher is applied to the sentenceKira is childish and he hates losingthe resulting string isariksihsidlihcdnaehsetahgnisolNow Yash is given some ciphered string and a list of words. Help him to find out any original sentence composed using only words from the list. Note, that any of the given words could be used in the sentence multiple times.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"327"During the programming classes Vasya was assigned a difficult problem. However, he doesn't know how to code and was unable to find the solution in the Internet, so he asks you to help.You are given a sequence $$$a$$$, consisting of $$$n$$$ distinct integers, that is used to construct the binary search tree. Below is the formal description of the construction process.  First element $$$a_1$$$ becomes the root of the tree.  Elements $$$a_2, a_3, \ldots, a_n$$$ are added one by one. To add element $$$a_i$$$ one needs to traverse the tree starting from the root and using the following rules:   The pointer to the current node is set to the root.  If $$$a_i$$$ is greater than the value in the current node, then its right child becomes the current node. Otherwise, the left child of the current node becomes the new current node.  If at some point there is no required child, the new node is created, it is assigned value $$$a_i$$$ and becomes the corresponding child of the current node.  ",['trees'],"[0, 0, 0, 0, 0, 0, 0, 1]"328You are given N points on a plane. Write a program which will find the sum of squares of distances between all pairs of points.,['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"329"Today on a lecture about strings Gerald learned a new definition of string equivalency. Two strings a and b of equal length are called equivalent in one of the two cases:   They are equal.  If we split string a into two halves of the same size a1 and a2, and string b into two halves of the same size b1 and b2, then one of the following is correct:   a1 is equivalent to b1, and a2 is equivalent to b2  a1 is equivalent to b2, and a2 is equivalent to b1  As a home task, the teacher gave two strings to his students and asked to determine if they are equivalent.Gerald has already completed this home task. Now it's your turn!",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"330"Kilani and Abd are neighbors for 3000 years, but then the day came and Kilani decided to move to another house. As a farewell gift, Kilani is going to challenge Abd with a problem written by their other neighbor with the same name Abd.  The problem is:You are given a connected tree rooted at node $$$1$$$.You should assign a character a or b to every node in the tree so that the total number of a's is equal to $$$x$$$ and the total number of b's is equal to $$$n - x$$$.Let's define a string for each node $$$v$$$ of the tree as follows:   if $$$v$$$ is root then the string is just one character assigned to $$$v$$$:  otherwise, let's take a string defined for the $$$v$$$'s parent $$$p_v$$$ and add to the end of it a character assigned to $$$v$$$. You should assign every node a character in a way that minimizes the number of distinct strings among the strings of all nodes.",['trees'],"[0, 0, 0, 0, 0, 0, 0, 1]"331"Mr. Chanek is currently participating in a science fair that is popular in town. He finds an exciting puzzle in the fair and wants to solve it.There are $$$N$$$ atoms numbered from $$$1$$$ to $$$N$$$. These atoms are especially quirky. Initially, each atom is in normal state. Each atom can be in an excited. Exciting atom $$$i$$$ requires $$$D_i$$$ energy. When atom $$$i$$$ is excited, it will give $$$A_i$$$ energy. You can excite any number of atoms (including zero).These atoms also form a peculiar one-way bond. For each $$$i$$$, $$$(1 \le i < N)$$$, if atom $$$i$$$ is excited, atom $$$E_i$$$ will also be excited at no cost. Initially, $$$E_i$$$ = $$$i+1$$$. Note that atom $$$N$$$ cannot form a bond to any atom.Mr. Chanek must change exactly $$$K$$$ bonds. Exactly $$$K$$$ times, Mr. Chanek chooses an atom $$$i$$$, $$$(1 \le i < N)$$$ and changes $$$E_i$$$ to a different value other than $$$i$$$ and the current $$$E_i$$$. Note that an atom's bond can remain unchanged or changed more than once. Help Mr. Chanek determine the maximum energy that he can achieve!note: You must first change exactly $$$K$$$ bonds before you can start exciting atoms.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"332"You are given a multiset of n integers. You should select exactly k of them in a such way that the difference between any two of them is divisible by m, or tell that it is impossible.Numbers can be repeated in the original multiset and in the multiset of selected numbers, but number of occurrences of any number in multiset of selected numbers should not exceed the number of its occurrences in the original multiset. ","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"333"Physicist Woll likes to play one relaxing game in between his search of the theory of everything.Game interface consists of a rectangular n × m playing field and a dashboard. Initially some cells of the playing field are filled while others are empty. Dashboard contains images of all various connected (we mean connectivity by side) figures of 2, 3, 4 and 5 cells, with all their rotations and reflections. Player can copy any figure from the dashboard and place it anywhere at the still empty cells of the playing field. Of course any figure can be used as many times as needed.Woll's aim is to fill the whole field in such a way that there are no empty cells left, and also... just have some fun.Every initially empty cell should be filled with exactly one cell of some figure. Every figure should be entirely inside the board.  In the picture black cells stand for initially filled cells of the field, and one-colour regions represent the figures.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"334"In mathematics, a subsequence is a sequence that can be derived from another sequence by deleting some elements without changing the order of the remaining elements. For example, the sequence BDF is a subsequence of ABCDEF. A substring of a string is a continuous subsequence of the string. For example, BCD is a substring of ABCDEF.You are given two strings s1, s2 and another string called virus. Your task is to find the longest common subsequence of s1 and s2, such that it doesn't contain virus as a substring.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"335"Everyone knows that DNA strands consist of nucleotides. There are four types of nucleotides: ""A"", ""T"", ""G"", ""C"". A DNA strand is a sequence of nucleotides. Scientists decided to track evolution of a rare species, which DNA strand was string s initially. Evolution of the species is described as a sequence of changes in the DNA. Every change is a change of some nucleotide, for example, the following change can happen in DNA strand ""AAGC"": the second nucleotide can change to ""T"" so that the resulting DNA strand is ""ATGC"".Scientists know that some segments of the DNA strand can be affected by some unknown infections. They can represent an infection as a sequence of nucleotides. Scientists are interested if there are any changes caused by some infections. Thus they sometimes want to know the value of impact of some infection to some segment of the DNA. This value is computed as follows:  Let the infection be represented as a string e, and let scientists be interested in DNA strand segment starting from position l to position r, inclusive.  Prefix of the string eee... (i.e. the string that consists of infinitely many repeats of string e) is written under the string s from position l to position r, inclusive.  The value of impact is the number of positions where letter of string s coincided with the letter written under it. Being a developer, Innokenty is interested in bioinformatics also, so the scientists asked him for help. Innokenty is busy preparing VK Cup, so he decided to delegate the problem to the competitors. Help the scientists!",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"336"You are given set of n points in 5-dimensional space. The points are labeled from 1 to n. No two points coincide.We will call point a bad if there are different points b and c, not equal to a, from the given set such that angle between vectors  and  is acute (i.e. strictly less than ). Otherwise, the point is called good.The angle between vectors  and  in 5-dimensional space is defined as , where  is the scalar product and  is length of .Given the list of points, print the indices of the good points in ascending order.","['geometry', 'math']","[0, 1, 0, 1, 0, 0, 0, 0]"337"Okabe likes to take walks but knows that spies from the Organization could be anywhere; that's why he wants to know how many different walks he can take in his city safely. Okabe's city can be represented as all points (x, y) such that x and y are non-negative. Okabe starts at the origin (point (0, 0)), and needs to reach the point (k, 0). If Okabe is currently at the point (x, y), in one step he can go to (x + 1, y + 1), (x + 1, y), or (x + 1, y - 1).Additionally, there are n horizontal line segments, the i-th of which goes from x = ai to x = bi inclusive, and is at y = ci. It is guaranteed that a1 = 0, an ≤ k ≤ bn, and ai = bi - 1 for 2 ≤ i ≤ n. The i-th line segment forces Okabe to walk with y-value in the range 0 ≤ y ≤ ci when his x value satisfies ai ≤ x ≤ bi, or else he might be spied on. This also means he is required to be under two line segments when one segment ends and another begins.Okabe now wants to know how many walks there are from the origin to the point (k, 0) satisfying these conditions, modulo 109 + 7.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"338"The store sells $$$n$$$ items, the price of the $$$i$$$-th item is $$$p_i$$$. The store's management is going to hold a promotion: if a customer purchases at least $$$x$$$ items, $$$y$$$ cheapest of them are free.The management has not yet decided on the exact values of $$$x$$$ and $$$y$$$. Therefore, they ask you to process $$$q$$$ queries: for the given values of $$$x$$$ and $$$y$$$, determine the maximum total value of items received for free, if a customer makes one purchase.Note that all queries are independent; they don't affect the store's stock.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"339"Dima took up the biology of bacteria, as a result of his experiments, he invented k types of bacteria. Overall, there are n bacteria at his laboratory right now, and the number of bacteria of type i equals ci. For convenience, we will assume that all the bacteria are numbered from 1 to n. The bacteria of type ci are numbered from  to .With the help of special equipment Dima can move energy from some bacteria into some other one. Of course, the use of such equipment is not free. Dima knows m ways to move energy from some bacteria to another one. The way with number i can be described with integers ui, vi and xi mean that this way allows moving energy from bacteria with number ui to bacteria with number vi or vice versa for xi dollars.Dima's Chef (Inna) calls the type-distribution correct if there is a way (may be non-direct) to move energy from any bacteria of the particular type to any other bacteria of the same type (between any two bacteria of the same type) for zero cost.As for correct type-distribution the cost of moving the energy depends only on the types of bacteria help Inna to determine is the type-distribution correct? If it is, print the matrix d with size k × k. Cell d[i][j] of this matrix must be equal to the minimal possible cost of energy-moving from bacteria with type i to bacteria with type j.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"340"A palindrome is a string $$$t$$$ which reads the same backward as forward (formally, $$$t[i] = t[|t| + 1 - i]$$$ for all $$$i \in [1, |t|]$$$). Here $$$|t|$$$ denotes the length of a string $$$t$$$. For example, the strings 010, 1001 and 0 are palindromes.You have $$$n$$$ binary strings $$$s_1, s_2, \dots, s_n$$$ (each $$$s_i$$$ consists of zeroes and/or ones). You can swap any pair of characters any number of times (possibly, zero). Characters can be either from the same string or from different strings — there are no restrictions.Formally, in one move you:  choose four integer numbers $$$x, a, y, b$$$ such that $$$1 \le x, y \le n$$$ and $$$1 \le a \le |s_x|$$$ and $$$1 \le b \le |s_y|$$$ (where $$$x$$$ and $$$y$$$ are string indices and $$$a$$$ and $$$b$$$ are positions in strings $$$s_x$$$ and $$$s_y$$$ respectively),  swap (exchange) the characters $$$s_x[a]$$$ and $$$s_y[b]$$$. What is the maximum number of strings you can make palindromic simultaneously?",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"341"One day student Vasya was sitting on a lecture and mentioned a string s1s2... sn, consisting of letters ""a"", ""b"" and ""c"" that was written on his desk. As the lecture was boring, Vasya decided to complete the picture by composing a graph G with the following properties:   G has exactly n vertices, numbered from 1 to n.  For all pairs of vertices i and j, where i ≠ j, there is an edge connecting them if and only if characters si and sj are either equal or neighbouring in the alphabet. That is, letters in pairs ""a""-""b"" and ""b""-""c"" are neighbouring, while letters ""a""-""c"" are not. Vasya painted the resulting graph near the string and then erased the string. Next day Vasya's friend Petya came to a lecture and found some graph at his desk. He had heard of Vasya's adventure and now he wants to find out whether it could be the original graph G, painted by Vasya. In order to verify this, Petya needs to know whether there exists a string s, such that if Vasya used this s he would produce the given graph G.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"342"Once Bob saw a string. It contained so many different letters, that the letters were marked by numbers, but at the same time each letter could be met in the string at most 10 times. Bob didn't like that string, because it contained repeats: a repeat of length x is such a substring of length 2x, that its first half coincides character by character with its second half. Bob started deleting all the repeats from the string. He does it as follows: while it's possible, Bob takes the shortest repeat, if it is not unique, he takes the leftmost one, and deletes its left half and everything that is to the left of this repeat.You're given the string seen by Bob. Find out, what it will look like after Bob deletes all the repeats in the way described above.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"343"We have a point $$$A$$$ with coordinate $$$x = n$$$ on $$$OX$$$-axis. We'd like to find an integer point $$$B$$$ (also on $$$OX$$$-axis), such that the absolute difference between the distance from $$$O$$$ to $$$B$$$ and the distance from $$$A$$$ to $$$B$$$ is equal to $$$k$$$.  The description of the first test case. Since sometimes it's impossible to find such point $$$B$$$, we can, in one step, increase or decrease the coordinate of $$$A$$$ by $$$1$$$. What is the minimum number of steps we should do to make such point $$$B$$$ exist?",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"344"You are given two integers $$$n$$$ and $$$m$$$. Calculate the number of pairs of arrays $$$(a, b)$$$ such that:  the length of both arrays is equal to $$$m$$$;  each element of each array is an integer between $$$1$$$ and $$$n$$$ (inclusive);  $$$a_i \le b_i$$$ for any index $$$i$$$ from $$$1$$$ to $$$m$$$;  array $$$a$$$ is sorted in non-descending order;  array $$$b$$$ is sorted in non-ascending order. As the result can be very large, you should print it modulo $$$10^9+7$$$.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"345"  William has an array of $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$. In one move he can swap two neighboring items. Two items $$$a_i$$$ and $$$a_j$$$ are considered neighboring if the condition $$$|i - j| = 1$$$ is satisfied.William wants you to calculate the minimal number of swaps he would need to perform to make it so that the array does not contain two neighboring items with the same parity.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"346"Everyone knows that computers become faster and faster. Recently Berland scientists have built a machine that can move itself back in time!More specifically, it works as follows. It has an infinite grid and a robot which stands on one of the cells. Each cell of the grid can either be empty or contain 0 or 1. The machine also has a program which consists of instructions, which are being handled one by one. Each instruction is represented by exactly one symbol (letter or digit) and takes exactly one unit of time (say, second) to be performed, except the last type of operation (it's described below). Here they are:  0 or 1: the robot places this number into the cell he is currently at. If this cell wasn't empty before the operation, its previous number is replaced anyway.  e: the robot erases the number into the cell he is at.  l, r, u or d: the robot goes one cell to the left/right/up/down.  s: the robot stays where he is for a unit of time.  t: let $$$x$$$ be $$$0$$$, if the cell with the robot is empty, otherwise let $$$x$$$ be one more than the digit in this cell (that is, $$$x = 1$$$ if the digit in this cell is $$$0$$$, and $$$x = 2$$$ if the digit is $$$1$$$). Then the machine travels $$$x$$$ seconds back in time. Note that this doesn't change the instructions order, but it changes the position of the robot and the numbers in the grid as they were $$$x$$$ units of time ago. You can consider this instruction to be equivalent to a Ctrl-Z pressed $$$x$$$ times. For example, let the board be completely empty, and the program be sr1t0. Let the robot initially be at $$$(0, 0)$$$.  [now is the moment $$$0$$$, the command is s]: we do nothing.  [now is the moment $$$1$$$, the command is r]: we are now at $$$(1, 0)$$$.  [now is the moment $$$2$$$, the command is 1]: we are at $$$(1, 0)$$$, and this cell contains $$$1$$$.  [now is the moment $$$3$$$, the command is t]: we travel $$$1 + 1 = 2$$$ moments back, that is, to the moment $$$1$$$.  [now is the moment $$$1$$$, the command is 0]: we are again at $$$(0, 0)$$$, and the board is clear again, but after we follow this instruction, this cell has $$$0$$$ in it. We've just rewritten the history. The consequences of the third instruction have never happened. Now Berland scientists want to use their machine in practice. For example, they want to be able to add two integers.Assume that the initial state of the machine is as follows:  One positive integer is written in binary on the grid in such a way that its right bit is at the cell $$$(0, 1)$$$, from left to right from the highest bit to the lowest bit.  The other positive integer is written in binary on the grid in such a way that its right bit is at the cell $$$(0, 0)$$$, from left to right from the highest bit to the lowest bit.  All the other cells are empty.  The robot is at $$$(0, 0)$$$.  We consider this state to be always in the past; that is, if you manage to travel to any negative moment, the board was always as described above, and the robot was at $$$(0, 0)$$$ for eternity. You are asked to write a program after which  The robot stands on a non-empty cell,  If we read the number starting from the cell with the robot and moving to the right until the first empty cell, this will be $$$a + b$$$ in binary, from the highest bit to the lowest bit. Note that there are no restrictions on other cells. In particular, there may be a digit just to the left to the robot after all instructions.In each test you are given up to $$$1000$$$ pairs $$$(a, b)$$$, and your program must work for all these pairs. Also since the machine's memory is not very big, your program must consist of no more than $$$10^5$$$ instructions.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"347"Sereja loves number sequences very much. That's why he decided to make himself a new one following a certain algorithm.Sereja takes a blank piece of paper. Then he starts writing out the sequence in m stages. Each time he either adds a new number to the end of the sequence or takes l first elements of the current sequence and adds them c times to the end. More formally, if we represent the current sequence as a1, a2, ..., an, then after we apply the described operation, the sequence transforms into a1, a2, ..., an[, a1, a2, ..., al] (the block in the square brackets must be repeated c times). A day has passed and Sereja has completed the sequence. He wonders what are the values of some of its elements. Help Sereja.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"348"Recently Irina arrived to one of the most famous cities of Berland — the Berlatov city. There are n showplaces in the city, numbered from 1 to n, and some of them are connected by one-directional roads. The roads in Berlatov are designed in a way such that there are no cyclic routes between showplaces.Initially Irina stands at the showplace 1, and the endpoint of her journey is the showplace n. Naturally, Irina wants to visit as much showplaces as she can during her journey. However, Irina's stay in Berlatov is limited and she can't be there for more than T time units.Help Irina determine how many showplaces she may visit during her journey from showplace 1 to showplace n within a time not exceeding T. It is guaranteed that there is at least one route from showplace 1 to showplace n such that Irina will spend no more than T time units passing it.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"349"The Red Kingdom is attacked by the White King and the Black King!The Kingdom is guarded by $$$n$$$ castles, the $$$i$$$-th castle is defended by $$$a_i$$$ soldiers. To conquer the Red Kingdom, the Kings have to eliminate all the defenders. Each day the White King launches an attack on one of the castles. Then, at night, the forces of the Black King attack a castle (possibly the same one). Then the White King attacks a castle, then the Black King, and so on. The first attack is performed by the White King.Each attack must target a castle with at least one alive defender in it. There are three types of attacks:  a mixed attack decreases the number of defenders in the targeted castle by $$$x$$$ (or sets it to $$$0$$$ if there are already less than $$$x$$$ defenders);  an infantry attack decreases the number of defenders in the targeted castle by $$$y$$$ (or sets it to $$$0$$$ if there are already less than $$$y$$$ defenders);  a cavalry attack decreases the number of defenders in the targeted castle by $$$z$$$ (or sets it to $$$0$$$ if there are already less than $$$z$$$ defenders). The mixed attack can be launched at any valid target (at any castle with at least one soldier). However, the infantry attack cannot be launched if the previous attack on the targeted castle had the same type, no matter when and by whom it was launched. The same applies to the cavalry attack. A castle that was not attacked at all can be targeted by any type of attack.The King who launches the last attack will be glorified as the conqueror of the Red Kingdom, so both Kings want to launch the last attack (and they are wise enough to find a strategy that allows them to do it no matter what are the actions of their opponent, if such strategy exists). The White King is leading his first attack, and you are responsible for planning it. Can you calculate the number of possible options for the first attack that allow the White King to launch the last attack? Each option for the first attack is represented by the targeted castle and the type of attack, and two options are different if the targeted castles or the types of attack are different.",['games'],"[1, 0, 0, 0, 0, 0, 0, 0]"350"All cinema halls in Berland are rectangles with K rows of K seats each, and K is an odd number. Rows and seats are numbered from 1 to K. For safety reasons people, who come to the box office to buy tickets, are not allowed to choose seats themselves. Formerly the choice was made by a cashier, but now this is the responsibility of a special seating program. It was found out that the large majority of Berland's inhabitants go to the cinema in order to watch a movie, that's why they want to sit as close to the hall center as possible. Moreover, a company of M people, who come to watch a movie, want necessarily to occupy M successive seats in one row. Let's formulate the algorithm, according to which the program chooses seats and sells tickets. As the request for M seats comes, the program should determine the row number x and the segment [yl, yr] of the seats numbers in this row, where yr - yl + 1 = M. From all such possible variants as a final result the program should choose the one with the minimum function value of total seats remoteness from the center. Say,  — the row and the seat numbers of the most ""central"" seat. Then the function value of seats remoteness from the hall center is . If the amount of minimum function values is more than one, the program should choose the one that is closer to the screen (i.e. the row number x is lower). If the variants are still multiple, it should choose the one with the minimum yl. If you did not get yet, your task is to simulate the work of this program. ",[],"[0, 0, 0, 0, 0, 0, 0, 0]"351"One evening Rainbow Dash and Fluttershy have come up with a game. Since the ponies are friends, they have decided not to compete in the game but to pursue a common goal. The game starts on a square flat grid, which initially has the outline borders built up. Rainbow Dash and Fluttershy have flat square blocks with size $$$1\times1$$$, Rainbow Dash has an infinite amount of light blue blocks, Fluttershy has an infinite amount of yellow blocks. The blocks are placed according to the following rule: each newly placed block must touch the built on the previous turns figure by a side (note that the outline borders of the grid are built initially). At each turn, one pony can place any number of blocks of her color according to the game rules.Rainbow and Fluttershy have found out that they can build patterns on the grid of the game that way. They have decided to start with something simple, so they made up their mind to place the blocks to form a chess coloring. Rainbow Dash is well-known for her speed, so she is interested in the minimum number of turns she and Fluttershy need to do to get a chess coloring, covering the whole grid with blocks. Please help her find that number!Since the ponies can play many times on different boards, Rainbow Dash asks you to find the minimum numbers of turns for several grids of the games.The chess coloring in two colors is the one in which each square is neighbor by side only with squares of different colors.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"352"Vitya has learned that the answer for The Ultimate Question of Life, the Universe, and Everything is not the integer 54 42, but an increasing integer sequence $$$a_1, \ldots, a_n$$$. In order to not reveal the secret earlier than needed, Vitya encrypted the answer and obtained the sequence $$$b_1, \ldots, b_n$$$ using the following rules:  $$$b_1 = a_1$$$; $$$b_i = a_i \oplus a_{i - 1}$$$ for all $$$i$$$ from 2 to $$$n$$$, where $$$x \oplus y$$$ is the bitwise XOR of $$$x$$$ and $$$y$$$. It is easy to see that the original sequence can be obtained using the rule $$$a_i = b_1 \oplus \ldots \oplus b_i$$$.However, some time later Vitya discovered that the integers $$$b_i$$$ in the cypher got shuffled, and it can happen that when decrypted using the rule mentioned above, it can produce a sequence that is not increasing. In order to save his reputation in the scientific community, Vasya decided to find some permutation of integers $$$b_i$$$ so that the sequence $$$a_i = b_1 \oplus \ldots \oplus b_i$$$ is strictly increasing. Help him find such a permutation or determine that it is impossible.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"353"A and B are preparing themselves for programming contests.After several years of doing sports programming and solving many problems that require calculating all sorts of abstract objects, A and B also developed rather peculiar tastes.A likes lowercase letters of the Latin alphabet. He has assigned to each letter a number that shows how much he likes that letter (he has assigned negative numbers to the letters he dislikes). B likes substrings. He especially likes the ones that start and end with the same letter (their length must exceed one).Also, A and B have a string s. Now they are trying to find out how many substrings t of a string s are interesting to B (that is, t starts and ends with the same letter and its length is larger than one), and also the sum of values of all letters (assigned by A), except for the first and the last one is equal to zero.Naturally, A and B have quickly found the number of substrings t that are interesting to them. Can you do it? ",[],"[0, 0, 0, 0, 0, 0, 0, 0]"354"In this problem we consider a special type of an auction, which is called the second-price auction. As in regular auction n bidders place a bid which is price a bidder ready to pay. The auction is closed, that is, each bidder secretly informs the organizer of the auction price he is willing to pay. After that, the auction winner is the participant who offered the highest price. However, he pay not the price he offers, but the highest price among the offers of other participants (hence the name: the second-price auction).Write a program that reads prices offered by bidders and finds the winner and the price he will pay. Consider that all of the offered prices are different.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"355"This is the easy version of the problem. The difference is the constraint on the sum of lengths of strings and the number of test cases. You can make hacks only if you solve all versions of this task.You are given a string $$$s$$$, consisting of lowercase English letters. Find the longest string, $$$t$$$, which satisfies the following conditions:   The length of $$$t$$$ does not exceed the length of $$$s$$$.  $$$t$$$ is a palindrome.  There exists two strings $$$a$$$ and $$$b$$$ (possibly empty), such that $$$t = a + b$$$ ( ""$$$+$$$"" represents concatenation), and $$$a$$$ is prefix of $$$s$$$ while $$$b$$$ is suffix of $$$s$$$. ",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"356"Recently, on a programming lesson little Petya showed how quickly he can create files and folders on the computer. But he got soon fed up with this activity, and he decided to do a much more useful thing. He decided to calculate what folder contains most subfolders (including nested folders, nested folders of nested folders, and so on) and what folder contains most files (including the files in the subfolders).More formally, the subfolders of the folder are all its directly nested folders and the subfolders of these nested folders. The given folder is not considered the subfolder of itself. A file is regarded as lying in a folder, if and only if it either lies directly in this folder, or lies in some subfolder of the folder.For a better understanding of how to count subfolders and files for calculating the answer, see notes and answers to the samples.You are given a few files that Petya has managed to create. The path to each file looks as follows:diskName:\folder1\folder2\...\ foldern\fileName  diskName is single capital letter from the set {C,D,E,F,G}. folder1, ..., foldern are folder names. Each folder name is nonempty sequence of lowercase Latin letters and digits from 0 to 9. (n ≥ 1) fileName is a file name in the form of name.extension, where the name and the extension are nonempty sequences of lowercase Latin letters and digits from 0 to 9. It is also known that there is no file whose path looks like diskName:\fileName. That is, each file is stored in some folder, but there are no files directly in the root. Also let us assume that the disk root is not a folder.Help Petya to find the largest number of subfolders, which can be in some folder, and the largest number of files that can be in some folder, counting all its subfolders.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"357"You are both a shop keeper and a shop assistant at a small nearby shop. You have $$$n$$$ goods, the $$$i$$$-th good costs $$$a_i$$$ coins.You got tired of remembering the price of each product when customers ask for it, thus you decided to simplify your life. More precisely you decided to set the same price for all $$$n$$$ goods you have.However, you don't want to lose any money so you want to choose the price in such a way that the sum of new prices is not less than the sum of the initial prices. It means that if you sell all $$$n$$$ goods for the new price, you will receive at least the same (or greater) amount of money as if you sell them for their initial prices.On the other hand, you don't want to lose customers because of big prices so among all prices you can choose you need to choose the minimum one.So you need to find the minimum possible equal price of all $$$n$$$ goods so if you sell them for this price, you will receive at least the same (or greater) amount of money as if you sell them for their initial prices.You have to answer $$$q$$$ independent queries.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"358"The Little Elephant loves sortings.He has an array a consisting of n integers. Let's number the array elements from 1 to n, then the i-th element will be denoted as ai. The Little Elephant can make one move to choose an arbitrary pair of integers l and r (1 ≤ l ≤ r ≤ n) and increase ai by 1 for all i such that l ≤ i ≤ r.Help the Little Elephant find the minimum number of moves he needs to convert array a to an arbitrary array sorted in the non-decreasing order. Array a, consisting of n elements, is sorted in the non-decreasing order if for any i (1 ≤ i < n) ai ≤ ai + 1 holds.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"359"You are given array $$$a_1, a_2, \ldots, a_n$$$, consisting of non-negative integers.Let's define operation of ""elimination"" with integer parameter $$$k$$$ ($$$1 \leq k \leq n$$$) as follows:  Choose $$$k$$$ distinct array indices $$$1 \leq i_1 < i_2 < \ldots < i_k \le n$$$.  Calculate $$$x = a_{i_1} ~ \& ~ a_{i_2} ~ \& ~ \ldots ~ \& ~ a_{i_k}$$$, where $$$\&$$$ denotes the bitwise AND operation (notes section contains formal definition).  Subtract $$$x$$$ from each of $$$a_{i_1}, a_{i_2}, \ldots, a_{i_k}$$$; all other elements remain untouched. Find all possible values of $$$k$$$, such that it's possible to make all elements of array $$$a$$$ equal to $$$0$$$ using a finite number of elimination operations with parameter $$$k$$$. It can be proven that exists at least one possible $$$k$$$ for any array $$$a$$$.Note that you firstly choose $$$k$$$ and only after that perform elimination operations with value $$$k$$$ you've chosen initially.","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"360"You are given an array consisting of all integers from $$$[l, r]$$$ inclusive. For example, if $$$l = 2$$$ and $$$r = 5$$$, the array would be $$$[2, 3, 4, 5]$$$. What's the minimum number of elements you can delete to make the bitwise AND of the array non-zero?A bitwise AND is a binary operation that takes two equal-length binary representations and performs the AND operation on each pair of the corresponding bits.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"361"You've got string s, consisting of only lowercase English letters. Find its lexicographically maximum subsequence.We'll call a non-empty string s[p1p2... pk] = sp1sp2... spk(1 ≤ p1 < p2 < ... < pk ≤ |s|) a subsequence of string s = s1s2... s|s|.String x = x1x2... x|x| is lexicographically larger than string y = y1y2... y|y|, if either |x| > |y| and x1 = y1, x2 = y2, ... , x|y| = y|y|, or exists such number r (r < |x|, r < |y|), that x1 = y1, x2 = y2, ... , xr = yr and xr + 1 > yr + 1. Characters in lines are compared like their ASCII codes.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"362"You are given an array of integers $$$a_1,a_2,\ldots,a_n$$$. Find the maximum possible value of $$$a_ia_ja_ka_la_t$$$ among all five indices $$$(i, j, k, l, t)$$$ ($$$i<j<k<l<t$$$).",[],"[0, 0, 0, 0, 0, 0, 0, 0]"363"In an attempt to escape the Mischievous Mess Makers' antics, Farmer John has abandoned his farm and is traveling to the other side of Bovinia. During the journey, he and his k cows have decided to stay at the luxurious Grand Moo-dapest Hotel. The hotel consists of n rooms located in a row, some of which are occupied.Farmer John wants to book a set of k + 1 currently unoccupied rooms for him and his cows. He wants his cows to stay as safe as possible, so he wishes to minimize the maximum distance from his room to the room of his cow. The distance between rooms i and j is defined as |j - i|. Help Farmer John protect his cows by calculating this minimum possible distance.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"364"There are $$$n$$$ cars on a coordinate axis $$$OX$$$. Each car is located at an integer point initially and no two cars are located at the same point. Also, each car is oriented either left or right, and they can move at any constant positive speed in that direction at any moment.More formally, we can describe the $$$i$$$-th car with a letter and an integer: its orientation $$$ori_i$$$ and its location $$$x_i$$$. If $$$ori_i = L$$$, then $$$x_i$$$ is decreasing at a constant rate with respect to time. Similarly, if $$$ori_i = R$$$, then $$$x_i$$$ is increasing at a constant rate with respect to time. We call two cars irrelevant if they never end up in the same point regardless of their speed. In other words, they won't share the same coordinate at any moment.We call two cars destined if they always end up in the same point regardless of their speed. In other words, they must share the same coordinate at some moment.Unfortunately, we lost all information about our cars, but we do remember $$$m$$$ relationships. There are two types of relationships:$$$1$$$ $$$i$$$ $$$j$$$ —$$$i$$$-th car and $$$j$$$-th car are irrelevant.$$$2$$$ $$$i$$$ $$$j$$$ —$$$i$$$-th car and $$$j$$$-th car are destined.Restore the orientations and the locations of the cars satisfying the relationships, or report that it is impossible. If there are multiple solutions, you can output any.Note that if two cars share the same coordinate, they will intersect, but at the same moment they will continue their movement in their directions.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"365"Vasya has found a piece of paper with an array written on it. The array consists of n integers a1, a2, ..., an. Vasya noticed that the following condition holds for the array ai ≤ ai + 1 ≤ 2·ai for any positive integer i (i < n).Vasya wants to add either a ""+"" or a ""-"" before each number of array. Thus, Vasya will get an expression consisting of n summands. The value of the resulting expression is the sum of all its elements. The task is to add signs ""+"" and ""-"" before each number so that the value of expression s meets the limits 0 ≤ s ≤ a1. Print a sequence of signs ""+"" and ""-"", satisfying the given limits. It is guaranteed that the solution for the problem exists.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"366"You received as a gift a very clever robot walking on a rectangular board. Unfortunately, you understood that it is broken and behaves rather strangely (randomly). The board consists of N rows and M columns of cells. The robot is initially at some cell on the i-th row and the j-th column. Then at every step the robot could go to some another cell. The aim is to go to the bottommost (N-th) row. The robot can stay at it's current cell, move to the left, move to the right, or move to the cell below the current. If the robot is in the leftmost column it cannot move to the left, and if it is in the rightmost column it cannot move to the right. At every step all possible moves are equally probable. Return the expected number of step to reach the bottommost row.","['math', 'probabilities']","[0, 0, 0, 1, 0, 1, 0, 0]"367"Nearly each project of the F company has a whole team of developers working on it. They often are in different rooms of the office in different cities and even countries. To keep in touch and track the results of the project, the F company conducts shared online meetings in a Spyke chat.One day the director of the F company got hold of the records of a part of an online meeting of one successful team. The director watched the record and wanted to talk to the team leader. But how can he tell who the leader is? The director logically supposed that the leader is the person who is present at any conversation during a chat meeting. In other words, if at some moment of time at least one person is present on the meeting, then the leader is present on the meeting.You are the assistant director. Given the 'user logged on'/'user logged off' messages of the meeting in the chronological order, help the director determine who can be the leader. Note that the director has the record of only a continuous part of the meeting (probably, it's not the whole meeting).",[],"[0, 0, 0, 0, 0, 0, 0, 0]"368"«Next please», — the princess called and cast an estimating glance at the next groom.The princess intends to choose the most worthy groom, this is, the richest one. Whenever she sees a groom who is more rich than each of the previous ones, she says a measured «Oh...». Whenever the groom is richer than all previous ones added together, she exclaims «Wow!» (no «Oh...» in this case). At the sight of the first groom the princess stays calm and says nothing.The fortune of each groom is described with an integer between 1 and 50000. You know that during the day the princess saw n grooms, said «Oh...» exactly a times and exclaimed «Wow!» exactly b times. Your task is to output a sequence of n integers t1, t2, ..., tn, where ti describes the fortune of i-th groom. If several sequences are possible, output any of them. If no sequence exists that would satisfy all the requirements, output a single number -1.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"369"Caisa solved the problem with the sugar and now he is on the way back to home. Caisa is playing a mobile game during his path. There are (n + 1) pylons numbered from 0 to n in this game. The pylon with number 0 has zero height, the pylon with number i (i > 0) has height hi. The goal of the game is to reach n-th pylon, and the only move the player can do is to jump from the current pylon (let's denote its number as k) to the next one (its number will be k + 1). When the player have made such a move, its energy increases by hk - hk + 1 (if this value is negative the player loses energy). The player must have non-negative amount of energy at any moment of the time. Initially Caisa stand at 0 pylon and has 0 energy. The game provides a special opportunity: one can pay a single dollar and increase the height of anyone pylon by one. Caisa may use that opportunity several times, but he doesn't want to spend too much money. What is the minimal amount of money he must paid to reach the goal of the game?",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"370"Note that the memory limit is unusual.You are given an integer $$$n$$$ and two sequences $$$a_1, a_2, \dots, a_n$$$ and $$$b_1, b_2, \dots, b_n$$$.Let's call a set of integers $$$S$$$ such that $$$S \subseteq \{1, 2, 3, \dots, n\}$$$ strange, if, for every element $$$i$$$ of $$$S$$$, the following condition is met: for every $$$j \in [1, i - 1]$$$, if $$$a_j$$$ divides $$$a_i$$$, then $$$j$$$ is also included in $$$S$$$. An empty set is always strange.The cost of the set $$$S$$$ is $$$\sum\limits_{i \in S} b_i$$$. You have to calculate the maximum possible cost of a strange set.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"371"You are given $$$n$$$ distinct points on a plane. The coordinates of the $$$i$$$-th point are $$$(x_i, y_i)$$$.For each point $$$i$$$, find the nearest (in terms of Manhattan distance) point with integer coordinates that is not among the given $$$n$$$ points. If there are multiple such points — you can choose any of them.The Manhattan distance between two points $$$(x_1, y_1)$$$ and $$$(x_2, y_2)$$$ is $$$|x_1 - x_2| + |y_1 - y_2|$$$.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"372"This problem consists of three subproblems: for solving subproblem F1 you will receive 8 points, for solving subproblem F2 you will receive 15 points, and for solving subproblem F3 you will receive 10 points.Manao has developed a model to predict the stock price of a company over the next n days and wants to design a profit-maximizing trading algorithm to make use of these predictions. Unfortunately, Manao's trading account has the following restrictions:   It only allows owning either zero or one shares of stock at a time;  It only allows buying or selling a share of this stock once per day;  It allows a maximum of k buy orders over the next n days; For the purposes of this problem, we define a trade to a be the act of buying one share of stock on day i, then holding the stock until some day j > i at which point the share is sold. To restate the above constraints, Manao is permitted to make at most k non-overlapping trades during the course of an n-day trading period for which Manao's model has predictions about the stock price.Even though these restrictions limit the amount of profit Manao can make compared to what would be achievable with an unlimited number of trades or the ability to hold more than one share at a time, Manao still has the potential to make a lot of money because Manao's model perfectly predicts the daily price of the stock. For example, using this model, Manao could wait until the price is low, then buy one share and hold until the price reaches a high value, then sell for a profit, and repeat this process up to k times until n days have passed.Nevertheless, Manao is not satisfied by having a merely good trading algorithm, and wants to develop an optimal strategy for trading subject to these constraints. Help Manao achieve this goal by writing a program that will determine when to buy and sell stock to achieve the greatest possible profit during the n-day trading period subject to the above constraints.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"373"Nastya owns too many arrays now, so she wants to delete the least important of them. However, she discovered that this array is magic! Nastya now knows that the array has the following properties:  In one second we can add an arbitrary (possibly negative) integer to all elements of the array that are not equal to zero.  When all elements of the array become equal to zero, the array explodes. Nastya is always busy, so she wants to explode the array as fast as possible. Compute the minimum time in which the array can be exploded.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"374"The cuteness of a binary string is the number of $$$\texttt{1}$$$s divided by the length of the string. For example, the cuteness of $$$\texttt{01101}$$$ is $$$\frac{3}{5}$$$.Juju has a binary string $$$s$$$ of length $$$n$$$. She wants to choose some non-intersecting subsegments of $$$s$$$ such that their concatenation has length $$$m$$$ and it has the same cuteness as the string $$$s$$$. More specifically, she wants to find two arrays $$$l$$$ and $$$r$$$ of equal length $$$k$$$ such that $$$1 \leq l_1 \leq r_1 < l_2 \leq r_2 < \ldots < l_k \leq r_k \leq n$$$, and also:  $$$\sum\limits_{i=1}^k (r_i - l_i + 1) = m$$$;  The cuteness of $$$s[l_1,r_1]+s[l_2,r_2]+\ldots+s[l_k,r_k]$$$ is equal to the cuteness of $$$s$$$, where $$$s[x, y]$$$ denotes the subsegment $$$s_x s_{x+1} \ldots s_y$$$, and $$$+$$$ denotes string concatenation. Juju does not like splitting the string into many parts, so she also wants to minimize the value of $$$k$$$. Find the minimum value of $$$k$$$ such that there exist $$$l$$$ and $$$r$$$ that satisfy the constraints above or determine that it is impossible to find such $$$l$$$ and $$$r$$$ for any $$$k$$$.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"375"Andrewid the Android is a galaxy-famous detective. He is now chasing a criminal hiding on the planet Oxa-5, the planet almost fully covered with water.The only dry land there is an archipelago of n narrow islands located in a row. For more comfort let's represent them as non-intersecting segments on a straight line: island i has coordinates [li, ri], besides, ri < li + 1 for 1 ≤ i ≤ n - 1.To reach the goal, Andrewid needs to place a bridge between each pair of adjacent islands. A bridge of length a can be placed between the i-th and the (i + 1)-th islads, if there are such coordinates of x and y, that li ≤ x ≤ ri, li + 1 ≤ y ≤ ri + 1 and y - x = a. The detective was supplied with m bridges, each bridge can be used at most once. Help him determine whether the bridges he got are enough to connect each pair of adjacent islands.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"376"Nezzar's favorite digit among $$$1,\ldots,9$$$ is $$$d$$$. He calls a positive integer lucky if $$$d$$$ occurs at least once in its decimal representation. Given $$$q$$$ integers $$$a_1,a_2,\ldots,a_q$$$, for each $$$1 \le i \le q$$$ Nezzar would like to know if $$$a_i$$$ can be equal to a sum of several (one or more) lucky numbers.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"377"The GCD table G of size n × n for an array of positive integers a of length n is defined by formula   Let us remind you that the greatest common divisor (GCD) of two positive integers x and y is the greatest integer that is divisor of both x and y, it is denoted as . For example, for array a = {4, 3, 6, 2} of length 4 the GCD table will look as follows:  Given all the numbers of the GCD table G, restore array a.",['number theory'],"[0, 0, 0, 0, 1, 0, 0, 0]"378"You have an n × m rectangle table, its cells are not initially painted. Your task is to paint all cells of the table. The resulting picture should be a tiling of the table with squares. More formally:  each cell must be painted some color (the colors are marked by uppercase Latin letters);  we will assume that two cells of the table are connected if they are of the same color and share a side; each connected region of the table must form a square. Given n and m, find lexicographically minimum coloring of the table that meets the described properties.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"379"Kuroni is very angry at the other setters for using him as a theme! As a punishment, he forced them to solve the following problem:You have an array $$$a$$$ consisting of $$$n$$$ positive integers. An operation consists of choosing an element and either adding $$$1$$$ to it or subtracting $$$1$$$ from it, such that the element remains positive. We say the array is good if the greatest common divisor of all its elements is not $$$1$$$. Find the minimum number of operations needed to make the array good.Unable to match Kuroni's intellect, the setters failed to solve the problem. Help them escape from Kuroni's punishment!","['math', 'number theory', 'probabilities']","[0, 0, 0, 1, 1, 1, 0, 0]"380"You play your favourite game yet another time. You chose the character you didn't play before. It has $$$str$$$ points of strength and $$$int$$$ points of intelligence. Also, at start, the character has $$$exp$$$ free experience points you can invest either in strength or in intelligence (by investing one point you can either raise strength by $$$1$$$ or raise intelligence by $$$1$$$).Since you'd like to make some fun you want to create a jock character, so it has more strength than intelligence points (resulting strength is strictly greater than the resulting intelligence).Calculate the number of different character builds you can create (for the purpose of replayability) if you must invest all free points. Two character builds are different if their strength and/or intellect are different.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"381"After the contest in comparing numbers, Shapur's teacher found out that he is a real genius and that no one could possibly do the calculations faster than him even using a super computer!Some days before the contest, the teacher took a very simple-looking exam and all his n students took part in the exam. The teacher gave them 3 strings and asked them to concatenate them. Concatenating strings means to put them in some arbitrary order one after the other. For example from concatenating Alireza and Amir we can get to AlirezaAmir or AmirAlireza depending on the order of concatenation.Unfortunately enough, the teacher forgot to ask students to concatenate their strings in a pre-defined order so each student did it the way he/she liked.Now the teacher knows that Shapur is such a fast-calculating genius boy and asks him to correct the students' papers.Shapur is not good at doing such a time-taking task. He rather likes to finish up with it as soon as possible and take his time to solve 3-SAT in polynomial time. Moreover, the teacher has given some advice that Shapur has to follow. Here's what the teacher said:   As I expect you know, the strings I gave to my students (including you) contained only lowercase and uppercase Persian Mikhi-Script letters. These letters are too much like Latin letters, so to make your task much harder I converted all the initial strings and all of the students' answers to Latin.  As latin alphabet has much less characters than Mikhi-Script, I added three odd-looking characters to the answers, these include ""-"", "";"" and ""_"". These characters are my own invention of course! And I call them Signs.  The length of all initial strings was less than or equal to 100 and the lengths of my students' answers are less than or equal to 600  My son, not all students are genius as you are. It is quite possible that they make minor mistakes changing case of some characters. For example they may write ALiReZaAmIR instead of AlirezaAmir. Don't be picky and ignore these mistakes.  Those signs which I previously talked to you about are not important. You can ignore them, since many students are in the mood for adding extra signs or forgetting about a sign. So something like Iran;;-- is the same as --;IRAN  You should indicate for any of my students if his answer was right or wrong. Do this by writing ""WA"" for Wrong answer or ""ACC"" for a correct answer.  I should remind you that none of the strings (initial strings or answers) are empty.  Finally, do these as soon as possible. You have less than 2 hours to complete this.  ",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"382"You are given a tree (an undirected connected acyclic graph) consisting of $$$n$$$ vertices and $$$n - 1$$$ edges. A number is written on each edge, each number is either $$$0$$$ (let's call such edges $$$0$$$-edges) or $$$1$$$ (those are $$$1$$$-edges).Let's call an ordered pair of vertices $$$(x, y)$$$ ($$$x \ne y$$$) valid if, while traversing the simple path from $$$x$$$ to $$$y$$$, we never go through a $$$0$$$-edge after going through a $$$1$$$-edge. Your task is to calculate the number of valid pairs in the tree.",['trees'],"[0, 0, 0, 0, 0, 0, 0, 1]"383"$$$n$$$ heroes fight against each other in the Arena. Initially, the $$$i$$$-th hero has level $$$a_i$$$.Each minute, a fight between two different heroes occurs. These heroes can be chosen arbitrarily (it's even possible that it is the same two heroes that were fighting during the last minute).When two heroes of equal levels fight, nobody wins the fight. When two heroes of different levels fight, the one with the higher level wins, and his level increases by $$$1$$$.The winner of the tournament is the first hero that wins in at least $$$100^{500}$$$ fights (note that it's possible that the tournament lasts forever if no hero wins this number of fights, then there is no winner). A possible winner is a hero such that there exists a sequence of fights that this hero becomes the winner of the tournament.Calculate the number of possible winners among $$$n$$$ heroes.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"384"This is the easy version of the problem. The only difference is that here $$$k=2$$$. You can make hacks only if both the versions of the problem are solved.This is an interactive problem.Every decimal number has a base $$$k$$$ equivalent. The individual digits of a base $$$k$$$ number are called $$$k$$$-its. Let's define the $$$k$$$-itwise XOR of two $$$k$$$-its $$$a$$$ and $$$b$$$ as $$$(a + b)\bmod k$$$.The $$$k$$$-itwise XOR of two base $$$k$$$ numbers is equal to the new number formed by taking the $$$k$$$-itwise XOR of their corresponding $$$k$$$-its. The $$$k$$$-itwise XOR of two decimal numbers $$$a$$$ and $$$b$$$ is denoted by $$$a\oplus_{k} b$$$ and is equal to the decimal representation of the $$$k$$$-itwise XOR of the base $$$k$$$ representations of $$$a$$$ and $$$b$$$. All further numbers used in the statement below are in decimal unless specified. When $$$k = 2$$$ (it is always true in this version), the $$$k$$$-itwise XOR is the same as the bitwise XOR.You have hacked the criminal database of Rockport Police Department (RPD), also known as the Rap Sheet. But in order to access it, you require a password. You don't know it, but you are quite sure that it lies between $$$0$$$ and $$$n-1$$$ inclusive. So, you have decided to guess it. Luckily, you can try at most $$$n$$$ times without being blocked by the system. But the system is adaptive. Each time you make an incorrect guess, it changes the password. Specifically, if the password before the guess was $$$x$$$, and you guess a different number $$$y$$$, then the system changes the password to a number $$$z$$$ such that $$$x\oplus_{k} z=y$$$. Guess the password and break into the system.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"385"You are given a sequence $$$a_1, a_2, \dots, a_n$$$ consisting of $$$n$$$ pairwise distinct positive integers.Find $$$\left\lfloor \frac n 2 \right\rfloor$$$ different pairs of integers $$$x$$$ and $$$y$$$ such that:   $$$x \neq y$$$;  $$$x$$$ and $$$y$$$ appear in $$$a$$$;  $$$x~mod~y$$$ doesn't appear in $$$a$$$. Note that some $$$x$$$ or $$$y$$$ can belong to multiple pairs.$$$\lfloor x \rfloor$$$ denotes the floor function — the largest integer less than or equal to $$$x$$$. $$$x~mod~y$$$ denotes the remainder from dividing $$$x$$$ by $$$y$$$.If there are multiple solutions, print any of them. It can be shown that at least one solution always exists.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"386"Even a cat has things it can do that AI cannot.— Fei-Fei LiYou are given $$$m$$$ arrays of positive integers. Each array is of even length.You need to split all these integers into two equal multisets $$$L$$$ and $$$R$$$, that is, each element of each array should go into one of two multisets (but not both). Additionally, for each of the $$$m$$$ arrays, exactly half of its elements should go into $$$L$$$, and the rest should go into $$$R$$$.Give an example of such a division or determine that no such division exists.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"387"It is the hard version of the problem. The difference is that in this version, there are nodes with already chosen colors.Theofanis is starving, and he wants to eat his favorite food, sheftalia. However, he should first finish his homework. Can you help him with this problem?You have a perfect binary tree of $$$2^k - 1$$$ nodes — a binary tree where all vertices $$$i$$$ from $$$1$$$ to $$$2^{k - 1} - 1$$$ have exactly two children: vertices $$$2i$$$ and $$$2i + 1$$$. Vertices from $$$2^{k - 1}$$$ to $$$2^k - 1$$$ don't have any children. You want to color its vertices with the $$$6$$$ Rubik's cube colors (White, Green, Red, Blue, Orange and Yellow).Let's call a coloring good when all edges connect nodes with colors that are neighboring sides in the Rubik's cube.  A picture of Rubik's cube and its 2D map. More formally:   a white node can not be neighboring with white and yellow nodes;  a yellow node can not be neighboring with white and yellow nodes;  a green node can not be neighboring with green and blue nodes;  a blue node can not be neighboring with green and blue nodes;  a red node can not be neighboring with red and orange nodes;  an orange node can not be neighboring with red and orange nodes; However, there are $$$n$$$ special nodes in the tree, colors of which are already chosen.You want to calculate the number of the good colorings of the binary tree. Two colorings are considered different if at least one node is colored with a different color.The answer may be too large, so output the answer modulo $$$10^9+7$$$.","['math', 'trees']","[0, 0, 0, 1, 0, 0, 0, 1]"388"Polycarp has $$$3$$$ positive integers $$$a$$$, $$$b$$$ and $$$c$$$. He can perform the following operation exactly once.  Choose a positive integer $$$m$$$ and multiply exactly one of the integers $$$a$$$, $$$b$$$ or $$$c$$$ by $$$m$$$. Can Polycarp make it so that after performing the operation, the sequence of three numbers $$$a$$$, $$$b$$$, $$$c$$$ (in this order) forms an arithmetic progression? Note that you cannot change the order of $$$a$$$, $$$b$$$ and $$$c$$$.Formally, a sequence $$$x_1, x_2, \dots, x_n$$$ is called an arithmetic progression (AP) if there exists a number $$$d$$$ (called ""common difference"") such that $$$x_{i+1}=x_i+d$$$ for all $$$i$$$ from $$$1$$$ to $$$n-1$$$. In this problem, $$$n=3$$$.For example, the following sequences are AP: $$$[5, 10, 15]$$$, $$$[3, 2, 1]$$$, $$$[1, 1, 1]$$$, and $$$[13, 10, 7]$$$. The following sequences are not AP: $$$[1, 2, 4]$$$, $$$[0, 1, 0]$$$ and $$$[1, 3, 2]$$$.You need to answer $$$t$$$ independent test cases.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"389"You must have heard all about the Foolland on your Geography lessons. Specifically, you must know that federal structure of this country has been the same for many centuries. The country consists of n cities, some pairs of cities are connected by bidirectional roads, each road is described by its length li.The fools lived in their land joyfully, but a recent revolution changed the king. Now the king is Vasily the Bear. Vasily divided the country cities into regions, so that any two cities of the same region have a path along the roads between them and any two cities of different regions don't have such path. Then Vasily decided to upgrade the road network and construct exactly p new roads in the country. Constructing a road goes like this:  We choose a pair of distinct cities u, v that will be connected by a new road (at that, it is possible that there already is a road between these cities).  We define the length of the new road: if cities u, v belong to distinct regions, then the length is calculated as min(109, S + 1) (S — the total length of all roads that exist in the linked regions), otherwise we assume that the length equals 1000.  We build a road of the specified length between the chosen cities. If the new road connects two distinct regions, after construction of the road these regions are combined into one new region. Vasily wants the road constructing process to result in the country that consists exactly of q regions. Your task is to come up with such road constructing plan for Vasily that it meets the requirement and minimizes the total length of the built roads.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"390"Petya has a string of length n consisting of small and large English letters and digits.He performs m operations. Each operation is described with two integers l and r and a character c: Petya removes from the string all characters c on positions between l and r, inclusive. It's obvious that the length of the string remains the same or decreases after each operation.Find how the string will look like after Petya performs all m operations.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"391"You are given the set of vectors on the plane, each of them starting at the origin. Your task is to find a pair of vectors with the minimal non-oriented angle between them.Non-oriented angle is non-negative value, minimal between clockwise and counterclockwise direction angles. Non-oriented angle is always between 0 and π. For example, opposite directions vectors have angle equals to π.",['geometry'],"[0, 1, 0, 0, 0, 0, 0, 0]"392"Petya has k matches, placed in n matchboxes lying in a line from left to right. We know that k is divisible by n. Petya wants all boxes to have the same number of matches inside. For that, he can move a match from its box to the adjacent one in one move. How many such moves does he need to achieve the desired configuration?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"393"Squirrel Liss is interested in sequences. She also has preferences of integers. She thinks n integers a1, a2, ..., an are good.Now she is interested in good sequences. A sequence x1, x2, ..., xk is called good if it satisfies the following three conditions:  The sequence is strictly increasing, i.e. xi < xi + 1 for each i (1 ≤ i ≤ k - 1).  No two adjacent elements are coprime, i.e. gcd(xi, xi + 1) > 1 for each i (1 ≤ i ≤ k - 1) (where gcd(p, q) denotes the greatest common divisor of the integers p and q).  All elements of the sequence are good integers. Find the length of the longest good sequence.",['number theory'],"[0, 0, 0, 0, 1, 0, 0, 0]"394"Kirill lives on a connected undirected graph of $$$n$$$ vertices and $$$m$$$ edges at vertex $$$1$$$. One fine evening he gathered $$$f$$$ friends, the $$$i$$$-th friend lives at the vertex $$$h_i$$$. So all friends are now in the vertex $$$1$$$, the $$$i$$$-th friend must get to his home to the vertex $$$h_i$$$.The evening is about to end and it is time to leave. It turned out that $$$k$$$ ($$$k \le 6$$$) of his friends have no cars, and they would have to walk if no one gives them a ride. One friend with a car can give a ride to any number of friends without cars, but only if he can give them a ride by driving along one of the shortest paths to his house.For example, in the graph below, a friend from vertex $$$h_i=5$$$ can give a ride to friends from the following sets of vertices: $$$[2, 3]$$$, $$$[2, 4]$$$, $$$[2]$$$, $$$[3]$$$, $$$[4]$$$, but can't give a ride to friend from vertex $$$6$$$ or a set $$$[3, 4]$$$.    The vertices where friends without cars live are highlighted in green, and with cars — in red. Kirill wants as few friends as possible to have to walk. Help him find the minimum possible number.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"395"During the loading of the game ""Dungeons and Candies"" you are required to get descriptions of k levels from the server. Each description is a map of an n × m checkered rectangular field. Some cells of the field contain candies (each cell has at most one candy). An empty cell is denoted as ""."" on the map, but if a cell has a candy, it is denoted as a letter of the English alphabet. A level may contain identical candies, in this case the letters in the corresponding cells of the map will be the same.  When you transmit information via a network, you want to minimize traffic — the total size of the transferred data. The levels can be transmitted in any order. There are two ways to transmit the current level A:  You can transmit the whole level A. Then you need to transmit n·m bytes via the network.  You can transmit the difference between level A and some previously transmitted level B (if it exists); this operation requires to transmit dA, B·w bytes, where dA, B is the number of cells of the field that are different for A and B, and w is a constant. Note, that you should compare only the corresponding cells of levels A and B to calculate dA, B. You cannot transform the maps of levels, i.e. rotate or shift them relatively to each other. Your task is to find a way to transfer all the k levels and minimize the traffic.","['graphs', 'trees']","[0, 0, 1, 0, 0, 0, 0, 1]"396"You have a board represented as a grid with $$$2 \times n$$$ cells.The first $$$k_1$$$ cells on the first row and first $$$k_2$$$ cells on the second row are colored in white. All other cells are colored in black.You have $$$w$$$ white dominoes ($$$2 \times 1$$$ tiles, both cells are colored in white) and $$$b$$$ black dominoes ($$$2 \times 1$$$ tiles, both cells are colored in black).You can place a white domino on the board if both board's cells are white and not occupied by any other domino. In the same way, you can place a black domino if both cells are black and not occupied by any other domino.Can you place all $$$w + b$$$ dominoes on the board if you can place dominoes both horizontally and vertically?",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"397"Alice has a cute cat. To keep her cat fit, Alice wants to design an exercising walk for her cat! Initially, Alice's cat is located in a cell $$$(x,y)$$$ of an infinite grid. According to Alice's theory, cat needs to move:   exactly $$$a$$$ steps left: from $$$(u,v)$$$ to $$$(u-1,v)$$$;  exactly $$$b$$$ steps right: from $$$(u,v)$$$ to $$$(u+1,v)$$$;  exactly $$$c$$$ steps down: from $$$(u,v)$$$ to $$$(u,v-1)$$$;  exactly $$$d$$$ steps up: from $$$(u,v)$$$ to $$$(u,v+1)$$$. Note that the moves can be performed in an arbitrary order. For example, if the cat has to move $$$1$$$ step left, $$$3$$$ steps right and $$$2$$$ steps down, then the walk right, down, left, right, right, down is valid.Alice, however, is worrying that her cat might get lost if it moves far away from her. So she hopes that her cat is always in the area $$$[x_1,x_2]\times [y_1,y_2]$$$, i.e. for every cat's position $$$(u,v)$$$ of a walk $$$x_1 \le u \le x_2$$$ and $$$y_1 \le v \le y_2$$$ holds.Also, note that the cat can visit the same cell multiple times.Can you help Alice find out if there exists a walk satisfying her wishes?Formally, the walk should contain exactly $$$a+b+c+d$$$ unit moves ($$$a$$$ to the left, $$$b$$$ to the right, $$$c$$$ to the down, $$$d$$$ to the up). Alice can do the moves in any order. Her current position $$$(u, v)$$$ should always satisfy the constraints: $$$x_1 \le u \le x_2$$$, $$$y_1 \le v \le y_2$$$. The staring point is $$$(x, y)$$$.You are required to answer $$$t$$$ test cases independently.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"398"Ivan wants to make a necklace as a present to his beloved girl. A necklace is a cyclic sequence of beads of different colors. Ivan says that necklace is beautiful relative to the cut point between two adjacent beads, if the chain of beads remaining after this cut is a palindrome (reads the same forward and backward).  Ivan has beads of n colors. He wants to make a necklace, such that it's beautiful relative to as many cuts as possible. He certainly wants to use all the beads. Help him to make the most beautiful necklace.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"399"You are given four integers $$$n$$$, $$$c_0$$$, $$$c_1$$$ and $$$h$$$ and a binary string $$$s$$$ of length $$$n$$$.A binary string is a string consisting of characters $$$0$$$ and $$$1$$$.You can change any character of the string $$$s$$$ (the string should be still binary after the change). You should pay $$$h$$$ coins for each change.After some changes (possibly zero) you want to buy the string. To buy the string you should buy all its characters. To buy the character $$$0$$$ you should pay $$$c_0$$$ coins, to buy the character $$$1$$$ you should pay $$$c_1$$$ coins.Find the minimum number of coins needed to buy the string.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"400"You are given two strings s and t consisting of small Latin letters, string s can also contain '?' characters. Suitability of string s is calculated by following metric:Any two letters can be swapped positions, these operations can be performed arbitrary number of times over any pair of positions. Among all resulting strings s, you choose the one with the largest number of non-intersecting occurrences of string t. Suitability is this number of occurrences.You should replace all '?' characters with small Latin letters in such a way that the suitability of string s is maximal.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"401"Omkar is creating a mosaic using colored square tiles, which he places in an $$$n \times n$$$ grid. When the mosaic is complete, each cell in the grid will have either a glaucous or sinoper tile. However, currently he has only placed tiles in some cells. A completed mosaic will be a mastapeece if and only if each tile is adjacent to exactly $$$2$$$ tiles of the same color ($$$2$$$ tiles are adjacent if they share a side.) Omkar wants to fill the rest of the tiles so that the mosaic becomes a mastapeece. Now he is wondering, is the way to do this unique, and if it is, what is it?",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"402"The developers of Looksery have to write an efficient algorithm that detects faces on a picture. Unfortunately, they are currently busy preparing a contest for you, so you will have to do it for them. In this problem an image is a rectangular table that consists of lowercase Latin letters. A face on the image is a 2 × 2 square, such that from the four letters of this square you can make word ""face"". You need to write a program that determines the number of faces on the image. The squares that correspond to the faces can overlap.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"403"Vova had a pretty weird sleeping schedule. There are $$$h$$$ hours in a day. Vova will sleep exactly $$$n$$$ times. The $$$i$$$-th time he will sleep exactly after $$$a_i$$$ hours from the time he woke up. You can assume that Vova woke up exactly at the beginning of this story (the initial time is $$$0$$$). Each time Vova sleeps exactly one day (in other words, $$$h$$$ hours).Vova thinks that the $$$i$$$-th sleeping time is good if he starts to sleep between hours $$$l$$$ and $$$r$$$ inclusive.Vova can control himself and before the $$$i$$$-th time can choose between two options: go to sleep after $$$a_i$$$ hours or after $$$a_i - 1$$$ hours.Your task is to say the maximum number of good sleeping times Vova can obtain if he acts optimally.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"404"You have an array $$$a$$$ consisting of $$$n$$$ positive integers and you have to handle $$$q$$$ queries of the following types:  $$$1$$$ $$$i$$$ $$$x$$$: change $$$a_{i}$$$ to $$$x$$$,  $$$2$$$ $$$l$$$ $$$r$$$ $$$k$$$: check if the number of occurrences of every positive integer in the subarray $$$a_{l}, a_{l+1}, \ldots a_{r}$$$ is a multiple of $$$k$$$ (check the example for better understanding). ",['probabilities'],"[0, 0, 0, 0, 0, 1, 0, 0]"405"""Duel!""Betting on the lovely princess Claris, the duel between Tokitsukaze and Quailty has started.There are $$$n$$$ cards in a row. Each card has two sides, one of which has color. At first, some of these cards are with color sides facing up and others are with color sides facing down. Then they take turns flipping cards, in which Tokitsukaze moves first. In each move, one should choose exactly $$$k$$$ consecutive cards and flip them to the same side, which means to make their color sides all face up or all face down. If all the color sides of these $$$n$$$ cards face the same direction after one's move, the one who takes this move will win.Princess Claris wants to know who will win the game if Tokitsukaze and Quailty are so clever that they won't make mistakes.",['games'],"[1, 0, 0, 0, 0, 0, 0, 0]"406"Olya loves energy drinks. She loves them so much that her room is full of empty cans from energy drinks.Formally, her room can be represented as a field of n × m cells, each cell of which is empty or littered with cans.Olya drank a lot of energy drink, so now she can run k meters per second. Each second she chooses one of the four directions (up, down, left or right) and runs from 1 to k meters in this direction. Of course, she can only run through empty cells.Now Olya needs to get from cell (x1, y1) to cell (x2, y2). How many seconds will it take her if she moves optimally?It's guaranteed that cells (x1, y1) and (x2, y2) are empty. These cells can coincide.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"407"You are given $$$n$$$ points with integer coordinates on a coordinate axis $$$OX$$$. The coordinate of the $$$i$$$-th point is $$$x_i$$$. All points' coordinates are distinct and given in strictly increasing order.For each point $$$i$$$, you can do the following operation no more than once: take this point and move it by $$$1$$$ to the left or to the right (i..e., you can change its coordinate $$$x_i$$$ to $$$x_i - 1$$$ or to $$$x_i + 1$$$). In other words, for each point, you choose (separately) its new coordinate. For the $$$i$$$-th point, it can be either $$$x_i - 1$$$, $$$x_i$$$ or $$$x_i + 1$$$.Your task is to determine if you can move some points as described above in such a way that the new set of points forms a consecutive segment of integers, i. e. for some integer $$$l$$$ the coordinates of points should be equal to $$$l, l + 1, \ldots, l + n - 1$$$.Note that the resulting points should have distinct coordinates.You have to answer $$$t$$$ independent test cases.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"408"Bike is interested in permutations. A permutation of length n is an integer sequence such that each integer from 0 to (n - 1) appears exactly once in it. For example, [0, 2, 1] is a permutation of length 3 while both [0, 2, 2] and [1, 2, 3] is not.A permutation triple of permutations of length n (a, b, c) is called a Lucky Permutation Triple if and only if . The sign ai denotes the i-th element of permutation a. The modular equality described above denotes that the remainders after dividing ai + bi by n and dividing ci by n are equal.Now, he has an integer n and wants to find a Lucky Permutation Triple. Could you please help him?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"409"Bob decided to take a break from calculus homework and designed a game for himself. The game is played on a sequence of piles of stones, which can be described with a sequence of integers $$$s_1, \ldots, s_k$$$, where $$$s_i$$$ is the number of stones in the $$$i$$$-th pile. On each turn, Bob picks a pair of non-empty adjacent piles $$$i$$$ and $$$i+1$$$ and takes one stone from each. If a pile becomes empty, its adjacent piles do not become adjacent. The game ends when Bob can't make turns anymore. Bob considers himself a winner if at the end all piles are empty.We consider a sequence of piles winning if Bob can start with it and win with some sequence of moves.You are given a sequence $$$a_1, \ldots, a_n$$$, count the number of subsegments of $$$a$$$ that describe a winning sequence of piles. In other words find the number of segments $$$[l, r]$$$ ($$$1 \leq l \leq r \leq n$$$), such that the sequence $$$a_l, a_{l+1}, \ldots, a_r$$$ is winning.",['games'],"[1, 0, 0, 0, 0, 0, 0, 0]"410"You are given four integers $$$n$$$, $$$B$$$, $$$x$$$ and $$$y$$$. You should build a sequence $$$a_0, a_1, a_2, \dots, a_n$$$ where $$$a_0 = 0$$$ and for each $$$i \ge 1$$$ you can choose:   either $$$a_i = a_{i - 1} + x$$$  or $$$a_i = a_{i - 1} - y$$$. Your goal is to build such a sequence $$$a$$$ that $$$a_i \le B$$$ for all $$$i$$$ and $$$\sum\limits_{i=0}^{n}{a_i}$$$ is maximum possible.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"411"There are n piles of stones of sizes a1, a2, ..., an lying on the table in front of you.During one move you can take one pile and add it to the other. As you add pile i to pile j, the size of pile j increases by the current size of pile i, and pile i stops existing. The cost of the adding operation equals the size of the added pile.Your task is to determine the minimum cost at which you can gather all stones in one pile. To add some challenge, the stone piles built up conspiracy and decided that each pile will let you add to it not more than k times (after that it can only be added to another pile). Moreover, the piles decided to puzzle you completely and told you q variants (not necessarily distinct) of what k might equal. Your task is to find the minimum cost for each of q variants.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"412"In some country live wizards. They love playing with numbers. The blackboard has two numbers written on it — a and b. The order of the numbers is not important. Let's consider a ≤ b for the sake of definiteness. The players can cast one of the two spells in turns:  Replace b with b - ak. Number k can be chosen by the player, considering the limitations that k > 0 and b - ak ≥ 0. Number k is chosen independently each time an active player casts a spell.  Replace b with b mod a. If a > b, similar moves are possible.If at least one of the numbers equals zero, a player can't make a move, because taking a remainder modulo zero is considered somewhat uncivilized, and it is far too boring to subtract a zero. The player who cannot make a move, loses.To perform well in the magic totalizator, you need to learn to quickly determine which player wins, if both wizards play optimally: the one that moves first or the one that moves second.","['games', 'math']","[1, 0, 0, 1, 0, 0, 0, 0]"413"A bitstring is a string that contains only the characters 0 and 1.Koyomi Kanou is working hard towards her dream of becoming a writer. To practice, she decided to participate in the Binary Novel Writing Contest. The writing prompt for the contest consists of three bitstrings of length $$$2n$$$. A valid novel for the contest is a bitstring of length at most $$$3n$$$ that contains at least two of the three given strings as subsequences.Koyomi has just received the three prompt strings from the contest organizers. Help her write a valid novel for the contest.A string $$$a$$$ is a subsequence of a string $$$b$$$ if $$$a$$$ can be obtained from $$$b$$$ by deletion of several (possibly, zero) characters.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"414"Programmer Rostislav got seriously interested in the Link/Cut Tree data structure, which is based on Splay trees. Specifically, he is now studying the expose procedure.Unfortunately, Rostislav is unable to understand the definition of this procedure, so he decided to ask programmer Serezha to help him. Serezha agreed to help if Rostislav solves a simple task (and if he doesn't, then why would he need Splay trees anyway?)Given integers l, r and k, you need to print all powers of number k within range from l to r inclusive. However, Rostislav doesn't want to spent time doing this, as he got interested in playing a network game called Agar with Gleb. Help him!",[],"[0, 0, 0, 0, 0, 0, 0, 0]"415"There are $$$n$$$ bricks numbered from $$$1$$$ to $$$n$$$. Brick $$$i$$$ has a weight of $$$a_i$$$.Pak Chanek has $$$3$$$ bags numbered from $$$1$$$ to $$$3$$$ that are initially empty. For each brick, Pak Chanek must put it into one of the bags. After this, each bag must contain at least one brick.After Pak Chanek distributes the bricks, Bu Dengklek will take exactly one brick from each bag. Let $$$w_j$$$ be the weight of the brick Bu Dengklek takes from bag $$$j$$$. The score is calculated as $$$|w_1 - w_2| + |w_2 - w_3|$$$, where $$$|x|$$$ denotes the absolute value of $$$x$$$.It is known that Bu Dengklek will take the bricks in such a way that minimises the score. What is the maximum possible final score if Pak Chanek distributes the bricks optimally?",['games'],"[1, 0, 0, 0, 0, 0, 0, 0]"416"This is an interactive problem.The jury has a permutation $$$p$$$ of length $$$n$$$ and wants you to guess it. For this, the jury created another permutation $$$q$$$ of length $$$n$$$. Initially, $$$q$$$ is an identity permutation ($$$q_i = i$$$ for all $$$i$$$).You can ask queries to get $$$q_i$$$ for any $$$i$$$ you want. After each query, the jury will change $$$q$$$ in the following way:   At first, the jury will create a new permutation $$$q'$$$ of length $$$n$$$ such that $$$q'_i = q_{p_i}$$$ for all $$$i$$$.  Then the jury will replace permutation $$$q$$$ with pemutation $$$q'$$$. You can make no more than $$$2n$$$ queries in order to quess $$$p$$$.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"417"You are given a rooted tree. Each vertex contains $$$a_i$$$ tons of gold, which costs $$$c_i$$$ per one ton. Initially, the tree consists only a root numbered $$$0$$$ with $$$a_0$$$ tons of gold and price $$$c_0$$$ per ton.There are $$$q$$$ queries. Each query has one of two types:   Add vertex $$$i$$$ (where $$$i$$$ is an index of query) as a son to some vertex $$$p_i$$$; vertex $$$i$$$ will have $$$a_i$$$ tons of gold with $$$c_i$$$ per ton. It's guaranteed that $$$c_i > c_{p_i}$$$.  For a given vertex $$$v_i$$$ consider the simple path from $$$v_i$$$ to the root. We need to purchase $$$w_i$$$ tons of gold from vertices on this path, spending the minimum amount of money. If there isn't enough gold on the path, we buy all we can. If we buy $$$x$$$ tons of gold in some vertex $$$v$$$ the remaining amount of gold in it decreases by $$$x$$$ (of course, we can't buy more gold that vertex has at the moment). For each query of the second type, calculate the resulting amount of gold we bought and the amount of money we should spend.Note that you should solve the problem in online mode. It means that you can't read the whole input at once. You can read each query only after writing the answer for the last query, so don't forget to flush output after printing answers. You can use functions like fflush(stdout) in C++ and BufferedWriter.flush in Java or similar after each writing in your program. In standard (if you don't tweak I/O), endl flushes cout in C++ and System.out.println in Java (or println in Kotlin) makes automatic flush as well. ",['trees'],"[0, 0, 0, 0, 0, 0, 0, 1]"418"The only difference between easy and hard versions is constraints.Ivan plays a computer game that contains some microtransactions to make characters look cooler. Since Ivan wants his character to be really cool, he wants to use some of these microtransactions — and he won't start playing until he gets all of them.Each day (during the morning) Ivan earns exactly one burle.There are $$$n$$$ types of microtransactions in the game. Each microtransaction costs $$$2$$$ burles usually and $$$1$$$ burle if it is on sale. Ivan has to order exactly $$$k_i$$$ microtransactions of the $$$i$$$-th type (he orders microtransactions during the evening).Ivan can order any (possibly zero) number of microtransactions of any types during any day (of course, if he has enough money to do it). If the microtransaction he wants to order is on sale then he can buy it for $$$1$$$ burle and otherwise he can buy it for $$$2$$$ burles.There are also $$$m$$$ special offers in the game shop. The $$$j$$$-th offer $$$(d_j, t_j)$$$ means that microtransactions of the $$$t_j$$$-th type are on sale during the $$$d_j$$$-th day.Ivan wants to order all microtransactions as soon as possible. Your task is to calculate the minimum day when he can buy all microtransactions he want and actually start playing.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"419"A progress bar is an element of graphical interface that displays the progress of a process for this very moment before it is completed. Let's take a look at the following form of such a bar.A bar is represented as n squares, located in line. To add clarity, let's number them with positive integers from 1 to n from the left to the right. Each square has saturation (ai for the i-th square), which is measured by an integer from 0 to k. When the bar for some i (1 ≤ i ≤ n) is displayed, squares 1, 2, ... , i - 1 has the saturation k, squares i + 1, i + 2, ... , n has the saturation 0, and the saturation of the square i can have any value from 0 to k.So some first squares of the progress bar always have the saturation k. Some last squares always have the saturation 0. And there is no more than one square that has the saturation different from 0 and k.The degree of the process's completion is measured in percents. Let the process be t% completed. Then the following inequation is fulfilled: An example of such a bar can be seen on the picture.  For the given n, k, t determine the measures of saturation for all the squares ai of the progress bar.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"420"There is a grid with $$$n$$$ rows and $$$m$$$ columns. Some cells are colored black, and the rest of the cells are colored white.In one operation, you can select some black cell and do exactly one of the following:   color all cells in its row black, or  color all cells in its column black. You are given two integers $$$r$$$ and $$$c$$$. Find the minimum number of operations required to make the cell in row $$$r$$$ and column $$$c$$$ black, or determine that it is impossible.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"421"As a big fan of Formula One, Charlie is really happy with the fact that he has to organize ticket sells for the next Grand Prix race in his own city. Unfortunately, the finacial crisis is striking everywhere and all the banknotes left in his country are valued either 10 euros or 20 euros. The price of all tickets for the race is 10 euros, so whenever someone comes to the ticket store only with 20 euro banknote Charlie must have a 10 euro banknote to give them change. Charlie realize that with the huge deficit of banknotes this could be a problem. Charlie has some priceless information but couldn't make use of it, so he needs your help. Exactly n + m people will come to buy a ticket. n of them will have only a single 10 euro banknote, and m of them will have only a single 20 euro banknote. Currently Charlie has k 10 euro banknotes, which he can use for change if needed. All n + m people will come to the ticket store in random order, all orders are equiprobable. Return the probability that the ticket selling process will run smoothly, i.e. Charlie will have change for every person with 20 euro banknote.","['math', 'probabilities']","[0, 0, 0, 1, 0, 1, 0, 0]"422"Let's call an array $$$a_1, a_2, \dots, a_m$$$ of nonnegative integer numbers good if $$$a_1 + a_2 + \dots + a_m = 2\cdot(a_1 \oplus a_2 \oplus \dots \oplus a_m)$$$, where $$$\oplus$$$ denotes the bitwise XOR operation.For example, array $$$[1, 2, 3, 6]$$$ is good, as $$$1 + 2 + 3 + 6 = 12 = 2\cdot 6 = 2\cdot (1\oplus 2 \oplus 3 \oplus 6)$$$. At the same time, array $$$[1, 2, 1, 3]$$$ isn't good, as $$$1 + 2 + 1 + 3 = 7 \neq 2\cdot 1 = 2\cdot(1\oplus 2 \oplus 1 \oplus 3)$$$.You are given an array of length $$$n$$$: $$$a_1, a_2, \dots, a_n$$$. Append at most $$$3$$$ elements to it to make it good. Appended elements don't have to be different. It can be shown that the solution always exists under the given constraints. If there are different solutions, you are allowed to output any of them. Note that you don't have to minimize the number of added elements!. So, if an array is good already you are allowed to not append elements.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"423"Vasya takes part in the orienteering competition. There are n checkpoints located along the line at coordinates x1, x2, ..., xn. Vasya starts at the point with coordinate a. His goal is to visit at least n - 1 checkpoint in order to finish the competition. Participant are allowed to visit checkpoints in arbitrary order.Vasya wants to pick such checkpoints and the order of visiting them that the total distance travelled is minimized. He asks you to calculate this minimum possible value.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"424"You are given an image, that can be represented with a 2-d n by m grid of pixels. Each pixel of the image is either on or off, denoted by the characters ""0"" or ""1"", respectively. You would like to compress this image. You want to choose an integer k > 1 and split the image into k by k blocks. If n and m are not divisible by k, the image is padded with only zeros on the right and bottom so that they are divisible by k. Each pixel in each individual block must have the same value. The given image may not be compressible in its current state. Find the minimum number of pixels you need to toggle (after padding) in order for the image to be compressible for some k. More specifically, the steps are to first choose k, then the image is padded with zeros, then, we can toggle the pixels so it is compressible for this k. The image must be compressible in that state.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"425"You are given two binary square matrices $$$a$$$ and $$$b$$$ of size $$$n \times n$$$. A matrix is called binary if each of its elements is equal to $$$0$$$ or $$$1$$$. You can do the following operations on the matrix $$$a$$$ arbitrary number of times (0 or more):   vertical xor. You choose the number $$$j$$$ ($$$1 \le j \le n$$$) and for all $$$i$$$ ($$$1 \le i \le n$$$) do the following: $$$a_{i, j} := a_{i, j} \oplus 1$$$ ($$$\oplus$$$ — is the operation xor (exclusive or)).  horizontal xor. You choose the number $$$i$$$ ($$$1 \le i \le n$$$) and for all $$$j$$$ ($$$1 \le j \le n$$$) do the following: $$$a_{i, j} := a_{i, j} \oplus 1$$$. Note that the elements of the $$$a$$$ matrix change after each operation.For example, if $$$n=3$$$ and the matrix $$$a$$$ is: $$$$$$ \begin{pmatrix} 1 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 0 \end{pmatrix} $$$$$$ Then the following sequence of operations shows an example of transformations:   vertical xor, $$$j=1$$$. $$$$$$ a= \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} $$$$$$  horizontal xor, $$$i=2$$$. $$$$$$ a= \begin{pmatrix} 0 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 0 \end{pmatrix} $$$$$$  vertical xor, $$$j=2$$$. $$$$$$ a= \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} $$$$$$ Check if there is a sequence of operations such that the matrix $$$a$$$ becomes equal to the matrix $$$b$$$.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"426The winter in Berland lasts n days. For each day we know the forecast for the average air temperature that day. Vasya has a new set of winter tires which allows him to drive safely no more than k days at any average air temperature. After k days of using it (regardless of the temperature of these days) the set of winter tires wears down and cannot be used more. It is not necessary that these k days form a continuous segment of days.Before the first winter day Vasya still uses summer tires. It is possible to drive safely on summer tires any number of days when the average air temperature is non-negative. It is impossible to drive on summer tires at days when the average air temperature is negative. Vasya can change summer tires to winter tires and vice versa at the beginning of any day.Find the minimum number of times Vasya needs to change summer tires to winter tires and vice versa to drive safely during the winter. At the end of the winter the car can be with any set of tires.,[],"[0, 0, 0, 0, 0, 0, 0, 0]"427"Vasya follows a basketball game and marks the distances from which each team makes a throw. He knows that each successful throw has value of either 2 or 3 points. A throw is worth 2 points if the distance it was made from doesn't exceed some value of d meters, and a throw is worth 3 points if the distance is larger than d meters, where d is some non-negative integer.Vasya would like the advantage of the points scored by the first team (the points of the first team minus the points of the second team) to be maximum. For that he can mentally choose the value of d. Help him to do that.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"428"You are given $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$. Find the maximum value of $$$max(a_l, a_{l + 1}, \ldots, a_r) \cdot min(a_l, a_{l + 1}, \ldots, a_r)$$$ over all pairs $$$(l, r)$$$ of integers for which $$$1 \le l < r \le n$$$.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"429"The subsequence is a sequence that can be derived from another sequence by deleting some elements without changing the order of the remaining elements.You are given an integer $$$n$$$. You have to find a sequence $$$s$$$ consisting of digits $$$\{1, 3, 7\}$$$ such that it has exactly $$$n$$$ subsequences equal to $$$1337$$$.For example, sequence $$$337133377$$$ has $$$6$$$ subsequences equal to $$$1337$$$:   $$$337\underline{1}3\underline{3}\underline{3}7\underline{7}$$$ (you can remove the second and fifth characters);  $$$337\underline{1}\underline{3}3\underline{3}7\underline{7}$$$ (you can remove the third and fifth characters);  $$$337\underline{1}\underline{3}\underline{3}37\underline{7}$$$ (you can remove the fourth and fifth characters);  $$$337\underline{1}3\underline{3}\underline{3}\underline{7}7$$$ (you can remove the second and sixth characters);  $$$337\underline{1}\underline{3}3\underline{3}\underline{7}7$$$ (you can remove the third and sixth characters);  $$$337\underline{1}\underline{3}\underline{3}3\underline{7}7$$$ (you can remove the fourth and sixth characters). Note that the length of the sequence $$$s$$$ must not exceed $$$10^5$$$.You have to answer $$$t$$$ independent queries.","['math', 'strings']","[0, 0, 0, 1, 0, 0, 1, 0]"430"The game of Berland poker is played with a deck of $$$n$$$ cards, $$$m$$$ of which are jokers. $$$k$$$ players play this game ($$$n$$$ is divisible by $$$k$$$).At the beginning of the game, each player takes $$$\frac{n}{k}$$$ cards from the deck (so each card is taken by exactly one player). The player who has the maximum number of jokers is the winner, and he gets the number of points equal to $$$x - y$$$, where $$$x$$$ is the number of jokers in the winner's hand, and $$$y$$$ is the maximum number of jokers among all other players. If there are two or more players with maximum number of jokers, all of them are winners and they get $$$0$$$ points.Here are some examples:  $$$n = 8$$$, $$$m = 3$$$, $$$k = 2$$$. If one player gets $$$3$$$ jokers and $$$1$$$ plain card, and another player gets $$$0$$$ jokers and $$$4$$$ plain cards, then the first player is the winner and gets $$$3 - 0 = 3$$$ points;  $$$n = 4$$$, $$$m = 2$$$, $$$k = 4$$$. Two players get plain cards, and the other two players get jokers, so both of them are winners and get $$$0$$$ points;  $$$n = 9$$$, $$$m = 6$$$, $$$k = 3$$$. If the first player gets $$$3$$$ jokers, the second player gets $$$1$$$ joker and $$$2$$$ plain cards, and the third player gets $$$2$$$ jokers and $$$1$$$ plain card, then the first player is the winner, and he gets $$$3 - 2 = 1$$$ point;  $$$n = 42$$$, $$$m = 0$$$, $$$k = 7$$$. Since there are no jokers, everyone gets $$$0$$$ jokers, everyone is a winner, and everyone gets $$$0$$$ points. Given $$$n$$$, $$$m$$$ and $$$k$$$, calculate the maximum number of points a player can get for winning the game.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"431"Omkar has received a message from Anton saying ""Your story for problem A is confusing. Just make a formal statement."" Because of this, Omkar gives you an array $$$a = [a_1, a_2, \ldots, a_n]$$$ of $$$n$$$ distinct integers. An array $$$b = [b_1, b_2, \ldots, b_k]$$$ is called nice if for any two distinct elements $$$b_i, b_j$$$ of $$$b$$$, $$$|b_i-b_j|$$$ appears in $$$b$$$ at least once. In addition, all elements in $$$b$$$ must be distinct. Can you add several (maybe, $$$0$$$) integers to $$$a$$$ to create a nice array $$$b$$$ of size at most $$$300$$$? If $$$a$$$ is already nice, you don't have to add any elements.For example, array $$$[3, 6, 9]$$$ is nice, as $$$|6-3|=|9-6| = 3$$$, which appears in the array, and $$$|9-3| = 6$$$, which appears in the array, while array $$$[4, 2, 0, 6, 9]$$$ is not nice, as $$$|9-4| = 5$$$ is not present in the array.For integers $$$x$$$ and $$$y$$$, $$$|x-y| = x-y$$$ if $$$x > y$$$ and $$$|x-y| = y-x$$$ otherwise.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"432"You are given an array $$$a$$$ consisting of $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$. Friends asked you to make the greatest common divisor (GCD) of all numbers in the array equal to $$$1$$$. In one operation, you can do the following: Select an arbitrary index in the array $$$1 \leq i \leq n$$$; Make $$$a_i = \gcd(a_i, i)$$$, where $$$\gcd(x, y)$$$ denotes the GCD of integers $$$x$$$ and $$$y$$$. The cost of such an operation is $$$n - i + 1$$$.You need to find the minimum total cost of operations we need to perform so that the GCD of the all array numbers becomes equal to $$$1$$$.","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"433"You have received data from a Bubble bot. You know your task is to make factory facilities, but before you even start, you need to know how big the factory is and how many rooms it has. When you look at the data you see that you have the dimensions of the construction, which is in rectangle shape: N x M. Then in the next N lines you have M numbers. These numbers represent factory tiles and they can go from 0 to 15. Each of these numbers should be looked in its binary form. Because from each number you know on which side the tile has walls. For example number 10 in it's binary form is 1010, which means that it has a wall from the North side, it doesn't have a wall from the East, it has a wall on the South side and it doesn't have a wall on the West side. So it goes North, East, South, West. It is guaranteed that the construction always has walls on it's edges. The input will be correct. Your task is to print the size of the rooms from biggest to smallest. ",[],"[0, 0, 0, 0, 0, 0, 0, 0]"434"The sequence $$$a$$$ is sent over the network as follows:  sequence $$$a$$$ is split into segments (each element of the sequence belongs to exactly one segment, each segment is a group of consecutive elements of sequence);  for each segment, its length is written next to it, either to the left of it or to the right of it;  the resulting sequence $$$b$$$ is sent over the network. For example, we needed to send the sequence $$$a = [1, 2, 3, 1, 2, 3]$$$. Suppose it was split into segments as follows: $$$[\color{red}{1}] + [\color{blue}{2, 3, 1}] + [\color{green}{2, 3}]$$$. Then we could have the following sequences:   $$$b = [1, \color{red}{1}, 3, \color{blue}{2, 3, 1}, \color{green}{2, 3}, 2]$$$,  $$$b = [\color{red}{1}, 1, 3, \color{blue}{2, 3, 1}, 2, \color{green}{2, 3}]$$$,  $$$b = [\color{red}{1}, 1, \color{blue}{2, 3, 1}, 3, 2, \color{green}{2, 3}]$$$,  $$$b = [\color{red}{1}, 1,\color{blue}{2, 3, 1}, 3, \color{green}{2, 3}, 2]$$$. If a different segmentation had been used, the sent sequence might have been different.The sequence $$$b$$$ is given. Could the sequence $$$b$$$ be sent over the network? In other words, is there such a sequence $$$a$$$ that converting $$$a$$$ to send it over the network could result in a sequence $$$b$$$?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"435"You are given array $$$a$$$ of length $$$n$$$. You can choose one segment $$$[l, r]$$$ ($$$1 \le l \le r \le n$$$) and integer value $$$k$$$ (positive, negative or even zero) and change $$$a_l, a_{l + 1}, \dots, a_r$$$ by $$$k$$$ each (i.e. $$$a_i := a_i + k$$$ for each $$$l \le i \le r$$$).What is the maximum possible number of elements with value $$$c$$$ that can be obtained after one such operation?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"436"Musicians of a popular band ""Flayer"" have announced that they are going to ""make their exit"" with a world tour. Of course, they will visit Berland as well.There are n cities in Berland. People can travel between cities using two-directional train routes; there are exactly m routes, i-th route can be used to go from city vi to city ui (and from ui to vi), and it costs wi coins to use this route.Each city will be visited by ""Flayer"", and the cost of the concert ticket in i-th city is ai coins.You have friends in every city of Berland, and they, knowing about your programming skills, asked you to calculate the minimum possible number of coins they have to pay to visit the concert. For every city i you have to compute the minimum number of coins a person from city i has to spend to travel to some city j (or possibly stay in city i), attend a concert there, and return to city i (if j ≠ i).Formally, for every  you have to calculate , where d(i, j) is the minimum number of coins you have to spend to travel from city i to city j. If there is no way to reach city j from city i, then we consider d(i, j) to be infinitely large.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"437"Alice and Bob are playing a game. Initially, they are given a non-empty string $$$s$$$, consisting of lowercase Latin letters. The length of the string is even. Each player also has a string of their own, initially empty.Alice starts, then they alternate moves. In one move, a player takes either the first or the last letter of the string $$$s$$$, removes it from $$$s$$$ and prepends (adds to the beginning) it to their own string.The game ends when the string $$$s$$$ becomes empty. The winner is the player with a lexicographically smaller string. If the players' strings are equal, then it's a draw.A string $$$a$$$ is lexicographically smaller than a string $$$b$$$ if there exists such position $$$i$$$ that $$$a_j = b_j$$$ for all $$$j < i$$$ and $$$a_i < b_i$$$.What is the result of the game if both players play optimally (e. g. both players try to win; if they can't, then try to draw)?",['games'],"[1, 0, 0, 0, 0, 0, 0, 0]"438"Once upon a time Petya and Gena gathered after another programming competition and decided to play some game. As they consider most modern games to be boring, they always try to invent their own games. They have only stickers and markers, but that won't stop them.The game they came up with has the following rules. Initially, there are n stickers on the wall arranged in a row. Each sticker has some number written on it. Now they alternate turn, Petya moves first.One move happens as follows. Lets say there are m ≥ 2 stickers on the wall. The player, who makes the current move, picks some integer k from 2 to m and takes k leftmost stickers (removes them from the wall). After that he makes the new sticker, puts it to the left end of the row, and writes on it the new integer, equal to the sum of all stickers he took on this move. Game ends when there is only one sticker left on the wall. The score of the player is equal to the sum of integers written on all stickers he took during all his moves. The goal of each player is to maximize the difference between his score and the score of his opponent.Given the integer n and the initial sequence of stickers on the wall, define the result of the game, i.e. the difference between the Petya's and Gena's score if both players play optimally. ",['games'],"[1, 0, 0, 0, 0, 0, 0, 0]"439"There are $$$n$$$ children numbered from $$$1$$$ to $$$n$$$ in a kindergarten. Kindergarten teacher gave $$$a_i$$$ ($$$1 \leq a_i \leq n$$$) candies to the $$$i$$$-th child. Children were seated in a row in order from $$$1$$$ to $$$n$$$ from left to right and started eating candies. While the $$$i$$$-th child was eating candies, he calculated two numbers $$$l_i$$$ and $$$r_i$$$ — the number of children seating to the left of him that got more candies than he and the number of children seating to the right of him that got more candies than he, respectively.Formally, $$$l_i$$$ is the number of indices $$$j$$$ ($$$1 \leq j < i$$$), such that $$$a_i < a_j$$$ and $$$r_i$$$ is the number of indices $$$j$$$ ($$$i < j \leq n$$$), such that $$$a_i < a_j$$$.Each child told to the kindergarten teacher the numbers $$$l_i$$$ and $$$r_i$$$ that he calculated. Unfortunately, she forgot how many candies she has given to each child. So, she asks you for help: given the arrays $$$l$$$ and $$$r$$$ determine whether she could have given the candies to the children such that all children correctly calculated their values $$$l_i$$$ and $$$r_i$$$, or some of them have definitely made a mistake. If it was possible, find any way how she could have done it.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"440"Alice and Bob are playing a game on a sequence $$$a_1, a_2, \dots, a_n$$$ of length $$$n$$$. They move in turns and Alice moves first.In the turn of each player, he or she should select an integer and remove it from the sequence. The game ends when there is no integer left in the sequence. Alice wins if the sum of her selected integers is even; otherwise, Bob wins. Your task is to determine who will win the game, if both players play optimally.","['games', 'math']","[1, 0, 0, 1, 0, 0, 0, 0]"441"A number is called powerful if it is a power of two or a factorial. In other words, the number $$$m$$$ is powerful if there exists a non-negative integer $$$d$$$ such that $$$m=2^d$$$ or $$$m=d!$$$, where $$$d!=1\cdot 2\cdot \ldots \cdot d$$$ (in particular, $$$0! = 1$$$). For example $$$1$$$, $$$4$$$, and $$$6$$$ are powerful numbers, because $$$1=1!$$$, $$$4=2^2$$$, and $$$6=3!$$$ but $$$7$$$, $$$10$$$, or $$$18$$$ are not.You are given a positive integer $$$n$$$. Find the minimum number $$$k$$$ such that $$$n$$$ can be represented as the sum of $$$k$$$ distinct powerful numbers, or say that there is no such $$$k$$$.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"442"Young wilderness explorers set off to their first expedition led by senior explorer Russell. Explorers went into a forest, set up a camp and decided to split into groups to explore as much interesting locations as possible. Russell was trying to form groups, but ran into some difficulties...Most of the young explorers are inexperienced, and sending them alone would be a mistake. Even Russell himself became senior explorer not long ago. Each of young explorers has a positive integer parameter $$$e_i$$$ — his inexperience. Russell decided that an explorer with inexperience $$$e$$$ can only join the group of $$$e$$$ or more people.Now Russell needs to figure out how many groups he can organize. It's not necessary to include every explorer in one of the groups: some can stay in the camp. Russell is worried about this expedition, so he asked you to help him.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"443"A permutation is a sequence of integers from 1 to n of length n containing each number exactly once. For example, (1), (4, 3, 5, 1, 2), (3, 2, 1) are permutations, and (1, 1), (4, 3, 1), (2, 3, 4) are not. There are many tasks on permutations. Today you are going to solve one of them. Let’s imagine that somebody took several permutations (perhaps, with a different number of elements), wrote them down consecutively as one array and then shuffled the resulting array. The task is to restore the initial permutations if it is possible.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"444"You have a Petri dish with bacteria and you are preparing to dive into the harsh micro-world. But, unfortunately, you don't have any microscope nearby, so you can't watch them.You know that you have $$$n$$$ bacteria in the Petri dish and size of the $$$i$$$-th bacteria is $$$a_i$$$. Also you know intergalactic positive integer constant $$$K$$$.The $$$i$$$-th bacteria can swallow the $$$j$$$-th bacteria if and only if $$$a_i > a_j$$$ and $$$a_i \le a_j + K$$$. The $$$j$$$-th bacteria disappear, but the $$$i$$$-th bacteria doesn't change its size. The bacteria can perform multiple swallows. On each swallow operation any bacteria $$$i$$$ can swallow any bacteria $$$j$$$ if $$$a_i > a_j$$$ and $$$a_i \le a_j + K$$$. The swallow operations go one after another.For example, the sequence of bacteria sizes $$$a=[101, 53, 42, 102, 101, 55, 54]$$$ and $$$K=1$$$. The one of possible sequences of swallows is: $$$[101, 53, 42, 102, \underline{101}, 55, 54]$$$ $$$\to$$$ $$$[101, \underline{53}, 42, 102, 55, 54]$$$ $$$\to$$$ $$$[\underline{101}, 42, 102, 55, 54]$$$ $$$\to$$$ $$$[42, 102, 55, \underline{54}]$$$ $$$\to$$$ $$$[42, 102, 55]$$$. In total there are $$$3$$$ bacteria remained in the Petri dish.Since you don't have a microscope, you can only guess, what the minimal possible number of bacteria can remain in your Petri dish when you finally will find any microscope.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"445"Piegirl was asked to implement two table join operation for distributed database system, minimizing the network traffic.Suppose she wants to join two tables, A and B. Each of them has certain number of rows which are distributed on different number of partitions. Table A is distributed on the first cluster consisting of m partitions. Partition with index i has ai rows from A. Similarly, second cluster containing table B has n partitions, i-th one having bi rows from B. In one network operation she can copy one row from any partition to any other partition. At the end, for each row from A and each row from B there should be a partition that has both rows. Determine the minimal number of network operations to achieve this.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"446"You are given an array $$$a$$$ consisting of $$$500000$$$ integers (numbered from $$$1$$$ to $$$500000$$$). Initially all elements of $$$a$$$ are zero.You have to process two types of queries to this array:  $$$1$$$ $$$x$$$ $$$y$$$ — increase $$$a_x$$$ by $$$y$$$;  $$$2$$$ $$$x$$$ $$$y$$$ — compute $$$\sum\limits_{i \in R(x, y)} a_i$$$, where $$$R(x, y)$$$ is the set of all integers from $$$1$$$ to $$$500000$$$ which have remainder $$$y$$$ modulo $$$x$$$. Can you process all the queries?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"447"You are given two very long integers a, b (leading zeroes are allowed). You should check what number a or b is greater or determine that they are equal.The input size is very large so don't use the reading of symbols one by one. Instead of that use the reading of a whole line or token.As input/output can reach huge size it is recommended to use fast input/output methods: for example, prefer to use scanf/printf instead of cin/cout in C++, prefer to use BufferedReader/PrintWriter instead of Scanner/System.out in Java. Don't use the function input() in Python2 instead of it use the function raw_input().",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"448"British mathematician John Littlewood once said about Indian mathematician Srinivasa Ramanujan that ""every positive integer was one of his personal friends.""It turns out that positive integers can also be friends with each other! You are given an array $$$a$$$ of distinct positive integers. Define a subarray $$$a_i, a_{i+1}, \ldots, a_j$$$ to be a friend group if and only if there exists an integer $$$m \ge 2$$$ such that $$$a_i \bmod m = a_{i+1} \bmod m = \ldots = a_j \bmod m$$$, where $$$x \bmod y$$$ denotes the remainder when $$$x$$$ is divided by $$$y$$$.Your friend Gregor wants to know the size of the largest friend group in $$$a$$$.","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"449"Manao's friends often send him new songs. He never listens to them right away. Instead, he compiles them into a playlist. When he feels that his mind is open to new music, he opens the playlist and starts to listen to the songs.Of course, there are some songs that Manao doesn't particuarly enjoy. To get more pleasure from the received songs, he invented the following procedure of listening to the playlist:  If after listening to some song Manao realizes that he liked it, then he remembers it and starts to listen to the next unlistened song.  If after listening to some song Manao realizes that he did not like it, he listens to all the songs he liked up to this point and then begins to listen to the next unlistened song. For example, if Manao has four songs in the playlist, A, B, C, D (in the corresponding order) and he is going to like songs A and C in the end, then the order of listening is the following:  Manao listens to A, he likes it, he remembers it.  Manao listens to B, he does not like it, so he listens to A, again.  Manao listens to C, he likes the song and he remembers it, too.  Manao listens to D, but does not enjoy it and re-listens to songs A and C. That is, in the end Manao listens to song A three times, to song C twice and songs B and D once. Note that if Manao once liked a song, he will never dislike it on a subsequent listening.Manao has received n songs: the i-th of them is li seconds long and Manao may like it with a probability of pi percents. The songs could get on Manao's playlist in any order, so Manao wants to know the maximum expected value of the number of seconds after which the listening process will be over, for all possible permutations of the songs in the playlist.","['math', 'probabilities']","[0, 0, 0, 1, 0, 1, 0, 0]"450"This is an interactive problem. Refer to the Interaction section below for better understanding.Ithea and Chtholly want to play a game in order to determine who can use the kitchen tonight.Initially, Ithea puts n clear sheets of paper in a line. They are numbered from 1 to n from left to right.This game will go on for m rounds. In each round, Ithea will give Chtholly an integer between 1 and c, and Chtholly needs to choose one of the sheets to write down this number (if there is already a number before, she will erase the original one and replace it with the new one).Chtholly wins if, at any time, all the sheets are filled with a number and the n numbers are in non-decreasing order looking from left to right from sheet 1 to sheet n, and if after m rounds she still doesn't win, she loses the game.Chtholly really wants to win the game as she wants to cook something for Willem. But she doesn't know how to win the game. So Chtholly finds you, and your task is to write a program to receive numbers that Ithea gives Chtholly and help her make the decision on which sheet of paper write this number.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"451"The score of an array $$$v_1,v_2,\ldots,v_n$$$ is defined as the number of indices $$$i$$$ ($$$1 \le i \le n$$$) such that $$$v_1+v_2+\ldots+v_i = 0$$$.You are given an array $$$a_1,a_2,\ldots,a_n$$$ of length $$$n$$$. You can perform the following operation multiple times:  select an index $$$i$$$ ($$$1 \le i \le n$$$) such that $$$a_i=0$$$;  then replace $$$a_i$$$ by an arbitrary integer.  What is the maximum possible score of $$$a$$$ that can be obtained by performing a sequence of such operations?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"452"You are given an array $$$a$$$ consisting of $$$n$$$ integers. You can perform the following operations arbitrary number of times (possibly, zero):  Choose a pair of indices $$$(i, j)$$$ such that $$$|i-j|=1$$$ (indices $$$i$$$ and $$$j$$$ are adjacent) and set $$$a_i := a_i + |a_i - a_j|$$$;  Choose a pair of indices $$$(i, j)$$$ such that $$$|i-j|=1$$$ (indices $$$i$$$ and $$$j$$$ are adjacent) and set $$$a_i := a_i - |a_i - a_j|$$$. The value $$$|x|$$$ means the absolute value of $$$x$$$. For example, $$$|4| = 4$$$, $$$|-3| = 3$$$.Your task is to find the minimum number of operations required to obtain the array of equal elements and print the order of operations to do it.It is guaranteed that you always can obtain the array of equal elements using such operations.Note that after each operation each element of the current array should not exceed $$$10^{18}$$$ by absolute value.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"453"Santa Claus has Robot which lives on the infinite grid and can move along its lines. He can also, having a sequence of m points p1, p2, ..., pm with integer coordinates, do the following: denote its initial location by p0. First, the robot will move from p0 to p1 along one of the shortest paths between them (please notice that since the robot moves only along the grid lines, there can be several shortest paths). Then, after it reaches p1, it'll move to p2, again, choosing one of the shortest ways, then to p3, and so on, until he has visited all points in the given order. Some of the points in the sequence may coincide, in that case Robot will visit that point several times according to the sequence order.While Santa was away, someone gave a sequence of points to Robot. This sequence is now lost, but Robot saved the protocol of its unit movements. Please, find the minimum possible length of the sequence.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"454"You are given two binary strings $$$a$$$ and $$$b$$$ of the same length. You can perform the following two operations on the string $$$a$$$:  Swap any two bits at indices $$$i$$$ and $$$j$$$ respectively ($$$1 \le i, j \le n$$$), the cost of this operation is $$$|i - j|$$$, that is, the absolute difference between $$$i$$$ and $$$j$$$.  Select any arbitrary index $$$i$$$ ($$$1 \le i \le n$$$) and flip (change $$$0$$$ to $$$1$$$ or $$$1$$$ to $$$0$$$) the bit at this index. The cost of this operation is $$$1$$$. Find the minimum cost to make the string $$$a$$$ equal to $$$b$$$. It is not allowed to modify string $$$b$$$.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"455"Lindsey Buckingham told Stevie Nicks ""Go your own way"". Nicks is now sad and wants to go away as quickly as possible, but she lives in a 2D hexagonal world.Consider a hexagonal tiling of the plane as on the picture below.  Nicks wishes to go from the cell marked $$$(0, 0)$$$ to a certain cell given by the coordinates. She may go from a hexagon to any of its six neighbors you want, but there is a cost associated with each of them. The costs depend only on the direction in which you travel. Going from $$$(0, 0)$$$ to $$$(1, 1)$$$ will take the exact same cost as going from $$$(-2, -1)$$$ to $$$(-1, 0)$$$. The costs are given in the input in the order $$$c_1$$$, $$$c_2$$$, $$$c_3$$$, $$$c_4$$$, $$$c_5$$$, $$$c_6$$$ as in the picture below.  Print the smallest cost of a path from the origin which has coordinates $$$(0, 0)$$$ to the given cell.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"456"For a permutation P[1... N] of integers from 1 to N, function f is defined as follows:  Let g(i) be the minimum positive integer j such that f(i, j) = i. We can show such j always exists.For given N, A, B, find a permutation P of integers from 1 to N such that for 1 ≤ i ≤ N, g(i) equals either A or B.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"457"You are given $$$n$$$ segments on a number line; each endpoint of every segment has integer coordinates. Some segments can degenerate to points. Segments can intersect with each other, be nested in each other or even coincide.The intersection of a sequence of segments is such a maximal set of points (not necesserily having integer coordinates) that each point lies within every segment from the sequence. If the resulting set isn't empty, then it always forms some continuous segment. The length of the intersection is the length of the resulting segment or $$$0$$$ in case the intersection is an empty set.For example, the intersection of segments $$$[1;5]$$$ and $$$[3;10]$$$ is $$$[3;5]$$$ (length $$$2$$$), the intersection of segments $$$[1;5]$$$ and $$$[5;7]$$$ is $$$[5;5]$$$ (length $$$0$$$) and the intersection of segments $$$[1;5]$$$ and $$$[6;6]$$$ is an empty set (length $$$0$$$).Your task is to remove exactly one segment from the given sequence in such a way that the intersection of the remaining $$$(n - 1)$$$ segments has the maximal possible length.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"458"In the evening Polycarp decided to analyze his today's travel expenses on public transport.The bus system in the capital of Berland is arranged in such a way that each bus runs along the route between two stops. Each bus has no intermediate stops. So each of the buses continuously runs along the route from one stop to the other and back. There is at most one bus running between a pair of stops.Polycarp made n trips on buses. About each trip the stop where he started the trip and the the stop where he finished are known. The trips follow in the chronological order in Polycarp's notes.It is known that one trip on any bus costs a burles. In case when passenger makes a transshipment the cost of trip decreases to b burles (b < a). A passenger makes a transshipment if the stop on which he boards the bus coincides with the stop where he left the previous bus. Obviously, the first trip can not be made with transshipment.For example, if Polycarp made three consecutive trips: ""BerBank""  ""University"", ""University""  ""BerMall"", ""University""  ""BerBank"", then he payed a + b + a = 2a + b burles. From the BerBank he arrived to the University, where he made transshipment to the other bus and departed to the BerMall. Then he walked to the University and returned to the BerBank by bus.Also Polycarp can buy no more than k travel cards. Each travel card costs f burles. The travel card for a single bus route makes free of charge any trip by this route (in both directions). Once purchased, a travel card can be used any number of times in any direction.What is the smallest amount of money Polycarp could have spent today if he can buy no more than k travel cards?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"459"You are given an array a consisting of positive integers and q queries to this array. There are two types of queries:   1 l r x — for each index i such that l ≤ i ≤ r set ai = x.  2 l r — find the minimum among such ai that l ≤ i ≤ r. We decided that this problem is too easy. So the array a is given in a compressed form: there is an array b consisting of n elements and a number k in the input, and before all queries a is equal to the concatenation of k arrays b (so the size of a is n·k).",[],"[0, 0, 0, 0, 0, 0, 0, 0]"460"You are given an array $$$a_1, a_2, \dots, a_n$$$ and an array $$$b_1, b_2, \dots, b_n$$$.For one operation you can sort in non-decreasing order any subarray $$$a[l \dots r]$$$ of the array $$$a$$$.For example, if $$$a = [4, 2, 2, 1, 3, 1]$$$ and you choose subbarray $$$a[2 \dots 5]$$$, then the array turns into $$$[4, 1, 2, 2, 3, 1]$$$. You are asked to determine whether it is possible to obtain the array $$$b$$$ by applying this operation any number of times (possibly zero) to the array $$$a$$$.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"461"Petya loves lucky numbers. We all know that lucky numbers are the positive integers whose decimal representations contain only the lucky digits 4 and 7. For example, numbers 47, 744, 4 are lucky and 5, 17, 467 are not.Petya wonders eagerly what minimum lucky number has the sum of digits equal to n. Help him cope with the task.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"462"You've gotten an n × m sheet of squared paper. Some of its squares are painted. Let's mark the set of all painted squares as A. Set A is connected. Your task is to find the minimum number of squares that we can delete from set A to make it not connected.A set of painted squares is called connected, if for every two squares a and b from this set there is a sequence of squares from the set, beginning in a and ending in b, such that in this sequence any square, except for the last one, shares a common side with the square that follows next in the sequence. An empty set and a set consisting of exactly one square are connected by definition.","['graphs', 'trees']","[0, 0, 1, 0, 0, 0, 0, 1]"463"Notice: unusual memory limit!After the war, destroyed cities in the neutral zone were restored. And children went back to school.The war changed the world, as well as education. In those hard days, a new math concept was created.As we all know, logarithm function can be described as: $$$$$$ \log(p_1^{a_1}p_2^{a_2}...p_k^{a_2}) = a_1 \log p_1 + a_2 \log p_2 + ... + a_k \log p_k $$$$$$ Where $$$p_1^{a_1}p_2^{a_2}...p_k^{a_2}$$$ is the prime factorization of a integer. A problem is that the function uses itself in the definition. That is why it is hard to calculate.So, the mathematicians from the neutral zone invented this: $$$$$$ \text{exlog}_f(p_1^{a_1}p_2^{a_2}...p_k^{a_2}) = a_1 f(p_1) + a_2 f(p_2) + ... + a_k f(p_k) $$$$$$Notice that $$$\text{exlog}_f(1)$$$ is always equal to $$$0$$$.This concept for any function $$$f$$$ was too hard for children. So teachers told them that $$$f$$$ can only be a polynomial of degree no more than $$$3$$$ in daily uses (i.e., $$$f(x) = Ax^3+Bx^2+Cx+D$$$).""Class is over! Don't forget to do your homework!"" Here it is: $$$$$$ \sum_{i=1}^n \text{exlog}_f(i) $$$$$$Help children to do their homework. Since the value can be very big, you need to find the answer modulo $$$2^{32}$$$.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"464"Childan is making up a legendary story and trying to sell his forgery — a necklace with a strong sense of ""Wu"" to the Kasouras. But Mr. Kasoura is challenging the truth of Childan's story. So he is going to ask a few questions about Childan's so-called ""personal treasure"" necklace.This ""personal treasure"" is a multiset $$$S$$$ of $$$m$$$ ""01-strings"".A ""01-string"" is a string that contains $$$n$$$ characters ""0"" and ""1"". For example, if $$$n=4$$$, strings ""0110"", ""0000"", and ""1110"" are ""01-strings"", but ""00110"" (there are $$$5$$$ characters, not $$$4$$$) and ""zero"" (unallowed characters) are not.Note that the multiset $$$S$$$ can contain equal elements.Frequently, Mr. Kasoura will provide a ""01-string"" $$$t$$$ and ask Childan how many strings $$$s$$$ are in the multiset $$$S$$$ such that the ""Wu"" value of the pair $$$(s, t)$$$ is not greater than $$$k$$$. Mrs. Kasoura and Mr. Kasoura think that if $$$s_i = t_i$$$ ($$$1\leq i\leq n$$$) then the ""Wu"" value of the character pair equals to $$$w_i$$$, otherwise $$$0$$$. The ""Wu"" value of the ""01-string"" pair is the sum of the ""Wu"" values of every character pair. Note that the length of every ""01-string"" is equal to $$$n$$$.For example, if $$$w=[4, 5, 3, 6]$$$, ""Wu"" of (""1001"", ""1100"") is $$$7$$$ because these strings have equal characters only on the first and third positions, so $$$w_1+w_3=4+3=7$$$.You need to help Childan to answer Mr. Kasoura's queries. That is to find the number of strings in the multiset $$$S$$$ such that the ""Wu"" value of the pair is not greater than $$$k$$$.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"465"This is the easy version of the problem. In this version, $$$n \le 3000$$$, $$$x \ge y$$$ holds. You can make hacks only if both versions of the problem are solved.You are given two binary strings $$$a$$$ and $$$b$$$, both of length $$$n$$$. You can do the following operation any number of times (possibly zero).   Select two indices $$$l$$$ and $$$r$$$ ($$$l < r$$$).  Change $$$a_l$$$ to $$$(1 - a_l)$$$, and $$$a_r$$$ to $$$(1 - a_r)$$$.  If $$$l + 1 = r$$$, the cost of the operation is $$$x$$$. Otherwise, the cost is $$$y$$$. You have to find the minimum cost needed to make $$$a$$$ equal to $$$b$$$ or say there is no way to do so.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"466"Easy and hard versions are actually different problems, so read statements of both problems completely and carefully.Summer vacation has started so Alice and Bob want to play and joy, but... Their mom doesn't think so. She says that they have to read some amount of books before all entertainments. Alice and Bob will read each book together to end this exercise faster.There are $$$n$$$ books in the family library. The $$$i$$$-th book is described by three integers: $$$t_i$$$ — the amount of time Alice and Bob need to spend to read it, $$$a_i$$$ (equals $$$1$$$ if Alice likes the $$$i$$$-th book and $$$0$$$ if not), and $$$b_i$$$ (equals $$$1$$$ if Bob likes the $$$i$$$-th book and $$$0$$$ if not).So they need to choose some books from the given $$$n$$$ books in such a way that:  Alice likes at least $$$k$$$ books from the chosen set and Bob likes at least $$$k$$$ books from the chosen set;  the total reading time of these books is minimized (they are children and want to play and joy as soon a possible). The set they choose is the same for both Alice an Bob (it's shared between them) and they read all books together, so the total reading time is the sum of $$$t_i$$$ over all books that are in the chosen set.Your task is to help them and find any suitable set of books or determine that it is impossible to find such a set.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"467"Recently, Mike was very busy with studying for exams and contests. Now he is going to chill a bit by doing some sight seeing in the city.City consists of n intersections numbered from 1 to n. Mike starts walking from his house located at the intersection number 1 and goes along some sequence of intersections. Walking from intersection number i to intersection j requires |i - j| units of energy. The total energy spent by Mike to visit a sequence of intersections p1 = 1, p2, ..., pk is equal to  units of energy.Of course, walking would be boring if there were no shortcuts. A shortcut is a special path that allows Mike walking from one intersection to another requiring only 1 unit of energy. There are exactly n shortcuts in Mike's city, the ith of them allows walking from intersection i to intersection ai (i ≤ ai ≤ ai + 1) (but not in the opposite direction), thus there is exactly one shortcut starting at each intersection. Formally, if Mike chooses a sequence p1 = 1, p2, ..., pk then for each 1 ≤ i < k satisfying pi + 1 = api and api ≠ pi Mike will spend only 1 unit of energy instead of |pi - pi + 1| walking from the intersection pi to intersection pi + 1. For example, if Mike chooses a sequence p1 = 1, p2 = ap1, p3 = ap2, ..., pk = apk - 1, he spends exactly k - 1 units of total energy walking around them.Before going on his adventure, Mike asks you to find the minimum amount of energy required to reach each of the intersections from his home. Formally, for each 1 ≤ i ≤ n Mike is interested in finding minimum possible total energy of some sequence p1 = 1, p2, ..., pk = i.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"468"Consider a sequence of distinct integers $$$a_1, \ldots, a_n$$$, each representing one node of a graph. There is an edge between two nodes if the two values are not coprime, i. e. they have a common divisor greater than $$$1$$$.There are $$$q$$$ queries, in each query, you want to get from one given node $$$a_s$$$ to another $$$a_t$$$. In order to achieve that, you can choose an existing value $$$a_i$$$ and create new value $$$a_{n+1} = a_i \cdot (1 + a_i)$$$, with edges to all values that are not coprime with $$$a_{n+1}$$$. Also, $$$n$$$ gets increased by $$$1$$$. You can repeat that operation multiple times, possibly making the sequence much longer and getting huge or repeated values. What's the minimum possible number of newly created nodes so that $$$a_t$$$ is reachable from $$$a_s$$$?Queries are independent. In each query, you start with the initial sequence $$$a$$$ given in the input.","['graphs', 'math', 'number theory']","[0, 0, 1, 1, 1, 0, 0, 0]"469"There are $$$n$$$ segments drawn on a plane; the $$$i$$$-th segment connects two points ($$$x_{i, 1}$$$, $$$y_{i, 1}$$$) and ($$$x_{i, 2}$$$, $$$y_{i, 2}$$$). Each segment is non-degenerate, and is either horizontal or vertical — formally, for every $$$i \in [1, n]$$$ either $$$x_{i, 1} = x_{i, 2}$$$ or $$$y_{i, 1} = y_{i, 2}$$$ (but only one of these conditions holds). Only segments of different types may intersect: no pair of horizontal segments shares any common points, and no pair of vertical segments shares any common points.We say that four segments having indices $$$h_1$$$, $$$h_2$$$, $$$v_1$$$ and $$$v_2$$$ such that $$$h_1 < h_2$$$ and $$$v_1 < v_2$$$ form a rectangle if the following conditions hold:  segments $$$h_1$$$ and $$$h_2$$$ are horizontal;  segments $$$v_1$$$ and $$$v_2$$$ are vertical;  segment $$$h_1$$$ intersects with segment $$$v_1$$$;  segment $$$h_2$$$ intersects with segment $$$v_1$$$;  segment $$$h_1$$$ intersects with segment $$$v_2$$$;  segment $$$h_2$$$ intersects with segment $$$v_2$$$. Please calculate the number of ways to choose four segments so they form a rectangle. Note that the conditions $$$h_1 < h_2$$$ and $$$v_1 < v_2$$$ should hold.",['geometry'],"[0, 1, 0, 0, 0, 0, 0, 0]"470"You have an array $$$a_1, a_2, \dots, a_n$$$. All $$$a_i$$$ are positive integers.In one step you can choose three distinct indices $$$i$$$, $$$j$$$, and $$$k$$$ ($$$i \neq j$$$; $$$i \neq k$$$; $$$j \neq k$$$) and assign the sum of $$$a_j$$$ and $$$a_k$$$ to $$$a_i$$$, i. e. make $$$a_i = a_j + a_k$$$.Can you make all $$$a_i$$$ lower or equal to $$$d$$$ using the operation above any number of times (possibly, zero)?",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"471"Have you ever used the chat application QQ? Well, in a chat group of QQ, administrators can muzzle a user for days.In Boboniu's chat group, there's a person called Du Yi who likes to make fun of Boboniu every day.Du will chat in the group for $$$n$$$ days. On the $$$i$$$-th day:  If Du can speak, he'll make fun of Boboniu with fun factor $$$a_i$$$. But after that, he may be muzzled depending on Boboniu's mood.  Otherwise, Du won't do anything. Boboniu's mood is a constant $$$m$$$. On the $$$i$$$-th day:  If Du can speak and $$$a_i>m$$$, then Boboniu will be angry and muzzle him for $$$d$$$ days, which means that Du won't be able to speak on the $$$i+1, i+2, \cdots, \min(i+d,n)$$$-th days.  Otherwise, Boboniu won't do anything. The total fun factor is the sum of the fun factors on the days when Du can speak.Du asked you to find the maximum total fun factor among all possible permutations of $$$a$$$.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"472You're given a tree with $$$n$$$ vertices.Your task is to determine the maximum possible number of edges that can be removed in such a way that all the remaining connected components will have even size.,"['graphs', 'trees']","[0, 0, 1, 0, 0, 0, 0, 1]"473"There are many sunflowers in the Garden of the Sun.Garden of the Sun is a rectangular table with $$$n$$$ rows and $$$m$$$ columns, where the cells of the table are farmlands. All of the cells grow a sunflower on it. Unfortunately, one night, the lightning stroke some (possibly zero) cells, and sunflowers on those cells were burned into ashes. In other words, those cells struck by the lightning became empty. Magically, any two empty cells have no common points (neither edges nor corners).Now the owner wants to remove some (possibly zero) sunflowers to reach the following two goals:    When you are on an empty cell, you can walk to any other empty cell. In other words, those empty cells are connected.   There is exactly one simple path between any two empty cells. In other words, there is no cycle among the empty cells. You can walk from an empty cell to another if they share a common edge.Could you please give the owner a solution that meets all her requirements?Note that you are not allowed to plant sunflowers. You don't need to minimize the number of sunflowers you remove. It can be shown that the answer always exists.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"474Jury picked a polynomial $$$f(x) = a_0 + a_1 \cdot x + a_2 \cdot x^2 + \dots + a_k \cdot x^k$$$. $$$k \le 10$$$ and all $$$a_i$$$ are integer numbers and $$$0 \le a_i < 10^6 + 3$$$. It's guaranteed that there is at least one $$$i$$$ such that $$$a_i > 0$$$.Now jury wants you to find such an integer $$$x_0$$$ that $$$f(x_0) \equiv 0 \mod (10^6 + 3)$$$ or report that there is not such $$$x_0$$$.You can ask no more than $$$50$$$ queries: you ask value $$$x_q$$$ and jury tells you value $$$f(x_q) \mod (10^6 + 3)$$$.Note that printing the answer doesn't count as a query.,['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"475"Student Vladislav came to his programming exam completely unprepared as usual. He got a question about some strange algorithm on a graph — something that will definitely never be useful in real life. He asked a girl sitting next to him to lend him some cheat papers for this questions and found there the following definition:The minimum spanning tree T of graph G is such a tree that it contains all the vertices of the original graph G, and the sum of the weights of its edges is the minimum possible among all such trees.Vladislav drew a graph with n vertices and m edges containing no loops and multiple edges. He found one of its minimum spanning trees and then wrote for each edge its weight and whether it is included in the found tree or not. Unfortunately, the piece of paper where the graph was painted is gone and the teacher is getting very angry and demands to see the original graph. Help Vladislav come up with a graph so that the information about the minimum spanning tree remains correct.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"476"You are given an array a of length n. We define fa the following way:  Initially fa = 0, M = 1;  for every 2 ≤ i ≤ n if aM < ai then we set fa = fa + aM and then set M = i. Calculate the sum of fa over all n! permutations of the array a modulo 109 + 7.Note: two elements are considered different if their indices differ, so for every array a there are exactly n! permutations.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"477"You are given a permutation $$$p$$$ consisting of $$$n$$$ integers $$$1, 2, \dots, n$$$ (a permutation is an array where each element from $$$1$$$ to $$$n$$$ occurs exactly once).Let's call an array $$$a$$$ bipartite if the following undirected graph is bipartite:  the graph consists of $$$n$$$ vertices;  two vertices $$$i$$$ and $$$j$$$ are connected by an edge if $$$i < j$$$ and $$$a_i > a_j$$$. Your task is to find a bipartite array of integers $$$a$$$ of size $$$n$$$, such that $$$a_i = p_i$$$ or $$$a_i = -p_i$$$, or report that no such array exists. If there are multiple answers, print any of them.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"478"Arseniy is already grown-up and independent. His mother decided to leave him alone for m days and left on a vacation. She have prepared a lot of food, left some money and washed all Arseniy's clothes. Ten minutes before her leave she realized that it would be also useful to prepare instruction of which particular clothes to wear on each of the days she will be absent. Arseniy's family is a bit weird so all the clothes is enumerated. For example, each of Arseniy's n socks is assigned a unique integer from 1 to n. Thus, the only thing his mother had to do was to write down two integers li and ri for each of the days — the indices of socks to wear on the day i (obviously, li stands for the left foot and ri for the right). Each sock is painted in one of k colors.When mother already left Arseniy noticed that according to instruction he would wear the socks of different colors on some days. Of course, that is a terrible mistake cause by a rush. Arseniy is a smart boy, and, by some magical coincidence, he posses k jars with the paint — one for each of k colors.Arseniy wants to repaint some of the socks in such a way, that for each of m days he can follow the mother's instructions and wear the socks of the same color. As he is going to be very busy these days he will have no time to change the colors of any socks so he has to finalize the colors now.The new computer game Bota-3 was just realised and Arseniy can't wait to play it. What is the minimum number of socks that need their color to be changed in order to make it possible to follow mother's instructions and wear the socks of the same color during each of m days.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"479"You are given three positive (greater than zero) integers $$$c$$$, $$$d$$$ and $$$x$$$. You have to find the number of pairs of positive integers $$$(a, b)$$$ such that equality $$$c \cdot lcm(a, b) - d \cdot gcd(a, b) = x$$$ holds. Where $$$lcm(a, b)$$$ is the least common multiple of $$$a$$$ and $$$b$$$ and $$$gcd(a, b)$$$ is the greatest common divisor of $$$a$$$ and $$$b$$$.","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"480"Rebel spy Heidi has just obtained the plans for the Death Star from the Empire and, now on her way to safety, she is trying to break the encryption of the plans (of course they are encrypted – the Empire may be evil, but it is not stupid!). The encryption has several levels of security, and here is how the first one looks.Heidi is presented with a screen that shows her a sequence of integers A and a positive integer p. She knows that the encryption code is a single number S, which is defined as follows:Define the score of X to be the sum of the elements of X modulo p.Heidi is given a sequence A that consists of N integers, and also given an integer p. She needs to split A into 2 parts such that:   Each part contains at least 1 element of A, and each part consists of contiguous elements of A.  The two parts do not overlap.  The total sum S of the scores of those two parts is maximized. This is the encryption code. Output the sum S, which is the encryption code.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"481"Misha was interested in water delivery from childhood. That's why his mother sent him to the annual Innovative Olympiad in Irrigation (IOI). Pupils from all Berland compete there demonstrating their skills in watering. It is extremely expensive to host such an olympiad, so after the first $$$n$$$ olympiads the organizers introduced the following rule of the host city selection.The host cities of the olympiads are selected in the following way. There are $$$m$$$ cities in Berland wishing to host the olympiad, they are numbered from $$$1$$$ to $$$m$$$. The host city of each next olympiad is determined as the city that hosted the olympiad the smallest number of times before. If there are several such cities, the city with the smallest index is selected among them.Misha's mother is interested where the olympiad will be held in some specific years. The only information she knows is the above selection rule and the host cities of the first $$$n$$$ olympiads. Help her and if you succeed, she will ask Misha to avoid flooding your house.",['trees'],"[0, 0, 0, 0, 0, 0, 0, 1]"482"There are n games in a football tournament. Three teams are participating in it. Currently k games had already been played. You are an avid football fan, but recently you missed the whole k games. Fortunately, you remember a guess of your friend for these k games. Your friend did not tell exact number of wins of each team, instead he thought that absolute difference between number of wins of first and second team will be d1 and that of between second and third team will be d2.You don't want any of team win the tournament, that is each team should have the same number of wins after n games. That's why you want to know: does there exist a valid tournament satisfying the friend's guess such that no team will win this tournament?Note that outcome of a match can not be a draw, it has to be either win or loss.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"483There are $$$n$$$ programmers that you want to split into several non-empty teams. The skill of the $$$i$$$-th programmer is $$$a_i$$$. You want to assemble the maximum number of teams from them. There is a restriction for each team: the number of programmers in the team multiplied by the minimum skill among all programmers in the team must be at least $$$x$$$.Each programmer should belong to at most one team. Some programmers may be left without a team.Calculate the maximum number of teams that you can assemble.,[],"[0, 0, 0, 0, 0, 0, 0, 0]"484"Let's call an array $$$a$$$ consisting of $$$n$$$ positive (greater than $$$0$$$) integers beautiful if the following condition is held for every $$$i$$$ from $$$1$$$ to $$$n$$$: either $$$a_i = 1$$$, or at least one of the numbers $$$a_i - 1$$$ and $$$a_i - 2$$$ exists in the array as well.For example:   the array $$$[5, 3, 1]$$$ is beautiful: for $$$a_1$$$, the number $$$a_1 - 2 = 3$$$ exists in the array; for $$$a_2$$$, the number $$$a_2 - 2 = 1$$$ exists in the array; for $$$a_3$$$, the condition $$$a_3 = 1$$$ holds;  the array $$$[1, 2, 2, 2, 2]$$$ is beautiful: for $$$a_1$$$, the condition $$$a_1 = 1$$$ holds; for every other number $$$a_i$$$, the number $$$a_i - 1 = 1$$$ exists in the array;  the array $$$[1, 4]$$$ is not beautiful: for $$$a_2$$$, neither $$$a_2 - 2 = 2$$$ nor $$$a_2 - 1 = 3$$$ exists in the array, and $$$a_2 \ne 1$$$;  the array $$$[2]$$$ is not beautiful: for $$$a_1$$$, neither $$$a_1 - 1 = 1$$$ nor $$$a_1 - 2 = 0$$$ exists in the array, and $$$a_1 \ne 1$$$;  the array $$$[2, 1, 3]$$$ is beautiful: for $$$a_1$$$, the number $$$a_1 - 1 = 1$$$ exists in the array; for $$$a_2$$$, the condition $$$a_2 = 1$$$ holds; for $$$a_3$$$, the number $$$a_3 - 2 = 1$$$ exists in the array. You are given a positive integer $$$s$$$. Find the minimum possible size of a beautiful array with the sum of elements equal to $$$s$$$.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"485You are given two arrays of integers a and b. For each element of the second array bj you should find the number of elements in array a that are less than or equal to the value bj.,[],"[0, 0, 0, 0, 0, 0, 0, 0]"486"A new Berland businessman Vitaly is going to open a household appliances' store. All he's got to do now is to hire the staff.The store will work seven days a week, but not around the clock. Every day at least k people must work in the store.Berland has a law that determines the order of working days and non-working days. Namely, each employee must work for exactly n consecutive days, then rest for exactly m days, then work for n more days and rest for m more, and so on. Vitaly doesn't want to break the law. Fortunately, there is a loophole: the law comes into force on the day when the employee is hired. For example, if an employee is hired on day x, then he should work on days [x, x + 1, ..., x + n - 1], [x + m + n, x + m + n + 1, ..., x + m + 2n - 1], and so on. Day x can be chosen arbitrarily by Vitaly.There is one more thing: the key to the store. Berland law prohibits making copies of keys, so there is only one key. Vitaly is planning to entrust the key to the store employees. At the same time on each day the key must be with an employee who works that day — otherwise on this day no one can get inside the store. During the day the key holder can give the key to another employee, if he also works that day. The key will handed to the first hired employee at his first working day.Each employee has to be paid salary. Therefore, Vitaly wants to hire as few employees as possible provided that the store can operate normally on each day from 1 to infinity. In other words, on each day with index from 1 to infinity, the store must have at least k working employees, and one of the working employees should have the key to the store.Help Vitaly and determine the minimum required number of employees, as well as days on which they should be hired.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"487"You have an integer $$$n$$$. Let's define following tree generation as McDic's generation:  Make a complete and full binary tree of $$$2^{n} - 1$$$ vertices. Complete and full binary tree means a tree that exactly one vertex is a root, all leaves have the same depth (distance from the root), and all non-leaf nodes have exactly two child nodes.  Select a non-root vertex $$$v$$$ from that binary tree.  Remove $$$v$$$ from tree and make new edges between $$$v$$$'s parent and $$$v$$$'s direct children. If $$$v$$$ has no children, then no new edges will be made. You have a tree. Determine if this tree can be made by McDic's generation. If yes, then find the parent vertex of removed vertex in tree.",['trees'],"[0, 0, 0, 0, 0, 0, 0, 1]"488"You are given a board with $$$n$$$ rows and $$$n$$$ columns, numbered from $$$1$$$ to $$$n$$$. The intersection of the $$$a$$$-th row and $$$b$$$-th column is denoted by $$$(a, b)$$$.A half-queen attacks cells in the same row, same column, and on one diagonal. More formally, a half-queen on $$$(a, b)$$$ attacks the cell $$$(c, d)$$$ if $$$a=c$$$ or $$$b=d$$$ or $$$a-b=c-d$$$.    The blue cells are under attack.  What is the minimum number of half-queens that can be placed on that board so as to ensure that each square is attacked by at least one half-queen?Construct an optimal solution.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"489"You've got a string $$$a_1, a_2, \dots, a_n$$$, consisting of zeros and ones.Let's call a sequence of consecutive elements $$$a_i, a_{i + 1}, \ldots, a_j$$$ ($$$1\leq i\leq j\leq n$$$) a substring of string $$$a$$$. You can apply the following operations any number of times:  Choose some substring of string $$$a$$$ (for example, you can choose entire string) and reverse it, paying $$$x$$$ coins for it (for example, «0101101» $$$\to$$$ «0111001»);  Choose some substring of string $$$a$$$ (for example, you can choose entire string or just one symbol) and replace each symbol to the opposite one (zeros are replaced by ones, and ones — by zeros), paying $$$y$$$ coins for it (for example, «0101101» $$$\to$$$ «0110001»). You can apply these operations in any order. It is allowed to apply the operations multiple times to the same substring.What is the minimum number of coins you need to spend to get a string consisting only of ones?",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"490"Welcome! Everything is fine.You have arrived in The Medium Place, the place between The Good Place and The Bad Place. You are assigned a task that will either make people happier or torture them for eternity.You have a list of $$$k$$$ pairs of people who have arrived in a new inhabited neighborhood. You need to assign each of the $$$2k$$$ people into one of the $$$2k$$$ houses. Each person will be the resident of exactly one house, and each house will have exactly one resident.Of course, in the neighborhood, it is possible to visit friends. There are $$$2k - 1$$$ roads, each of which connects two houses. It takes some time to traverse a road. We will specify the amount of time it takes in the input. The neighborhood is designed in such a way that from anyone's house, there is exactly one sequence of distinct roads you can take to any other house. In other words, the graph with the houses as vertices and the roads as edges is a tree.The truth is, these $$$k$$$ pairs of people are actually soulmates. We index them from $$$1$$$ to $$$k$$$. We denote by $$$f(i)$$$ the amount of time it takes for the $$$i$$$-th pair of soulmates to go to each other's houses.As we have said before, you will need to assign each of the $$$2k$$$ people into one of the $$$2k$$$ houses. You have two missions, one from the entities in The Good Place and one from the entities of The Bad Place. Here they are:  The first mission, from The Good Place, is to assign the people into the houses such that the sum of $$$f(i)$$$ over all pairs $$$i$$$ is minimized. Let's define this minimized sum as $$$G$$$. This makes sure that soulmates can easily and efficiently visit each other;  The second mission, from The Bad Place, is to assign the people into the houses such that the sum of $$$f(i)$$$ over all pairs $$$i$$$ is maximized. Let's define this maximized sum as $$$B$$$. This makes sure that soulmates will have a difficult time to visit each other. What are the values of $$$G$$$ and $$$B$$$?","['graphs', 'trees']","[0, 0, 1, 0, 0, 0, 0, 1]"491"Sasha is taking part in a programming competition. In one of the problems she should check if some rooted trees are isomorphic or not. She has never seen this problem before, but, being an experienced participant, she guessed that she should match trees to some sequences and then compare these sequences instead of trees. Sasha wants to match each tree with a sequence a0, a1, ..., ah, where h is the height of the tree, and ai equals to the number of vertices that are at distance of i edges from root. Unfortunately, this time Sasha's intuition was wrong, and there could be several trees matching the same sequence. To show it, you need to write a program that, given the sequence ai, builds two non-isomorphic rooted trees that match that sequence, or determines that there is only one such tree.Two rooted trees are isomorphic, if you can reenumerate the vertices of the first one in such a way, that the index of the root becomes equal the index of the root of the second tree, and these two trees become equal.The height of a rooted tree is the maximum number of edges on a path from the root to any other vertex.",['trees'],"[0, 0, 0, 0, 0, 0, 0, 1]"492"You are given an integer array $$$a_1, a_2, \ldots, a_n$$$.The array $$$b$$$ is called to be a subsequence of $$$a$$$ if it is possible to remove some elements from $$$a$$$ to get $$$b$$$.Array $$$b_1, b_2, \ldots, b_k$$$ is called to be good if it is not empty and for every $$$i$$$ ($$$1 \le i \le k$$$) $$$b_i$$$ is divisible by $$$i$$$.Find the number of good subsequences in $$$a$$$ modulo $$$10^9 + 7$$$. Two subsequences are considered different if index sets of numbers included in them are different. That is, the values ​of the elements ​do not matter in the comparison of subsequences. In particular, the array $$$a$$$ has exactly $$$2^n - 1$$$ different subsequences (excluding an empty subsequence).","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"493"You went to the store, selling $$$n$$$ types of chocolates. There are $$$a_i$$$ chocolates of type $$$i$$$ in stock.You have unlimited amount of cash (so you are not restricted by any prices) and want to buy as many chocolates as possible. However if you buy $$$x_i$$$ chocolates of type $$$i$$$ (clearly, $$$0 \le x_i \le a_i$$$), then for all $$$1 \le j < i$$$ at least one of the following must hold:  $$$x_j = 0$$$ (you bought zero chocolates of type $$$j$$$) $$$x_j < x_i$$$ (you bought less chocolates of type $$$j$$$ than of type $$$i$$$) For example, the array $$$x = [0, 0, 1, 2, 10]$$$ satisfies the requirement above (assuming that all $$$a_i \ge x_i$$$), while arrays $$$x = [0, 1, 0]$$$, $$$x = [5, 5]$$$ and $$$x = [3, 2]$$$ don't.Calculate the maximum number of chocolates you can buy.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"494"A social network for dogs called DH (DogHouse) has k special servers to recompress uploaded videos of cute cats. After each video is uploaded, it should be recompressed on one (any) of the servers, and only after that it can be saved in the social network.We know that each server takes one second to recompress a one minute fragment. Thus, any server takes m seconds to recompress a m minute video.We know the time when each of the n videos were uploaded to the network (in seconds starting from the moment all servers started working). All videos appear at different moments of time and they are recompressed in the order they appear. If some video appeared at time s, then its recompressing can start at that very moment, immediately. Some videos can await recompressing when all the servers are busy. In this case, as soon as a server is available, it immediately starts recompressing another video. The videos that await recompressing go in a queue. If by the moment the videos started being recompressed some servers are available, then any of them starts recompressing the video.For each video find the moment it stops being recompressed.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"495"The employees of the R1 company often spend time together: they watch football, they go camping, they solve contests. So, it's no big deal that sometimes someone pays for someone else.Today is the day of giving out money rewards. The R1 company CEO will invite employees into his office one by one, rewarding each one for the hard work this month. The CEO knows who owes money to whom. And he also understands that if he invites person x to his office for a reward, and then immediately invite person y, who has lent some money to person x, then they can meet. Of course, in such a situation, the joy of person x from his brand new money reward will be much less. Therefore, the R1 CEO decided to invite the staff in such an order that the described situation will not happen for any pair of employees invited one after another.However, there are a lot of employees in the company, and the CEO doesn't have a lot of time. Therefore, the task has been assigned to you. Given the debt relationships between all the employees, determine in which order they should be invited to the office of the R1 company CEO, or determine that the described order does not exist.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"496"Bob is a pirate looking for the greatest treasure the world has ever seen. The treasure is located at the point $$$T$$$, which coordinates to be found out.Bob travelled around the world and collected clues of the treasure location at $$$n$$$ obelisks. These clues were in an ancient language, and he has only decrypted them at home. Since he does not know which clue belongs to which obelisk, finding the treasure might pose a challenge. Can you help him?As everyone knows, the world is a two-dimensional plane. The $$$i$$$-th obelisk is at integer coordinates $$$(x_i, y_i)$$$. The $$$j$$$-th clue consists of $$$2$$$ integers $$$(a_j, b_j)$$$ and belongs to the obelisk $$$p_j$$$, where $$$p$$$ is some (unknown) permutation on $$$n$$$ elements. It means that the treasure is located at $$$T=(x_{p_j} + a_j, y_{p_j} + b_j)$$$. This point $$$T$$$ is the same for all clues.In other words, each clue belongs to exactly one of the obelisks, and each obelisk has exactly one clue that belongs to it. A clue represents the vector from the obelisk to the treasure. The clues must be distributed among the obelisks in such a way that they all point to the same position of the treasure.Your task is to find the coordinates of the treasure. If there are multiple solutions, you may print any of them.Note that you don't need to find the permutation. Permutations are used only in order to explain the problem.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"497"Petya loves lucky numbers. Everybody knows that lucky numbers are positive integers whose decimal representation contains only the lucky digits 4 and 7. For example, numbers 47, 744, 4 are lucky and 5, 17, 467 are not.Petya has a number consisting of n digits without leading zeroes. He represented it as an array of digits without leading zeroes. Let's call it d. The numeration starts with 1, starting from the most significant digit. Petya wants to perform the following operation k times: find the minimum x (1 ≤ x < n) such that dx = 4 and dx + 1 = 7, if x is odd, then to assign dx = dx + 1 = 4, otherwise to assign dx = dx + 1 = 7. Note that if no x was found, then the operation counts as completed and the array doesn't change at all.You are given the initial number as an array of digits and the number k. Help Petya find the result of completing k operations.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"498"One particularly well-known fact about zombies is that they move and think terribly slowly. While we still don't know why their movements are so sluggish, the problem of laggy thinking has been recently resolved. It turns out that the reason is not (as previously suspected) any kind of brain defect – it's the opposite! Independent researchers confirmed that the nervous system of a zombie is highly complicated – it consists of n brains (much like a cow has several stomachs). They are interconnected by brain connectors, which are veins capable of transmitting thoughts between brains. There are two important properties such a brain network should have to function properly:   It should be possible to exchange thoughts between any two pairs of brains (perhaps indirectly, through other brains).  There should be no redundant brain connectors, that is, removing any brain connector would make property 1 false. If both properties are satisfied, we say that the nervous system is valid. Unfortunately (?), if the system is not valid, the zombie stops thinking and becomes (even more) dead. Your task is to analyze a given nervous system of a zombie and find out whether it is valid.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"499"Assume that you have $$$k$$$ one-dimensional segments $$$s_1, s_2, \dots s_k$$$ (each segment is denoted by two integers — its endpoints). Then you can build the following graph on these segments. The graph consists of $$$k$$$ vertexes, and there is an edge between the $$$i$$$-th and the $$$j$$$-th vertexes ($$$i \neq j$$$) if and only if the segments $$$s_i$$$ and $$$s_j$$$ intersect (there exists at least one point that belongs to both of them).For example, if $$$s_1 = [1, 6], s_2 = [8, 20], s_3 = [4, 10], s_4 = [2, 13], s_5 = [17, 18]$$$, then the resulting graph is the following:  A tree of size $$$m$$$ is good if it is possible to choose $$$m$$$ one-dimensional segments so that the graph built on these segments coincides with this tree.You are given a tree, you have to find its good subtree with maximum possible size. Recall that a subtree is a connected subgraph of a tree.Note that you have to answer $$$q$$$ independent queries.","['graphs', 'trees']","[0, 0, 1, 0, 0, 0, 0, 1]"500"Casimir has a rectangular piece of paper with a checkered field of size $$$n \times m$$$. Initially, all cells of the field are white.Let us denote the cell with coordinates $$$i$$$ vertically and $$$j$$$ horizontally by $$$(i, j)$$$. The upper left cell will be referred to as $$$(1, 1)$$$ and the lower right cell as $$$(n, m)$$$.Casimir draws ticks of different sizes on the field. A tick of size $$$d$$$ ($$$d > 0$$$) with its center in cell $$$(i, j)$$$ is drawn as follows:   First, the center cell $$$(i, j)$$$ is painted black.  Then exactly $$$d$$$ cells on the top-left diagonally to the center and exactly $$$d$$$ cells on the top-right diagonally to the center are also painted black.  That is all the cells with coordinates $$$(i - h, j \pm h)$$$ for all $$$h$$$ between $$$0$$$ and $$$d$$$ are painted. In particular, a tick consists of $$$2d + 1$$$ black cells. An already painted cell will remain black if painted again. Below you can find an example of the $$$4 \times 9$$$ box, with two ticks of sizes $$$2$$$ and $$$3$$$.  You are given a description of a checkered field of size $$$n \times m$$$. Casimir claims that this field came about after he drew some (possibly $$$0$$$) ticks on it. The ticks could be of different sizes, but the size of each tick is at least $$$k$$$ (that is, $$$d \ge k$$$ for all the ticks).Determine whether this field can indeed be obtained by drawing some (possibly none) ticks of sizes $$$d \ge k$$$ or not.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"501"Please note the non-standard memory limit.There are $$$n$$$ problems numbered with integers from $$$1$$$ to $$$n$$$. $$$i$$$-th problem has the complexity $$$c_i = 2^i$$$, tag $$$tag_i$$$ and score $$$s_i$$$.After solving the problem $$$i$$$ it's allowed to solve problem $$$j$$$ if and only if $$$\text{IQ} < |c_i - c_j|$$$ and $$$tag_i \neq tag_j$$$. After solving it your $$$\text{IQ}$$$ changes and becomes $$$\text{IQ} = |c_i - c_j|$$$ and you gain $$$|s_i - s_j|$$$ points.Any problem can be the first. You can solve problems in any order and as many times as you want.Initially your $$$\text{IQ} = 0$$$. Find the maximum number of points that can be earned.","['graphs', 'number theory']","[0, 0, 1, 0, 1, 0, 0, 0]"502"Pasha loves to send strictly positive integers to his friends. Pasha cares about security, therefore when he wants to send an integer $$$n$$$, he encrypts it in the following way: he picks three integers $$$a$$$, $$$b$$$ and $$$c$$$ such that $$$l \leq a,b,c \leq r$$$, and then he computes the encrypted value $$$m = n \cdot a + b - c$$$.Unfortunately, an adversary intercepted the values $$$l$$$, $$$r$$$ and $$$m$$$. Is it possible to recover the original values of $$$a$$$, $$$b$$$ and $$$c$$$ from this information? More formally, you are asked to find any values of $$$a$$$, $$$b$$$ and $$$c$$$ such that  $$$a$$$, $$$b$$$ and $$$c$$$ are integers,  $$$l \leq a, b, c \leq r$$$,  there exists a strictly positive integer $$$n$$$, such that $$$n \cdot a + b - c = m$$$. ","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"503"Amr lives in Lala Land. Lala Land is a very beautiful country that is located on a coordinate line. Lala Land is famous with its apple trees growing everywhere.Lala Land has exactly n apple trees. Tree number i is located in a position xi and has ai apples growing on it. Amr wants to collect apples from the apple trees. Amr currently stands in x = 0 position. At the beginning, he can choose whether to go right or left. He'll continue in his direction until he meets an apple tree he didn't visit before. He'll take all of its apples and then reverse his direction, continue walking in this direction until he meets another apple tree he didn't visit before and so on. In the other words, Amr reverses his direction when visiting each new apple tree. Amr will stop collecting apples when there are no more trees he didn't visit in the direction he is facing.What is the maximum number of apples he can collect?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"504"Rudolf is on his way to the castle. Before getting into the castle, the security staff asked him a question:Given two binary numbers $$$a$$$ and $$$b$$$ of length $$$n$$$. How many different ways of swapping two digits in $$$a$$$ (only in $$$a$$$, not $$$b$$$) so that bitwise OR of these two numbers will be changed? In other words, let $$$c$$$ be the bitwise OR of $$$a$$$ and $$$b$$$, you need to find the number of ways of swapping two bits in $$$a$$$ so that bitwise OR will not be equal to $$$c$$$.Note that binary numbers can contain leading zeros so that length of each number is exactly $$$n$$$.Bitwise OR is a binary operation. A result is a binary number which contains a one in each digit if there is a one in at least one of the two numbers. For example, $$$01010_2$$$ OR $$$10011_2$$$ = $$$11011_2$$$.Well, to your surprise, you are not Rudolf, and you don't need to help him$$$\ldots$$$ You are the security staff! Please find the number of ways of swapping two bits in $$$a$$$ so that bitwise OR will be changed.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"505"You are given an array $$$a$$$ consisting of $$$n$$$ integers numbered from $$$1$$$ to $$$n$$$.Let's define the $$$k$$$-amazing number of the array as the minimum number that occurs in all of the subsegments of the array having length $$$k$$$ (recall that a subsegment of $$$a$$$ of length $$$k$$$ is a contiguous part of $$$a$$$ containing exactly $$$k$$$ elements). If there is no integer occuring in all subsegments of length $$$k$$$ for some value of $$$k$$$, then the $$$k$$$-amazing number is $$$-1$$$.For each $$$k$$$ from $$$1$$$ to $$$n$$$ calculate the $$$k$$$-amazing number of the array $$$a$$$.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"506"A club plans to hold a party and will invite some of its $$$n$$$ members. The $$$n$$$ members are identified by the numbers $$$1, 2, \dots, n$$$. If member $$$i$$$ is not invited, the party will gain an unhappiness value of $$$a_i$$$.There are $$$m$$$ pairs of friends among the $$$n$$$ members. As per tradition, if both people from a friend pair are invited, they will share a cake at the party. The total number of cakes eaten will be equal to the number of pairs of friends such that both members have been invited.However, the club's oven can only cook two cakes at a time. So, the club demands that the total number of cakes eaten is an even number.What is the minimum possible total unhappiness value of the party, respecting the constraint that the total number of cakes eaten is even?",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"507"Maria is the most active old lady in her house. She was tired of sitting at home. She decided to organize a ceremony against the coronavirus.She has $$$n$$$ friends who are also grannies (Maria is not included in this number). The $$$i$$$-th granny is ready to attend the ceremony, provided that at the time of her appearance in the courtyard there will be at least $$$a_i$$$ other grannies there. Note that grannies can come into the courtyard at the same time. Formally, the granny $$$i$$$ agrees to come if the number of other grannies who came earlier or at the same time with her is greater than or equal to $$$a_i$$$.Grannies gather in the courtyard like that.  Initially, only Maria is in the courtyard (that is, the initial number of grannies in the courtyard is $$$1$$$). All the remaining $$$n$$$ grannies are still sitting at home. On each step Maria selects a subset of grannies, none of whom have yet to enter the courtyard. She promises each of them that at the time of her appearance there will be at least $$$a_i$$$ other grannies (including Maria) in the courtyard. Maria can call several grannies at once. In this case, the selected grannies will go out into the courtyard at the same moment of time. She cannot deceive grannies, that is, the situation when the $$$i$$$-th granny in the moment of appearing in the courtyard, finds that now there are strictly less than $$$a_i$$$ other grannies (except herself, but including Maria), is prohibited. Please note that if several grannies appeared in the yard at the same time, then each of them sees others at the time of appearance. Your task is to find what maximum number of grannies (including herself) Maria can collect in the courtyard for the ceremony. After all, the more people in one place during quarantine, the more effective the ceremony!Consider an example: if $$$n=6$$$ and $$$a=[1,5,4,5,1,9]$$$, then:  at the first step Maria can call grannies with numbers $$$1$$$ and $$$5$$$, each of them will see two grannies at the moment of going out into the yard (note that $$$a_1=1 \le 2$$$ and $$$a_5=1 \le 2$$$);  at the second step, Maria can call grannies with numbers $$$2$$$, $$$3$$$ and $$$4$$$, each of them will see five grannies at the moment of going out into the yard (note that $$$a_2=5 \le 5$$$, $$$a_3=4 \le 5$$$ and $$$a_4=5 \le 5$$$);  the $$$6$$$-th granny cannot be called into the yard  — therefore, the answer is $$$6$$$ (Maria herself and another $$$5$$$ grannies). ",[],"[0, 0, 0, 0, 0, 0, 0, 0]"508"Given $$$n$$$ strings, each of length $$$2$$$, consisting of lowercase Latin alphabet letters from 'a' to 'k', output the number of pairs of indices $$$(i, j)$$$ such that $$$i < j$$$ and the $$$i$$$-th string and the $$$j$$$-th string differ in exactly one position.In other words, count the number of pairs $$$(i, j)$$$ ($$$i < j$$$) such that the $$$i$$$-th string and the $$$j$$$-th string have exactly one position $$$p$$$ ($$$1 \leq p \leq 2$$$) such that $$${s_{i}}_{p} \neq {s_{j}}_{p}$$$.The answer may not fit into 32-bit integer type, so you should use 64-bit integers like long long in C++ to avoid integer overflow.","['math', 'strings']","[0, 0, 0, 1, 0, 0, 1, 0]"509"Monocarp has got an array $$$a$$$ consisting of $$$n$$$ integers. Let's denote $$$k$$$ as the mathematic mean of these elements (note that it's possible that $$$k$$$ is not an integer). The mathematic mean of an array of $$$n$$$ elements is the sum of elements divided by the number of these elements (i. e. sum divided by $$$n$$$).Monocarp wants to delete exactly two elements from $$$a$$$ so that the mathematic mean of the remaining $$$(n - 2)$$$ elements is still equal to $$$k$$$.Your task is to calculate the number of pairs of positions $$$[i, j]$$$ ($$$i < j$$$) such that if the elements on these positions are deleted, the mathematic mean of $$$(n - 2)$$$ remaining elements is equal to $$$k$$$ (that is, it is equal to the mathematic mean of $$$n$$$ elements of the original array $$$a$$$).",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"510"You are given $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$. You choose any subset of the given numbers (possibly, none or all numbers) and negate these numbers (i. e. change $$$x \to (-x)$$$). What is the maximum number of different values in the array you can achieve?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"511"In a Berland's zoo there is an enclosure with camels. It is known that camels like to spit. Bob watched these interesting animals for the whole day and registered in his notepad where each animal spitted. Now he wants to know if in the zoo there are two camels, which spitted at each other. Help him to solve this task.The trajectory of a camel's spit is an arc, i.e. if the camel in position x spits d meters right, he can hit only the camel in position x + d, if such a camel exists.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"512"Let's denote a function You are given an array a consisting of n integers. You have to calculate the sum of d(ai, aj) over all pairs (i, j) such that 1 ≤ i ≤ j ≤ n.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"513"Dreamoon likes sequences very much. So he created a problem about the sequence that you can't find in OEIS: You are given two integers $$$d, m$$$, find the number of arrays $$$a$$$, satisfying the following constraints:  The length of $$$a$$$ is $$$n$$$, $$$n \ge 1$$$  $$$1 \le a_1 < a_2 < \dots < a_n \le d$$$  Define an array $$$b$$$ of length $$$n$$$ as follows: $$$b_1 = a_1$$$, $$$\forall i > 1, b_i = b_{i - 1} \oplus a_i$$$, where $$$\oplus$$$ is the bitwise exclusive-or (xor). After constructing an array $$$b$$$, the constraint $$$b_1 < b_2 < \dots < b_{n - 1} < b_n$$$ should hold. Since the number of possible arrays may be too large, you need to find the answer modulo $$$m$$$.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"514"Ashish has an array $$$a$$$ of size $$$n$$$.A subsequence of $$$a$$$ is defined as a sequence that can be obtained from $$$a$$$ by deleting some elements (possibly none), without changing the order of the remaining elements.Consider a subsequence $$$s$$$ of $$$a$$$. He defines the cost of $$$s$$$ as the minimum between:   The maximum among all elements at odd indices of $$$s$$$.  The maximum among all elements at even indices of $$$s$$$. Note that the index of an element is its index in $$$s$$$, rather than its index in $$$a$$$. The positions are numbered from $$$1$$$. So, the cost of $$$s$$$ is equal to $$$min(max(s_1, s_3, s_5, \ldots), max(s_2, s_4, s_6, \ldots))$$$.For example, the cost of $$$\{7, 5, 6\}$$$ is $$$min( max(7, 6), max(5) ) = min(7, 5) = 5$$$.Help him find the minimum cost of a subsequence of size $$$k$$$.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"515You are given a directed acyclic graph with n vertices and m edges. There are no self-loops or multiple edges between any pair of vertices. Graph can be disconnected.You should assign labels to all vertices in such a way that:  Labels form a valid permutation of length n — an integer sequence such that each integer from 1 to n appears exactly once in it.  If there exists an edge from vertex v to vertex u then labelv should be smaller than labelu.  Permutation should be lexicographically smallest among all suitable. Find such sequence of labels to satisfy all the conditions.,['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"516"A permutation is a sequence of $$$n$$$ integers from $$$1$$$ to $$$n$$$, in which all the numbers occur exactly once. For example, $$$[1]$$$, $$$[3, 5, 2, 1, 4]$$$, $$$[1, 3, 2]$$$ are permutations, and $$$[2, 3, 2]$$$, $$$[4, 3, 1]$$$, $$$[0]$$$ are not.Polycarp was given four integers $$$n$$$, $$$l$$$, $$$r$$$ ($$$1 \le l \le r \le n)$$$ and $$$s$$$ ($$$1 \le s \le \frac{n (n+1)}{2}$$$) and asked to find a permutation $$$p$$$ of numbers from $$$1$$$ to $$$n$$$ that satisfies the following condition:   $$$s = p_l + p_{l+1} + \ldots + p_r$$$. For example, for $$$n=5$$$, $$$l=3$$$, $$$r=5$$$, and $$$s=8$$$, the following permutations are suitable (not all options are listed):   $$$p = [3, 4, 5, 2, 1]$$$;  $$$p = [5, 2, 4, 3, 1]$$$;  $$$p = [5, 2, 1, 3, 4]$$$.  But, for example, there is no permutation suitable for the condition above for $$$n=4$$$, $$$l=1$$$, $$$r=1$$$, and $$$s=5$$$.Help Polycarp, for the given $$$n$$$, $$$l$$$, $$$r$$$, and $$$s$$$, find a permutation of numbers from $$$1$$$ to $$$n$$$ that fits the condition above. If there are several suitable permutations, print any of them.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"517"Mahmoud has an array a consisting of n integers. He asked Ehab to find another array b of the same length such that:  b is lexicographically greater than or equal to a.  bi ≥ 2.  b is pairwise coprime: for every 1 ≤ i < j ≤ n, bi and bj are coprime, i. e. GCD(bi, bj) = 1, where GCD(w, z) is the greatest common divisor of w and z. Ehab wants to choose a special array so he wants the lexicographically minimal array between all the variants. Can you find it?An array x is lexicographically greater than an array y if there exists an index i such than xi > yi and xj = yj for all 1 ≤ j < i. An array x is equal to an array y if xi = yi for all 1 ≤ i ≤ n.","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"518"Today, Wet Shark is given n integers. Using any of these integers no more than once, Wet Shark wants to get maximum possible even (divisible by 2) sum. Please, calculate this value for Wet Shark. Note, that if Wet Shark uses no integers from the n integers, the sum is an even integer 0.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"519"You are given several queries. Each query consists of three integers $$$p$$$, $$$q$$$ and $$$b$$$. You need to answer whether the result of $$$p/q$$$ in notation with base $$$b$$$ is a finite fraction.A fraction in notation with base $$$b$$$ is finite if it contains finite number of numerals after the decimal point. It is also possible that a fraction has zero numerals after the decimal point.","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"520"An ant moves on the real line with constant speed of $$$1$$$ unit per second. It starts at $$$0$$$ and always moves to the right (so its position increases by $$$1$$$ each second).There are $$$n$$$ portals, the $$$i$$$-th of which is located at position $$$x_i$$$ and teleports to position $$$y_i < x_i$$$. Each portal can be either active or inactive. The initial state of the $$$i$$$-th portal is determined by $$$s_i$$$: if $$$s_i=0$$$ then the $$$i$$$-th portal is initially inactive, if $$$s_i=1$$$ then the $$$i$$$-th portal is initially active. When the ant travels through a portal (i.e., when its position coincides with the position of a portal):   if the portal is inactive, it becomes active (in this case the path of the ant is not affected);  if the portal is active, it becomes inactive and the ant is instantly teleported to the position $$$y_i$$$, where it keeps on moving as normal. How long (from the instant it starts moving) does it take for the ant to reach the position $$$x_n + 1$$$? It can be shown that this happens in a finite amount of time. Since the answer may be very large, compute it modulo $$$998\,244\,353$$$.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"521"Vasya and Kolya play a game with a string, using the following rules. Initially, Kolya creates a string s, consisting of small English letters, and uniformly at random chooses an integer k from a segment [0, len(s) - 1]. He tells Vasya this string s, and then shifts it k letters to the left, i. e. creates a new string t = sk + 1sk + 2... sns1s2... sk. Vasya does not know the integer k nor the string t, but he wants to guess the integer k. To do this, he asks Kolya to tell him the first letter of the new string, and then, after he sees it, open one more letter on some position, which Vasya can choose.Vasya understands, that he can't guarantee that he will win, but he wants to know the probability of winning, if he plays optimally. He wants you to compute this probability. Note that Vasya wants to know the value of k uniquely, it means, that if there are at least two cyclic shifts of s that fit the information Vasya knowns, Vasya loses. Of course, at any moment of the game Vasya wants to maximize the probability of his win.","['math', 'probabilities']","[0, 0, 0, 1, 0, 1, 0, 0]"522"Suppose $$$a_1, a_2, \dots, a_n$$$ is a sorted integer sequence of length $$$n$$$ such that $$$a_1 \leq a_2 \leq \dots \leq a_n$$$. For every $$$1 \leq i \leq n$$$, the prefix sum $$$s_i$$$ of the first $$$i$$$ terms $$$a_1, a_2, \dots, a_i$$$ is defined by $$$$$$ s_i = \sum_{k=1}^i a_k = a_1 + a_2 + \dots + a_i. $$$$$$Now you are given the last $$$k$$$ terms of the prefix sums, which are $$$s_{n-k+1}, \dots, s_{n-1}, s_{n}$$$. Your task is to determine whether this is possible. Formally, given $$$k$$$ integers $$$s_{n-k+1}, \dots, s_{n-1}, s_{n}$$$, the task is to check whether there is a sequence $$$a_1, a_2, \dots, a_n$$$ such that   $$$a_1 \leq a_2 \leq \dots \leq a_n$$$, and  $$$s_i = a_1 + a_2 + \dots + a_i$$$ for all $$$n-k+1 \leq i \leq n$$$. ",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"523"It is given a non-negative integer $$$x$$$, the decimal representation of which contains $$$n$$$ digits. You need to color each its digit in red or black, so that the number formed by the red digits is divisible by $$$A$$$, and the number formed by the black digits is divisible by $$$B$$$.At least one digit must be colored in each of two colors. Consider, the count of digits colored in red is $$$r$$$ and the count of digits colored in black is $$$b$$$. Among all possible colorings of the given number $$$x$$$, you need to output any such that the value of $$$|r - b|$$$ is the minimum possible.Note that the number $$$x$$$ and the numbers formed by digits of each color, may contain leading zeros.  Example of painting a number for $$$A = 3$$$ and $$$B = 13$$$ The figure above shows an example of painting the number $$$x = 02165$$$ of $$$n = 5$$$ digits for $$$A = 3$$$ and $$$B = 13$$$. The red digits form the number $$$015$$$, which is divisible by $$$3$$$, and the black ones — $$$26$$$, which is divisible by $$$13$$$. Note that the absolute value of the difference between the counts of red and black digits is $$$1$$$, it is impossible to achieve a smaller value.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"524"One day Masha was walking in the park and found a graph under a tree... Surprised? Did you think that this problem would have some logical and reasoned story? No way! So, the problem...Masha has an oriented graph which $$$i$$$-th vertex contains some positive integer $$$a_i$$$. Initially Masha can put a coin at some vertex. In one operation she can move a coin placed in some vertex $$$u$$$ to any other vertex $$$v$$$ such that there is an oriented edge $$$u \to v$$$ in the graph. Each time when the coin is placed in some vertex $$$i$$$, Masha write down an integer $$$a_i$$$ in her notebook (in particular, when Masha initially puts a coin at some vertex, she writes an integer written at this vertex in her notebook). Masha wants to make exactly $$$k - 1$$$ operations in such way that the maximum number written in her notebook is as small as possible.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"525"Two players play a game.Initially there are $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ written on the board. Each turn a player selects one number and erases it from the board. This continues until there is only one number left on the board, i. e. $$$n - 1$$$ turns are made. The first player makes the first move, then players alternate turns.The first player wants to minimize the last number that would be left on the board, while the second player wants to maximize it.You want to know what number will be left on the board after $$$n - 1$$$ turns if both players make optimal moves.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"526"Fishingprince loves trees. A tree is a connected undirected graph without cycles.Fishingprince has a tree of $$$n$$$ vertices. The vertices are numbered $$$1$$$ through $$$n$$$. Let $$$d(x,y)$$$ denote the shortest distance on the tree from vertex $$$x$$$ to vertex $$$y$$$, assuming that the length of each edge is $$$1$$$.However, the tree was lost in an accident. Fortunately, Fishingprince still remembers some information about the tree. More specifically, for every triple of integers $$$x,y,z$$$ ($$$1\le x<y\le n$$$, $$$1\le z\le n$$$) he remembers whether $$$d(x,z)=d(y,z)$$$ or not.Help him recover the structure of the tree, or report that no tree satisfying the constraints exists.","['graphs', 'trees']","[0, 0, 1, 0, 0, 0, 0, 1]"527"Pasha decided to invite his friends to a tea party. For that occasion, he has a large teapot with the capacity of w milliliters and 2n tea cups, each cup is for one of Pasha's friends. The i-th cup can hold at most ai milliliters of water.It turned out that among Pasha's friends there are exactly n boys and exactly n girls and all of them are going to come to the tea party. To please everyone, Pasha decided to pour the water for the tea as follows:  Pasha can boil the teapot exactly once by pouring there at most w milliliters of water;  Pasha pours the same amount of water to each girl;  Pasha pours the same amount of water to each boy;  if each girl gets x milliliters of water, then each boy gets 2x milliliters of water. In the other words, each boy should get two times more water than each girl does.Pasha is very kind and polite, so he wants to maximize the total amount of the water that he pours to his friends. Your task is to help him and determine the optimum distribution of cups between Pasha's friends.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"528"Let's call two strings $$$s$$$ and $$$t$$$ anagrams of each other if it is possible to rearrange symbols in the string $$$s$$$ to get a string, equal to $$$t$$$.Let's consider two strings $$$s$$$ and $$$t$$$ which are anagrams of each other. We say that $$$t$$$ is a reducible anagram of $$$s$$$ if there exists an integer $$$k \ge 2$$$ and $$$2k$$$ non-empty strings $$$s_1, t_1, s_2, t_2, \dots, s_k, t_k$$$ that satisfy the following conditions:  If we write the strings $$$s_1, s_2, \dots, s_k$$$ in order, the resulting string will be equal to $$$s$$$;  If we write the strings $$$t_1, t_2, \dots, t_k$$$ in order, the resulting string will be equal to $$$t$$$;  For all integers $$$i$$$ between $$$1$$$ and $$$k$$$ inclusive, $$$s_i$$$ and $$$t_i$$$ are anagrams of each other. If such strings don't exist, then $$$t$$$ is said to be an irreducible anagram of $$$s$$$. Note that these notions are only defined when $$$s$$$ and $$$t$$$ are anagrams of each other.For example, consider the string $$$s = $$$ ""gamegame"". Then the string $$$t = $$$ ""megamage"" is a reducible anagram of $$$s$$$, we may choose for example $$$s_1 = $$$ ""game"", $$$s_2 = $$$ ""gam"", $$$s_3 = $$$ ""e"" and $$$t_1 = $$$ ""mega"", $$$t_2 = $$$ ""mag"", $$$t_3 = $$$ ""e"":  On the other hand, we can prove that $$$t = $$$ ""memegaga"" is an irreducible anagram of $$$s$$$.You will be given a string $$$s$$$ and $$$q$$$ queries, represented by two integers $$$1 \le l \le r \le |s|$$$ (where $$$|s|$$$ is equal to the length of the string $$$s$$$). For each query, you should find if the substring of $$$s$$$ formed by characters from the $$$l$$$-th to the $$$r$$$-th has at least one irreducible anagram.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"529"An array $$$[b_1, b_2, \ldots, b_m]$$$ is a palindrome, if $$$b_i = b_{m+1-i}$$$ for each $$$i$$$ from $$$1$$$ to $$$m$$$. Empty array is also a palindrome.An array is called kalindrome, if the following condition holds: It's possible to select some integer $$$x$$$ and delete some of the elements of the array equal to $$$x$$$, so that the remaining array (after gluing together the remaining parts) is a palindrome. Note that you don't have to delete all elements equal to $$$x$$$, and you don't have to delete at least one element equal to $$$x$$$.For example :   $$$[1, 2, 1]$$$ is kalindrome because you can simply not delete a single element.  $$$[3, 1, 2, 3, 1]$$$ is kalindrome because you can choose $$$x = 3$$$ and delete both elements equal to $$$3$$$, obtaining array $$$[1, 2, 1]$$$, which is a palindrome.  $$$[1, 2, 3]$$$ is not kalindrome. You are given an array $$$[a_1, a_2, \ldots, a_n]$$$. Determine if $$$a$$$ is kalindrome or not.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"530"Inna loves sweets very much. She has n closed present boxes lines up in a row in front of her. Each of these boxes contains either a candy (Dima's work) or nothing (Sereja's work). Let's assume that the boxes are numbered from 1 to n, from left to right.As the boxes are closed, Inna doesn't know which boxes contain candies and which boxes contain nothing. Inna chose number k and asked w questions to Dima to find that out. Each question is characterised by two integers li, ri (1 ≤ li ≤ ri ≤ n; r - l + 1 is divisible by k), the i-th question is: ""Dima, is that true that among the boxes with numbers from li to ri, inclusive, the candies lie only in boxes with numbers li + k - 1, li + 2k - 1, li + 3k - 1, ..., ri?""Dima hates to say ""no"" to Inna. That's why he wonders, what number of actions he will have to make for each question to make the answer to the question positive. In one action, Dima can either secretly take the candy from any box or put a candy to any box (Dima has infinitely many candies). Help Dima count the number of actions for each Inna's question.Please note that Dima doesn't change the array during Inna's questions. That's why when you calculate the number of operations for the current question, please assume that the sequence of boxes didn't change.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"531"Students love to celebrate their holidays. Especially if the holiday is the day of the end of exams!Despite the fact that Igor K., unlike his groupmates, failed to pass a programming test, he decided to invite them to go to a cafe so that each of them could drink a bottle of... fresh cow milk. Having entered the cafe, the m friends found n different kinds of milk on the menu, that's why they ordered n bottles — one bottle of each kind. We know that the volume of milk in each bottle equals w.When the bottles were brought in, they decided to pour all the milk evenly among the m cups, so that each got a cup. As a punishment for not passing the test Igor was appointed the person to pour the milk. He protested that he was afraid to mix something up and suggested to distribute the drink so that the milk from each bottle was in no more than two different cups. His friends agreed but they suddenly faced the following problem — and what is actually the way to do it?Help them and write the program that will help to distribute the milk among the cups and drink it as quickly as possible!Note that due to Igor K.'s perfectly accurate eye and unswerving hands, he can pour any fractional amount of milk from any bottle to any cup.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"532"Marin feels exhausted after a long day of cosplay, so Gojou invites her to play a game!Marin and Gojou take turns to place one of their tokens on an $$$n \times n$$$ grid with Marin starting first. There are some restrictions and allowances on where to place tokens:   Apart from the first move, the token placed by a player must be more than Manhattan distance $$$k$$$ away from the previous token placed on the matrix. In other words, if a player places a token at $$$(x_1, y_1)$$$, then the token placed by the other player in the next move must be in a cell $$$(x_2, y_2)$$$ satisfying $$$|x_2 - x_1| + |y_2 - y_1| > k$$$.  Apart from the previous restriction, a token can be placed anywhere on the matrix, including cells where tokens were previously placed by any player. Whenever a player places a token on cell $$$(x, y)$$$, that player gets $$$v_{x,\ y}$$$ points. All values of $$$v$$$ on the grid are distinct. You still get points from a cell even if tokens were already placed onto the cell. The game finishes when each player makes $$$10^{100}$$$ moves.Marin and Gojou will play $$$n^2$$$ games. For each cell of the grid, there will be exactly one game where Marin places a token on that cell on her first move. Please answer for each game, if Marin and Gojou play optimally (after Marin's first move), who will have more points at the end? Or will the game end in a draw (both players have the same points at the end)?","['games', 'math', 'number theory']","[1, 0, 0, 1, 1, 0, 0, 0]"533"Cirno gives AquaMoon a problem. There are $$$m$$$ people numbered from $$$0$$$ to $$$m - 1$$$. They are standing on a coordinate axis in points with positive integer coordinates. They are facing right (i.e. in the direction of the coordinate increase). At this moment everyone will start running with the constant speed in the direction of coordinate increasing. The initial coordinate of the $$$i$$$-th person on the line is $$$x_i$$$, and the speed of the $$$i$$$-th person is $$$v_i$$$. So the coordinate of the $$$i$$$-th person at the moment $$$t$$$ will be $$$x_i + t \cdot v_i$$$.Cirno captured the coordinates of $$$m$$$ people in $$$k$$$ consecutive integer moments from $$$0$$$ to $$$k - 1$$$. In every moment, the coordinates of $$$m$$$ people were recorded in arbitrary order.To make the problem more funny, Cirno modified one coordinate at the moment $$$y$$$ ($$$0 < y < k-1$$$) to a different integer.AquaMoon wants to find the moment $$$y$$$ and the original coordinate $$$p$$$ before the modification. Actually, she is not a programmer at all. So she wasn't able to solve it. Can you help her?",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"534"Stepan had a favorite string s which consisted of the lowercase letters of the Latin alphabet. After graduation, he decided to remember it, but it was a long time ago, so he can't now remember it. But Stepan remembers some information about the string, namely the sequence of integers c1, c2, ..., cn, where n equals the length of the string s, and ci equals the number of substrings in the string s with the length i, consisting of the same letters. The substring is a sequence of consecutive characters in the string s.For example, if the Stepan's favorite string is equal to ""tttesst"", the sequence c looks like: c = [7, 3, 1, 0, 0, 0, 0].Stepan asks you to help to repair his favorite string s according to the given sequence c1, c2, ..., cn. ",[],"[0, 0, 0, 0, 0, 0, 0, 0]"535"You are given a picture consisting of $$$n$$$ rows and $$$m$$$ columns. Rows are numbered from $$$1$$$ to $$$n$$$ from the top to the bottom, columns are numbered from $$$1$$$ to $$$m$$$ from the left to the right. Each cell is painted either black or white. You think that this picture is not interesting enough. You consider a picture to be interesting if there is at least one cross in it. A cross is represented by a pair of numbers $$$x$$$ and $$$y$$$, where $$$1 \le x \le n$$$ and $$$1 \le y \le m$$$, such that all cells in row $$$x$$$ and all cells in column $$$y$$$ are painted black.For examples, each of these pictures contain crosses:  The fourth picture contains 4 crosses: at $$$(1, 3)$$$, $$$(1, 5)$$$, $$$(3, 3)$$$ and $$$(3, 5)$$$.Following images don't contain crosses:  You have a brush and a can of black paint, so you can make this picture interesting. Each minute you may choose a white cell and paint it black.What is the minimum number of minutes you have to spend so the resulting picture contains at least one cross?You are also asked to answer multiple independent queries.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"536"Wilbur the pig is tinkering with arrays again. He has the array a1, a2, ..., an initially consisting of n zeros. At one step, he can choose any index i and either add 1 to all elements ai, ai + 1, ... , an or subtract 1 from all elements ai, ai + 1, ..., an. His goal is to end up with the array b1, b2, ..., bn. Of course, Wilbur wants to achieve this goal in the minimum number of steps and asks you to compute this value.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"537"C*++ language is quite similar to C++. The similarity manifests itself in the fact that the programs written in C*++ sometimes behave unpredictably and lead to absolutely unexpected effects. For example, let's imagine an arithmetic expression in C*++ that looks like this (expression is the main term):  expression ::= summand | expression + summand | expression - summand  summand ::= increment | coefficient*increment  increment ::= a++ | ++a  coefficient ::= 0|1|2|...|1000 For example, ""5*a++-3*++a+a++"" is a valid expression in C*++.Thus, we have a sum consisting of several summands divided by signs ""+"" or ""-"". Every summand is an expression ""a++"" or ""++a"" multiplied by some integer coefficient. If the coefficient is omitted, it is suggested being equal to 1.The calculation of such sum in C*++ goes the following way. First all the summands are calculated one after another, then they are summed by the usual arithmetic rules. If the summand contains ""a++"", then during the calculation first the value of the ""a"" variable is multiplied by the coefficient, then value of ""a"" is increased by 1. If the summand contains ""++a"", then the actions on it are performed in the reverse order: first ""a"" is increased by 1, then — multiplied by the coefficient.The summands may be calculated in any order, that's why sometimes the result of the calculation is completely unpredictable! Your task is to find its largest possible value.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"538"Companies always have a lot of equipment, furniture and other things. All of them should be tracked. To do this, there is an inventory number assigned with each item. It is much easier to create a database by using those numbers and keep the track of everything.During an audit, you were surprised to find out that the items are not numbered sequentially, and some items even share the same inventory number! There is an urgent need to fix it. You have chosen to make the numbers of the items sequential, starting with 1. Changing a number is quite a time-consuming process, and you would like to make maximum use of the current numbering.You have been given information on current inventory numbers for n items in the company. Renumber items so that their inventory numbers form a permutation of numbers from 1 to n by changing the number of as few items as possible. Let us remind you that a set of n numbers forms a permutation if all the numbers are in the range from 1 to n, and no two numbers are equal.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"539"In a far away land, there are two cities near a river. One day, the cities decide that they have too little space and would like to reclaim some of the river area into land.The river area can be represented by a grid with r rows and exactly two columns — each cell represents a rectangular area. The rows are numbered 1 through r from top to bottom, while the columns are numbered 1 and 2.Initially, all of the cells are occupied by the river. The plan is to turn some of those cells into land one by one, with the cities alternately choosing a cell to reclaim, and continuing until no more cells can be reclaimed.However, the river is also used as a major trade route. The cities need to make sure that ships will still be able to sail from one end of the river to the other. More formally, if a cell (r, c) has been reclaimed, it is not allowed to reclaim any of the cells (r - 1, 3 - c), (r, 3 - c), or (r + 1, 3 - c).The cities are not on friendly terms, and each city wants to be the last city to reclaim a cell (they don't care about how many cells they reclaim, just who reclaims a cell last). The cities have already reclaimed n cells. Your job is to determine which city will be the last to reclaim a cell, assuming both choose cells optimally from current moment.",['games'],"[1, 0, 0, 0, 0, 0, 0, 0]"540"Polycarp is a music editor at the radio station. He received a playlist for tomorrow, that can be represented as a sequence a1, a2, ..., an, where ai is a band, which performs the i-th song. Polycarp likes bands with the numbers from 1 to m, but he doesn't really like others. We define as bj the number of songs the group j is going to perform tomorrow. Polycarp wants to change the playlist in such a way that the minimum among the numbers b1, b2, ..., bm will be as large as possible.Find this maximum possible value of the minimum among the bj (1 ≤ j ≤ m), and the minimum number of changes in the playlist Polycarp needs to make to achieve it. One change in the playlist is a replacement of the performer of the i-th song with any other group.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"541"Vasya has found a piece of paper with a coordinate system written on it. There are n distinct squares drawn in this coordinate system. Let's number the squares with integers from 1 to n. It turned out that points with coordinates (0, 0) and (ai, ai) are the opposite corners of the i-th square.Vasya wants to find such integer point (with integer coordinates) of the plane, that belongs to exactly k drawn squares. We'll say that a point belongs to a square, if the point is located either inside the square, or on its boundary. Help Vasya find a point that would meet the described limits.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"542"There are $$$n$$$ startups. Startups can be active or acquired. If a startup is acquired, then that means it has exactly one active startup that it is following. An active startup can have arbitrarily many acquired startups that are following it. An active startup cannot follow any other startup.The following steps happen until there is exactly one active startup. The following sequence of steps takes exactly 1 day.  Two distinct active startups $$$A$$$, $$$B$$$, are chosen uniformly at random.  A fair coin is flipped, and with equal probability, $$$A$$$ acquires $$$B$$$ or $$$B$$$ acquires $$$A$$$ (i.e. if $$$A$$$ acquires $$$B$$$, then that means $$$B$$$'s state changes from active to acquired, and its starts following $$$A$$$).  When a startup changes from active to acquired, all of its previously acquired startups become active. For example, the following scenario can happen: Let's say $$$A$$$, $$$B$$$ are active startups. $$$C$$$, $$$D$$$, $$$E$$$ are acquired startups under $$$A$$$, and $$$F$$$, $$$G$$$ are acquired startups under $$$B$$$: Active startups are shown in red. If $$$A$$$ acquires $$$B$$$, then the state will be $$$A$$$, $$$F$$$, $$$G$$$ are active startups. $$$C$$$, $$$D$$$, $$$E$$$, $$$B$$$ are acquired startups under $$$A$$$. $$$F$$$ and $$$G$$$ have no acquired startups:  If instead, $$$B$$$ acquires $$$A$$$, then the state will be $$$B$$$, $$$C$$$, $$$D$$$, $$$E$$$ are active startups. $$$F$$$, $$$G$$$, $$$A$$$ are acquired startups under $$$B$$$. $$$C$$$, $$$D$$$, $$$E$$$ have no acquired startups:  You are given the initial state of the startups. For each startup, you are told if it is either acquired or active. If it is acquired, you are also given the index of the active startup that it is following.You're now wondering, what is the expected number of days needed for this process to finish with exactly one active startup at the end.It can be shown the expected number of days can be written as a rational number $$$P/Q$$$, where $$$P$$$ and $$$Q$$$ are co-prime integers, and $$$Q \not= 0 \pmod{10^9+7}$$$. Return the value of $$$P \cdot Q^{-1}$$$ modulo $$$10^9+7$$$.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"543"  While trading on his favorite exchange trader William realized that he found a vulnerability. Using this vulnerability he could change the values of certain internal variables to his advantage. To play around he decided to change the values of all internal variables from $$$a_1, a_2, \ldots, a_n$$$ to $$$-a_1, -a_2, \ldots, -a_n$$$. For some unknown reason, the number of service variables is always an even number.William understands that with his every action he attracts more and more attention from the exchange's security team, so the number of his actions must not exceed $$$5\,000$$$ and after every operation no variable can have an absolute value greater than $$$10^{18}$$$. William can perform actions of two types for two chosen variables with indices $$$i$$$ and $$$j$$$, where $$$i < j$$$:  Perform assignment $$$a_i = a_i + a_j$$$  Perform assignment $$$a_j = a_j - a_i$$$  William wants you to develop a strategy that will get all the internal variables to the desired values.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"544"There is a field divided into $$$n$$$ rows and $$$m$$$ columns. Some cells are empty (denoted as E), other cells contain robots (denoted as R).You can send a command to all robots at the same time. The command can be of one of the four types:  move up;  move right;  move down;  move left. When you send a command, all robots at the same time attempt to take one step in the direction you picked. If a robot tries to move outside the field, it explodes; otherwise, every robot moves to an adjacent cell in the chosen direction.You can send as many commands as you want (possibly, zero), in any order. Your goal is to make at least one robot reach the upper left corner of the field. Can you do this without forcing any of the robots to explode?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"545"After overcoming the stairs Dasha came to classes. She needed to write a password to begin her classes. The password is a string of length n which satisfies the following requirements:  There is at least one digit in the string,  There is at least one lowercase (small) letter of the Latin alphabet in the string,  There is at least one of three listed symbols in the string: '#', '*', '&'.   Considering that these are programming classes it is not easy to write the password.For each character of the password we have a fixed string of length m, on each of these n strings there is a pointer on some character. The i-th character displayed on the screen is the pointed character in the i-th string. Initially, all pointers are on characters with indexes 1 in the corresponding strings (all positions are numbered starting from one).During one operation Dasha can move a pointer in one string one character to the left or to the right. Strings are cyclic, it means that when we move the pointer which is on the character with index 1 to the left, it moves to the character with the index m, and when we move it to the right from the position m it moves to the position 1.You need to determine the minimum number of operations necessary to make the string displayed on the screen a valid password. ",[],"[0, 0, 0, 0, 0, 0, 0, 0]"546"Some natural number was written on the board. Its sum of digits was not less than k. But you were distracted a bit, and someone changed this number to n, replacing some digits with others. It's known that the length of the number didn't change.You have to find the minimum number of digits in which these two numbers can differ.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"547"Berland Government decided to improve relations with neighboring countries. First of all, it was decided to build new roads so that from each city of Berland and neighboring countries it became possible to reach all the others. There are n cities in Berland and neighboring countries in total and exactly n - 1 two-way roads. Because of the recent financial crisis, the Berland Government is strongly pressed for money, so to build a new road it has to close some of the existing ones. Every day it is possible to close one existing road and immediately build a new one. Your task is to determine how many days would be needed to rebuild roads so that from each city it became possible to reach all the others, and to draw a plan of closure of old roads and building of new ones.","['graphs', 'trees']","[0, 0, 1, 0, 0, 0, 0, 1]"548"She does her utmost to flawlessly carry out a person's last rites and preserve the world's balance of yin and yang.Hu Tao, being the little prankster she is, has tried to scare you with this graph problem! You are given a connected undirected graph of $$$n$$$ nodes with $$$m$$$ edges. You also have $$$q$$$ queries. Each query consists of two nodes $$$a$$$ and $$$b$$$.Initially, all edges in the graph have a weight of $$$0$$$. For each query, you must choose a simple path starting from $$$a$$$ and ending at $$$b$$$. Then you add $$$1$$$ to every edge along this path. Determine if it's possible, after processing all $$$q$$$ queries, for all edges in this graph to have an even weight. If so, output the choice of paths for each query. If it is not possible, determine the smallest number of extra queries you could add to make it possible. It can be shown that this number will not exceed $$$10^{18}$$$ under the given constraints.A simple path is defined as any path that does not visit a node more than once.An edge is said to have an even weight if its value is divisible by $$$2$$$.","['graphs', 'trees']","[0, 0, 1, 0, 0, 0, 0, 1]"549"Today Adilbek is taking his probability theory test. Unfortunately, when Adilbek arrived at the university, there had already been a long queue of students wanting to take the same test. Adilbek has estimated that he will be able to start the test only $$$T$$$ seconds after coming. Fortunately, Adilbek can spend time without revising any boring theorems or formulas. He has an app on this smartphone which contains $$$n$$$ Japanese crosswords to solve. Adilbek has decided to solve them all one by one in the order they are listed in the app, without skipping any crossword. For each crossword, a number $$$t_i$$$ is given that represents the time it takes an average crossword expert to solve this crossword (the time is given in seconds).Adilbek is a true crossword expert, but, unfortunately, he is sometimes unlucky in choosing the way to solve the crossword. So, it takes him either $$$t_i$$$ seconds or $$$t_i + 1$$$ seconds to solve the $$$i$$$-th crossword, equiprobably (with probability $$$\frac{1}{2}$$$ he solves the crossword in exactly $$$t_i$$$ seconds, and with probability $$$\frac{1}{2}$$$ he has to spend an additional second to finish the crossword). All these events are independent.After $$$T$$$ seconds pass (or after solving the last crossword, if he manages to do it in less than $$$T$$$ seconds), Adilbek closes the app (if he finishes some crossword at the same moment, that crossword is considered solved; otherwise Adilbek does not finish solving the current crossword at all). He thinks it would be an interesting probability theory problem to calculate $$$E$$$ — the expected number of crosswords he will be able to solve completely. Can you calculate it? Recall that the expected value of a discrete random variable is the probability-weighted average of all possible values — in this problem it means that the expected value of the number of solved crosswords can be calculated as $$$E = \sum \limits_{i = 0}^{n} i p_i$$$, where $$$p_i$$$ is the probability that Adilbek will solve exactly $$$i$$$ crosswords. We can represent $$$E$$$ as rational fraction $$$\frac{P}{Q}$$$ with $$$Q > 0$$$. To give the answer, you should print $$$P \cdot Q^{-1} \bmod (10^9 + 7)$$$.","['number theory', 'probabilities']","[0, 0, 0, 0, 1, 1, 0, 0]"550"A permutation is a sequence of integers p1, p2, ..., pn, consisting of n distinct positive integers, each of them doesn't exceed n. Let's denote the i-th element of permutation p as pi. We'll call number n the size of permutation p1, p2, ..., pn.Nickolas adores permutations. He likes some permutations more than the others. He calls such permutations perfect. A perfect permutation is such permutation p that for any i (1 ≤ i ≤ n) (n is the permutation size) the following equations hold ppi = i and pi ≠ i. Nickolas asks you to print any perfect permutation of size n for the given n.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"551"Bob got a job as a system administrator in X corporation. His first task was to connect n servers with the help of m two-way direct connection so that it becomes possible to transmit data from one server to any other server via these connections. Each direct connection has to link two different servers, each pair of servers should have at most one direct connection. Y corporation, a business rival of X corporation, made Bob an offer that he couldn't refuse: Bob was asked to connect the servers in such a way, that when server with index v fails, the transmission of data between some other two servers becomes impossible, i.e. the system stops being connected. Help Bob connect the servers.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"552"Once Petya was in such a good mood that he decided to help his mum with the washing-up. There were n dirty bowls in the sink. From the geometrical point of view each bowl looks like a blunted cone. We can disregard the width of the walls and bottom. Petya puts the clean bowls one on another naturally, i. e. so that their vertical axes coincide (see the picture). You will be given the order in which Petya washes the bowls. Determine the height of the construction, i.e. the distance from the bottom of the lowest bowl to the top of the highest one.   ",['geometry'],"[0, 1, 0, 0, 0, 0, 0, 0]"553"Jeff has become friends with Furik. Now these two are going to play one quite amusing game.At the beginning of the game Jeff takes a piece of paper and writes down a permutation consisting of n numbers: p1, p2, ..., pn. Then the guys take turns to make moves, Jeff moves first. During his move, Jeff chooses two adjacent permutation elements and then the boy swaps them. During his move, Furic tosses a coin and if the coin shows ""heads"" he chooses a random pair of adjacent elements with indexes i and i + 1, for which an inequality pi > pi + 1 holds, and swaps them. But if the coin shows ""tails"", Furik chooses a random pair of adjacent elements with indexes i and i + 1, for which the inequality pi < pi + 1 holds, and swaps them. If the coin shows ""heads"" or ""tails"" and Furik has multiple ways of adjacent pairs to take, then he uniformly takes one of the pairs. If Furik doesn't have any pair to take, he tosses a coin one more time. The game ends when the permutation is sorted in the increasing order.Jeff wants the game to finish as quickly as possible (that is, he wants both players to make as few moves as possible). Help Jeff find the minimum mathematical expectation of the number of moves in the game if he moves optimally well.You can consider that the coin shows the heads (or tails) with the probability of 50 percent.",['probabilities'],"[0, 0, 0, 0, 0, 1, 0, 0]"554"You are organizing a cycling race on the streets of the city. The city contains n junctions, some pairs of them are connected by roads; on each road you can move in any direction. No two roads connect the same pair of intersections, and no road connects the intersection with itself.You want the race to be open to both professional athletes and beginner cyclists, and that's why you will organize the race in three nominations: easy, moderate and difficult; each participant will choose the more suitable nomination. For each nomination you must choose the route — the chain of junctions, consecutively connected by roads. Routes must meet the following conditions: all three routes should start at the same intersection, and finish at the same intersection (place of start and finish can't be the same); to avoid collisions, no two routes can have common junctions (except for the common start and finish), and can not go along the same road (irrespective of the driving direction on the road for those two routes); no route must pass twice through the same intersection or visit the same road twice (irrespective of the driving direction on the road for the first and second time of visit).Preparing for the competition is about to begin, and you need to determine the routes of the race as quickly as possible. The length of the routes is not important, it is only important that all the given requirements were met.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"555"A sequence a0, a1, ..., at - 1 is called increasing if ai - 1 < ai for each i: 0 < i < t.You are given a sequence b0, b1, ..., bn - 1 and a positive integer d. In each move you may choose one element of the given sequence and add d to it. What is the least number of moves required to make the given sequence increasing?",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"556"There is an infinite sequence consisting of all positive integers in the increasing order: p = {1, 2, 3, ...}. We performed n swap operations with this sequence. A swap(a, b) is an operation of swapping the elements of the sequence on positions a and b. Your task is to find the number of inversions in the resulting sequence, i.e. the number of such index pairs (i, j), that i < j and pi > pj.",['trees'],"[0, 0, 0, 0, 0, 0, 0, 1]"557"You are given a directed graph $$$G$$$ which can contain loops (edges from a vertex to itself). Multi-edges are absent in $$$G$$$ which means that for all ordered pairs $$$(u, v)$$$ exists at most one edge from $$$u$$$ to $$$v$$$. Vertices are numbered from $$$1$$$ to $$$n$$$.A path from $$$u$$$ to $$$v$$$ is a sequence of edges such that:  vertex $$$u$$$ is the start of the first edge in the path;  vertex $$$v$$$ is the end of the last edge in the path;  for all pairs of adjacent edges next edge starts at the vertex that the previous edge ends on. We will assume that the empty sequence of edges is a path from $$$u$$$ to $$$u$$$.For each vertex $$$v$$$ output one of four values:  $$$0$$$, if there are no paths from $$$1$$$ to $$$v$$$;  $$$1$$$, if there is only one path from $$$1$$$ to $$$v$$$;  $$$2$$$, if there is more than one path from $$$1$$$ to $$$v$$$ and the number of paths is finite;  $$$-1$$$, if the number of paths from $$$1$$$ to $$$v$$$ is infinite. Let's look at the example shown in the figure.  Then:  the answer for vertex $$$1$$$ is $$$1$$$: there is only one path from $$$1$$$ to $$$1$$$ (path with length $$$0$$$);  the answer for vertex $$$2$$$ is $$$0$$$: there are no paths from $$$1$$$ to $$$2$$$;  the answer for vertex $$$3$$$ is $$$1$$$: there is only one path from $$$1$$$ to $$$3$$$ (it is the edge $$$(1, 3)$$$);  the answer for vertex $$$4$$$ is $$$2$$$: there are more than one paths from $$$1$$$ to $$$4$$$ and the number of paths are finite (two paths: $$$[(1, 3), (3, 4)]$$$ and $$$[(1, 4)]$$$);  the answer for vertex $$$5$$$ is $$$-1$$$: the number of paths from $$$1$$$ to $$$5$$$ is infinite (the loop can be used in a path many times);  the answer for vertex $$$6$$$ is $$$-1$$$: the number of paths from $$$1$$$ to $$$6$$$ is infinite (the loop can be used in a path many times). ","['graphs', 'trees']","[0, 0, 1, 0, 0, 0, 0, 1]"558"A process RAM is a sequence of bytes that are indexed from 1 to n. Polycarpus's program contains such instructions as ""memset"", that is, the operations of filling memory cells on a segment with some value. The details are: the code only contains m instructions that look like ""set13 a_i l_i"". Instruction i fills a continuous memory segment of length li, starting from cell number ai, (that it cells with numbers ai, ai + 1, ..., ai + li - 1) with values 13.In Polycarpus's code, the optimizer's task is to remove the maximum number of instructions from his code in such a way that the remaining instructions set value 13 in all the memory bytes that got this value from the code before the optimization. Also, the value 13 should be set only in the memory bytes that got this value from the code before the optimization. Your task is to implement the optimizer for such program.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"559You are standing on the $$$\mathit{OX}$$$-axis at point $$$0$$$ and you want to move to an integer point $$$x > 0$$$.You can make several jumps. Suppose you're currently at point $$$y$$$ ($$$y$$$ may be negative) and jump for the $$$k$$$-th time. You can:   either jump to the point $$$y + k$$$  or jump to the point $$$y - 1$$$. What is the minimum number of jumps you need to reach the point $$$x$$$?,['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"560"Mark is cleaning a row of $$$n$$$ rooms. The $$$i$$$-th room has a nonnegative dust level $$$a_i$$$. He has a magical cleaning machine that can do the following three-step operation.   Select two indices $$$i<j$$$ such that the dust levels $$$a_i$$$, $$$a_{i+1}$$$, $$$\dots$$$, $$$a_{j-1}$$$ are all strictly greater than $$$0$$$.  Set $$$a_i$$$ to $$$a_i-1$$$.  Set $$$a_j$$$ to $$$a_j+1$$$.  Mark's goal is to make $$$a_1 = a_2 = \ldots = a_{n-1} = 0$$$ so that he can nicely sweep the $$$n$$$-th room. Determine the minimum number of operations needed to reach his goal.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"561"Petya loves lucky numbers very much. Everybody knows that lucky numbers are positive integers whose decimal record contains only the lucky digits 4 and 7. For example, numbers 47, 744, 4 are lucky and 5, 17, 467 are not.Petya has sequence a consisting of n integers.The subsequence of the sequence a is such subsequence that can be obtained from a by removing zero or more of its elements.Two sequences are considered different if index sets of numbers included in them are different. That is, the values ​of the elements ​do not matter in the comparison of subsequences. In particular, any sequence of length n has exactly 2n different subsequences (including an empty subsequence).A subsequence is considered lucky if it has a length exactly k and does not contain two identical lucky numbers (unlucky numbers can be repeated any number of times).Help Petya find the number of different lucky subsequences of the sequence a. As Petya's parents don't let him play with large numbers, you should print the result modulo prime number 1000000007 (109 + 7).",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"562"The Saratov State University Olympiad Programmers Training Center (SSU OPTC) has n students. For each student you know the number of times he/she has participated in the ACM ICPC world programming championship. According to the ACM ICPC rules, each person can participate in the world championship at most 5 times.The head of the SSU OPTC is recently gathering teams to participate in the world championship. Each team must consist of exactly three people, at that, any person cannot be a member of two or more teams. What maximum number of teams can the head make if he wants each team to participate in the world championship with the same members at least k times?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"563"Daemon Targaryen decided to stop looking like a Metin2 character. He turned himself into the most beautiful thing, a bracket sequence.For a bracket sequence, we can do two kind of operations:  Select one of its substrings$$$^\dagger$$$ and cyclic shift it to the right. For example, after a cyclic shift to the right, ""(())"" will become "")(()"";  Insert any bracket, opening '(' or closing ')', wherever you want in the sequence. We define the cost of a bracket sequence as the minimum number of such operations to make it balanced$$$^\ddagger$$$.Given a bracket sequence $$$s$$$ of length $$$n$$$, find the sum of costs across all its $$$\frac{n(n+1)}{2}$$$ non-empty substrings. Note that for each substring we calculate the cost independently.$$$^\dagger$$$ A string $$$a$$$ is a substring of a string $$$b$$$ if $$$a$$$ can be obtained from $$$b$$$ by deletion of several (possibly, zero or all) characters from the beginning and several (possibly, zero or all) characters from the end.$$$^\ddagger$$$ A sequence of brackets is called balanced if one can turn it into a valid math expression by adding characters $$$+$$$ and $$$1$$$. For example, sequences ""(())()"", ""()"", and ""(()(()))"" are balanced, while "")("", ""(()"", and ""(()))("" are not.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"564"This is the easy version of the problem. The only difference is the constraints on $$$n$$$ and $$$k$$$. You can make hacks only if all versions of the problem are solved.You have a string $$$s$$$, and you can do two types of operations on it:   Delete the last character of the string.  Duplicate the string: $$$s:=s+s$$$, where $$$+$$$ denotes concatenation. You can use each operation any number of times (possibly none).Your task is to find the lexicographically smallest string of length exactly $$$k$$$ that can be obtained by doing these operations on string $$$s$$$.A string $$$a$$$ is lexicographically smaller than a string $$$b$$$ if and only if one of the following holds:  $$$a$$$ is a prefix of $$$b$$$, but $$$a\ne b$$$;  In the first position where $$$a$$$ and $$$b$$$ differ, the string $$$a$$$ has a letter that appears earlier in the alphabet than the corresponding letter in $$$b$$$. ",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"565"Maksim has $$$n$$$ objects and $$$m$$$ boxes, each box has size exactly $$$k$$$. Objects are numbered from $$$1$$$ to $$$n$$$ in order from left to right, the size of the $$$i$$$-th object is $$$a_i$$$.Maksim wants to pack his objects into the boxes and he will pack objects by the following algorithm: he takes one of the empty boxes he has, goes from left to right through the objects, and if the $$$i$$$-th object fits in the current box (the remaining size of the box is greater than or equal to $$$a_i$$$), he puts it in the box, and the remaining size of the box decreases by $$$a_i$$$. Otherwise he takes the new empty box and continues the process above. If he has no empty boxes and there is at least one object not in some box then Maksim cannot pack the chosen set of objects.Maksim wants to know the maximum number of objects he can pack by the algorithm above. To reach this target, he will throw out the leftmost object from the set until the remaining set of objects can be packed in boxes he has. Your task is to say the maximum number of objects Maksim can pack in boxes he has.Each time when Maksim tries to pack the objects into the boxes, he will make empty all the boxes he has before do it (and the relative order of the remaining set of objects will not change).",[],"[0, 0, 0, 0, 0, 0, 0, 0]"566"Anton is playing a very interesting computer game, but now he is stuck at one of the levels. To pass to the next level he has to prepare n potions.Anton has a special kettle, that can prepare one potions in x seconds. Also, he knows spells of two types that can faster the process of preparing potions.  Spells of this type speed up the preparation time of one potion. There are m spells of this type, the i-th of them costs bi manapoints and changes the preparation time of each potion to ai instead of x.  Spells of this type immediately prepare some number of potions. There are k such spells, the i-th of them costs di manapoints and instantly create ci potions. Anton can use no more than one spell of the first type and no more than one spell of the second type, and the total number of manapoints spent should not exceed s. Consider that all spells are used instantly and right before Anton starts to prepare potions.Anton wants to get to the next level as fast as possible, so he is interested in the minimum number of time he needs to spent in order to prepare at least n potions.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"567"One not particularly beautiful evening Valera got very bored. To amuse himself a little bit, he found the following game.He took a checkered white square piece of paper, consisting of n × n cells. After that, he started to paint the white cells black one after the other. In total he painted m different cells on the piece of paper. Since Valera was keen on everything square, he wondered, how many moves (i.e. times the boy paints a square black) he should make till a black square with side 3 can be found on the piece of paper. But Valera does not know the answer to this question, so he asks you to help him.Your task is to find the minimum number of moves, till the checkered piece of paper has at least one black square with side of 3. Otherwise determine that such move does not exist.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"568"A binary string is a string that consists of characters $$$0$$$ and $$$1$$$.Let $$$\operatorname{MEX}$$$ of a binary string be the smallest digit among $$$0$$$, $$$1$$$, or $$$2$$$ that does not occur in the string. For example, $$$\operatorname{MEX}$$$ of $$$001011$$$ is $$$2$$$, because $$$0$$$ and $$$1$$$ occur in the string at least once, $$$\operatorname{MEX}$$$ of $$$1111$$$ is $$$0$$$, because $$$0$$$ and $$$2$$$ do not occur in the string and $$$0 < 2$$$.A binary string $$$s$$$ is given. You should cut it into any number of substrings such that each character is in exactly one substring. It is possible to cut the string into a single substring — the whole string.A string $$$a$$$ is a substring of a string $$$b$$$ if $$$a$$$ can be obtained from $$$b$$$ by deletion of several (possibly, zero or all) characters from the beginning and several (possibly, zero or all) characters from the end.What is the minimal sum of $$$\operatorname{MEX}$$$ of all substrings pieces can be?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"569"You are given two arrays $$$a$$$ and $$$b$$$, both of length $$$n$$$. All elements of both arrays are from $$$0$$$ to $$$n-1$$$.You can reorder elements of the array $$$b$$$ (if you want, you may leave the order of elements as it is). After that, let array $$$c$$$ be the array of length $$$n$$$, the $$$i$$$-th element of this array is $$$c_i = (a_i + b_i) \% n$$$, where $$$x \% y$$$ is $$$x$$$ modulo $$$y$$$.Your task is to reorder elements of the array $$$b$$$ to obtain the lexicographically minimum possible array $$$c$$$.Array $$$x$$$ of length $$$n$$$ is lexicographically less than array $$$y$$$ of length $$$n$$$, if there exists such $$$i$$$ ($$$1 \le i \le n$$$), that $$$x_i < y_i$$$, and for any $$$j$$$ ($$$1 \le j < i$$$) $$$x_j = y_j$$$.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"570"In the official contest this problem has a different statement, for which jury's solution was working incorrectly, and for this reason it was excluded from the contest. This mistake have been fixed and the current given problem statement and model solution corresponds to what jury wanted it to be during the contest.Vova and Lesha are friends. They often meet at Vova's place and compete against each other in a computer game named The Ancient Papyri: Swordsink. Vova always chooses a warrior as his fighter and Leshac chooses an archer. After that they should choose initial positions for their characters and start the fight. A warrior is good at melee combat, so Vova will try to make the distance between fighters as small as possible. An archer prefers to keep the enemy at a distance, so Lesha will try to make the initial distance as large as possible.There are n (n is always even) possible starting positions for characters marked along the Ox axis. The positions are given by their distinct coordinates x1, x2, ..., xn, two characters cannot end up at the same position.Vova and Lesha take turns banning available positions, Vova moves first. During each turn one of the guys bans exactly one of the remaining positions. Banned positions cannot be used by both Vova and Lesha. They continue to make moves until there are only two possible positions remaining (thus, the total number of moves will be n - 2). After that Vova's character takes the position with the lesser coordinate and Lesha's character takes the position with the bigger coordinate and the guys start fighting.Vova and Lesha are already tired by the game of choosing positions, as they need to play it before every fight, so they asked you (the developer of the The Ancient Papyri: Swordsink) to write a module that would automatically determine the distance at which the warrior and the archer will start fighting if both Vova and Lesha play optimally.",['games'],"[1, 0, 0, 0, 0, 0, 0, 0]"571"Alice received a set of Toy Train™ from Bob. It consists of one train and a connected railway network of $$$n$$$ stations, enumerated from $$$1$$$ through $$$n$$$. The train occupies one station at a time and travels around the network of stations in a circular manner. More precisely, the immediate station that the train will visit after station $$$i$$$ is station $$$i+1$$$ if $$$1 \leq i < n$$$ or station $$$1$$$ if $$$i = n$$$. It takes the train $$$1$$$ second to travel to its next station as described.Bob gave Alice a fun task before he left: to deliver $$$m$$$ candies that are initially at some stations to their independent destinations using the train. The candies are enumerated from $$$1$$$ through $$$m$$$. Candy $$$i$$$ ($$$1 \leq i \leq m$$$), now at station $$$a_i$$$, should be delivered to station $$$b_i$$$ ($$$a_i \neq b_i$$$).    The blue numbers on the candies correspond to $$$b_i$$$ values. The image corresponds to the $$$1$$$-st example. The train has infinite capacity, and it is possible to load off any number of candies at a station. However, only at most one candy can be loaded from a station onto the train before it leaves the station. You can choose any candy at this station. The time it takes to move the candies is negligible.Now, Alice wonders how much time is needed for the train to deliver all candies. Your task is to find, for each station, the minimum time the train would need to deliver all the candies were it to start from there.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"572"Valera loves to participate in competitions. Especially in programming contests. Today he has participated in the contest with his team, consisting of n students (including Valera). This contest was an individual competition, so each student in the team solved problems individually.After the contest was over, Valera was interested in results. He found out that:  each student in the team scored at least l points and at most r points;  in total, all members of the team scored exactly sall points;  the total score of the k members of the team who scored the most points is equal to exactly sk; more formally, if a1, a2, ..., an is the sequence of points earned by the team of students in the non-increasing order (a1 ≥ a2 ≥ ... ≥ an), then sk = a1 + a2 + ... + ak. However, Valera did not find out exactly how many points each of n students scored. Valera asked you to recover any distribution of scores between the students of the team, such that all the conditions above are met.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"573"Bertown is under siege! The attackers have blocked all the ways out and their cannon is bombarding the city. Fortunately, Berland intelligence managed to intercept the enemies' shooting plan. Let's introduce the Cartesian system of coordinates, the origin of which coincides with the cannon's position, the Ox axis is directed rightwards in the city's direction, the Oy axis is directed upwards (to the sky). The cannon will make n more shots. The cannon balls' initial speeds are the same in all the shots and are equal to V, so that every shot is characterized by only one number alphai which represents the angle at which the cannon fires. Due to the cannon's technical peculiarities this angle does not exceed 45 angles (π / 4). We disregard the cannon sizes and consider the firing made from the point (0, 0).The balls fly according to the known physical laws of a body thrown towards the horizon at an angle: vx(t) = V·cos(alpha) vy(t) = V·sin(alpha)  –  g·t x(t) = V·cos(alpha)·t y(t) = V·sin(alpha)·t  –  g·t2 / 2Think of the acceleration of gravity g as equal to 9.8.Bertown defends m walls. The i-th wall is represented as a vertical segment (xi, 0) - (xi, yi). When a ball hits a wall, it gets stuck in it and doesn't fly on. If a ball doesn't hit any wall it falls on the ground (y = 0) and stops. If the ball exactly hits the point (xi, yi), it is considered stuck. Your task is to find for each ball the coordinates of the point where it will be located in the end.",['geometry'],"[0, 1, 0, 0, 0, 0, 0, 0]"574"Nura wants to buy k gadgets. She has only s burles for that. She can buy each gadget for dollars or for pounds. So each gadget is selling only for some type of currency. The type of currency and the cost in that currency are not changing.Nura can buy gadgets for n days. For each day you know the exchange rates of dollar and pound, so you know the cost of conversion burles to dollars or to pounds.Each day (from 1 to n) Nura can buy some gadgets by current exchange rate. Each day she can buy any gadgets she wants, but each gadget can be bought no more than once during n days.Help Nura to find the minimum day index when she will have k gadgets. Nura always pays with burles, which are converted according to the exchange rate of the purchase day. Nura can't buy dollars or pounds, she always stores only burles. Gadgets are numbered with integers from 1 to m in order of their appearing in input.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"575"You are given an array $$$a$$$ of length $$$2^n$$$. You should process $$$q$$$ queries on it. Each query has one of the following $$$4$$$ types:   $$$Replace(x, k)$$$ — change $$$a_x$$$ to $$$k$$$;  $$$Reverse(k)$$$ — reverse each subarray $$$[(i-1) \cdot 2^k+1, i \cdot 2^k]$$$ for all $$$i$$$ ($$$i \ge 1$$$);  $$$Swap(k)$$$ — swap subarrays $$$[(2i-2) \cdot 2^k+1, (2i-1) \cdot 2^k]$$$ and $$$[(2i-1) \cdot 2^k+1, 2i \cdot 2^k]$$$ for all $$$i$$$ ($$$i \ge 1$$$);  $$$Sum(l, r)$$$ — print the sum of the elements of subarray $$$[l, r]$$$. Write a program that can quickly process given queries.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"576"Vasya tries to break in a safe. He knows that a code consists of n numbers, and every number is a 0 or a 1. Vasya has made m attempts to enter the code. After each attempt the system told him in how many position stand the right numbers. It is not said in which positions the wrong numbers stand. Vasya has been so unlucky that he hasn’t entered the code where would be more than 5 correct numbers. Now Vasya is completely bewildered: he thinks there’s a mistake in the system and it is self-contradictory. Help Vasya — calculate how many possible code variants are left that do not contradict the previous system responses.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"577"This is the easy version of the problem. The difference is that in this version the array can not contain zeros. You can make hacks only if both versions of the problem are solved.You are given an array $$$[a_1, a_2, \ldots a_n]$$$ consisting of integers $$$-1$$$ and $$$1$$$. You have to build a partition of this array into the set of segments $$$[l_1, r_1], [l_2, r_2], \ldots, [l_k, r_k]$$$ with the following property:  Denote the alternating sum of all elements of the $$$i$$$-th segment as $$$s_i$$$: $$$s_i$$$ = $$$a_{l_i} - a_{l_i+1} + a_{l_i+2} - a_{l_i+3} + \ldots \pm a_{r_i}$$$. For example, the alternating sum of elements of segment $$$[2, 4]$$$ in array $$$[1, 0, -1, 1, 1]$$$ equals to $$$0 - (-1) + 1 = 2$$$.  The sum of $$$s_i$$$ over all segments of partition should be equal to zero. Note that each $$$s_i$$$ does not have to be equal to zero, this property is about sum of $$$s_i$$$ over all segments of partition.The set of segments $$$[l_1, r_1], [l_2, r_2], \ldots, [l_k, r_k]$$$ is called a partition of the array $$$a$$$ of length $$$n$$$ if $$$1 = l_1 \le r_1, l_2 \le r_2, \ldots, l_k \le r_k = n$$$ and $$$r_i + 1 = l_{i+1}$$$ for all $$$i = 1, 2, \ldots k-1$$$. In other words, each element of the array must belong to exactly one segment.You have to build a partition of the given array with properties described above or determine that such partition does not exist.Note that it is not required to minimize the number of segments in the partition.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"578"Polycarp's workday lasts exactly $$$n$$$ minutes. He loves chocolate bars and can eat one bar in one minute. Today Polycarp has $$$k$$$ bars at the beginning of the workday.In some minutes of the workday Polycarp has important things to do and in such minutes he is not able to eat a chocolate bar. In other minutes he can either eat or not eat one chocolate bar. It is guaranteed, that in the first and in the last minutes of the workday Polycarp has no important things to do and he will always eat bars in this minutes to gladden himself at the begining and at the end of the workday. Also it is guaranteed, that $$$k$$$ is strictly greater than $$$1$$$.Your task is to determine such an order of eating chocolate bars that the maximum break time between eating bars is as minimum as possible.Consider that Polycarp eats a bar in the minute $$$x$$$ and the next bar in the minute $$$y$$$ ($$$x < y$$$). Then the break time is equal to $$$y - x - 1$$$ minutes. It is not necessary for Polycarp to eat all bars he has.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"579"You have $$$n$$$ rectangular wooden blocks, which are numbered from $$$1$$$ to $$$n$$$. The $$$i$$$-th block is $$$1$$$ unit high and $$$\lceil \frac{i}{2} \rceil$$$ units long.Here, $$$\lceil \frac{x}{2} \rceil$$$ denotes the result of division of $$$x$$$ by $$$2$$$, rounded up. For example, $$$\lceil \frac{4}{2} \rceil = 2$$$ and $$$\lceil \frac{5}{2} \rceil = \lceil 2.5 \rceil = 3$$$.For example, if $$$n=5$$$, then the blocks have the following sizes: $$$1 \times 1$$$, $$$1 \times 1$$$, $$$1 \times 2$$$, $$$1 \times 2$$$, $$$1 \times 3$$$.  The available blocks for $$$n=5$$$ Find the maximum possible side length of a square you can create using these blocks, without rotating any of them. Note that you don't have to use all of the blocks.  One of the ways to create $$$3 \times 3$$$ square using blocks $$$1$$$ through $$$5$$$ ",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"580"We're giving away nice huge bags containing number tiles! A bag we want to present to you contains $$$n$$$ tiles. Each of them has a single number written on it — either $$$1$$$ or $$$2$$$.However, there is one condition you must fulfill in order to receive the prize. You will need to put all the tiles from the bag in a sequence, in any order you wish. We will then compute the sums of all prefixes in the sequence, and then count how many of these sums are prime numbers. If you want to keep the prize, you will need to maximize the number of primes you get.Can you win the prize? Hurry up, the bags are waiting!","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"581"You have given tree consist of $$$n$$$ vertices. Select a vertex as root vertex that satisfies the condition below.  For all vertices $$$v_{1}$$$ and $$$v_{2}$$$, if $$$distance$$$($$$root$$$, $$$v_{1}$$$) $$$= distance$$$($$$root$$$, $$$v_{2})$$$ then $$$degree$$$($$$v_{1}$$$) $$$= degree$$$($$$v_{2}$$$), where $$$degree$$$ means the number of vertices connected to that vertex, and $$$distance$$$ means the number of edges between two vertices. Determine and find if there is such root vertex in the tree. If there are multiple answers, find any of them.",['trees'],"[0, 0, 0, 0, 0, 0, 0, 1]"582"Appleman has n cards. Each card has an uppercase letter written on it. Toastman must choose k cards from Appleman's cards. Then Appleman should give Toastman some coins depending on the chosen cards. Formally, for each Toastman's card i you should calculate how much Toastman's cards have the letter equal to letter on ith, then sum up all these quantities, such a number of coins Appleman should give to Toastman.Given the description of Appleman's cards. What is the maximum number of coins Toastman can get?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"583"In the Main Berland Bank n people stand in a queue at the cashier, everyone knows his/her height hi, and the heights of the other people in the queue. Each of them keeps in mind number ai — how many people who are taller than him/her and stand in queue in front of him.After a while the cashier has a lunch break and the people in the queue seat on the chairs in the waiting room in a random order.When the lunch break was over, it turned out that nobody can remember the exact order of the people in the queue, but everyone remembers his number ai.Your task is to restore the order in which the people stood in the queue if it is possible. There may be several acceptable orders, but you need to find any of them. Also, you need to print a possible set of numbers hi — the heights of people in the queue, so that the numbers ai are correct.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"584"You are given a table consisting of n rows and m columns. Each cell of the table contains a number, 0 or 1. In one move we can choose some row of the table and cyclically shift its values either one cell to the left, or one cell to the right.To cyclically shift a table row one cell to the right means to move the value of each cell, except for the last one, to the right neighboring cell, and to move the value of the last cell to the first cell. A cyclical shift of a row to the left is performed similarly, but in the other direction. For example, if we cyclically shift a row ""00110"" one cell to the right, we get a row ""00011"", but if we shift a row ""00110"" one cell to the left, we get a row ""01100"".Determine the minimum number of moves needed to make some table column consist only of numbers 1.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"585"Let's denote a $$$k$$$-step ladder as the following structure: exactly $$$k + 2$$$ wooden planks, of which  two planks of length at least $$$k+1$$$ — the base of the ladder;  $$$k$$$ planks of length at least $$$1$$$ — the steps of the ladder; Note that neither the base planks, nor the steps planks are required to be equal.For example, ladders $$$1$$$ and $$$3$$$ are correct $$$2$$$-step ladders and ladder $$$2$$$ is a correct $$$1$$$-step ladder. On the first picture the lengths of planks are $$$[3, 3]$$$ for the base and $$$[1]$$$ for the step. On the second picture lengths are $$$[3, 3]$$$ for the base and $$$[2]$$$ for the step. On the third picture lengths are $$$[3, 4]$$$ for the base and $$$[2, 3]$$$ for the steps.   You have $$$n$$$ planks. The length of the $$$i$$$-th planks is $$$a_i$$$. You don't have a saw, so you can't cut the planks you have. Though you have a hammer and nails, so you can assemble the improvised ""ladder"" from the planks.The question is: what is the maximum number $$$k$$$ such that you can choose some subset of the given planks and assemble a $$$k$$$-step ladder using them?",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"586"Vasya has recently learned at school what a number's divisor is and decided to determine a string's divisor. Here is what he came up with.String a is the divisor of string b if and only if there exists a positive integer x such that if we write out string a consecutively x times, we get string b. For example, string ""abab"" has two divisors — ""ab"" and ""abab"".Now Vasya wants to write a program that calculates the number of common divisors of two strings. Please help him.","['math', 'strings']","[0, 0, 0, 1, 0, 0, 1, 0]"587"All of us love treasures, right? That's why young Vasya is heading for a Treasure Island.Treasure Island may be represented as a rectangular table $$$n \times m$$$ which is surrounded by the ocean. Let us number rows of the field with consecutive integers from $$$1$$$ to $$$n$$$ from top to bottom and columns with consecutive integers from $$$1$$$ to $$$m$$$ from left to right. Denote the cell in $$$r$$$-th row and $$$c$$$-th column as $$$(r, c)$$$. Some of the island cells contain impassable forests, and some cells are free and passable. Treasure is hidden in cell $$$(n, m)$$$.Vasya got off the ship in cell $$$(1, 1)$$$. Now he wants to reach the treasure. He is hurrying up, so he can move only from cell to the cell in next row (downwards) or next column (rightwards), i.e. from cell $$$(x, y)$$$ he can move only to cells $$$(x+1, y)$$$ and $$$(x, y+1)$$$. Of course Vasya can't move through cells with impassable forests.Evil Witch is aware of Vasya's journey and she is going to prevent him from reaching the treasure. Before Vasya's first move she is able to grow using her evil magic impassable forests in previously free cells. Witch is able to grow a forest in any number of any free cells except cells $$$(1, 1)$$$ where Vasya got off his ship and $$$(n, m)$$$ where the treasure is hidden.Help Evil Witch by finding out the minimum number of cells she has to turn into impassable forests so that Vasya is no longer able to reach the treasure.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"588"There are n cities in the country where the Old Peykan lives. These cities are located on a straight line, we'll denote them from left to right as c1, c2, ..., cn. The Old Peykan wants to travel from city c1 to cn using roads. There are (n - 1) one way roads, the i-th road goes from city ci to city ci + 1 and is di kilometers long.The Old Peykan travels 1 kilometer in 1 hour and consumes 1 liter of fuel during this time.Each city ci (except for the last city cn) has a supply of si liters of fuel which immediately transfers to the Old Peykan if it passes the city or stays in it. This supply refreshes instantly k hours after it transfers. The Old Peykan can stay in a city for a while and fill its fuel tank many times. Initially (at time zero) the Old Peykan is at city c1 and s1 liters of fuel is transferred to it's empty tank from c1's supply. The Old Peykan's fuel tank capacity is unlimited. Old Peykan can not continue its travel if its tank is emptied strictly between two cities.Find the minimum time the Old Peykan needs to reach city cn.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"589"God's Blessing on This PermutationForces!A Random PebbleYou are given a permutation $$$p_1,p_2,\ldots,p_n$$$ of length $$$n$$$ and a positive integer $$$k \le n$$$. In one operation you can choose two indices $$$i$$$ and $$$j$$$ ($$$1 \le i < j \le n$$$) and swap $$$p_i$$$ with $$$p_j$$$.Find the minimum number of operations needed to make the sum $$$p_1 + p_2 + \ldots + p_k$$$ as small as possible.A permutation is an array consisting of $$$n$$$ distinct integers from $$$1$$$ to $$$n$$$ in arbitrary order. For example, $$$[2,3,1,5,4]$$$ is a permutation, but $$$[1,2,2]$$$ is not a permutation ($$$2$$$ appears twice in the array) and $$$[1,3,4]$$$ is also not a permutation ($$$n=3$$$ but there is $$$4$$$ in the array).",[],"[0, 0, 0, 0, 0, 0, 0, 0]"590"You are given an infinite periodic array a0, a1, ..., an - 1, ... with the period of length n. Formally, . A periodic subarray (l, s) (0 ≤ l < n, 1 ≤ s < n) of array a is an infinite periodic array with a period of length s that is a subsegment of array a, starting with position l.A periodic subarray (l, s) is superior, if when attaching it to the array a, starting from index l, any element of the subarray is larger than or equal to the corresponding element of array a. An example of attaching is given on the figure (top — infinite array a, bottom — its periodic subarray (l, s)):  Find the number of distinct pairs (l, s), corresponding to the superior periodic arrays.",['number theory'],"[0, 0, 0, 0, 1, 0, 0, 0]"591"One day little Vasya found mom's pocket book. The book had n names of her friends and unusually enough, each name was exactly m letters long. Let's number the names from 1 to n in the order in which they are written.As mom wasn't home, Vasya decided to play with names: he chose three integers i, j, k (1 ≤ i < j ≤ n, 1 ≤ k ≤ m), then he took names number i and j and swapped their prefixes of length k. For example, if we take names ""CBDAD"" and ""AABRD"" and swap their prefixes with the length of 3, the result will be names ""AABAD"" and ""CBDRD"".You wonder how many different names Vasya can write instead of name number 1, if Vasya is allowed to perform any number of the described actions. As Vasya performs each action, he chooses numbers i, j, k independently from the previous moves and his choice is based entirely on his will. The sought number can be very large, so you should only find it modulo 1000000007 (109 + 7).",[],"[0, 0, 0, 0, 0, 0, 0, 0]"592"The legend of the foundation of Vectorland talks of two integers $$$x$$$ and $$$y$$$. Centuries ago, the array king placed two markers at points $$$|x|$$$ and $$$|y|$$$ on the number line and conquered all the land in between (including the endpoints), which he declared to be Arrayland. Many years later, the vector king placed markers at points $$$|x - y|$$$ and $$$|x + y|$$$ and conquered all the land in between (including the endpoints), which he declared to be Vectorland. He did so in such a way that the land of Arrayland was completely inside (including the endpoints) the land of Vectorland.Here $$$|z|$$$ denotes the absolute value of $$$z$$$.Now, Jose is stuck on a question of his history exam: ""What are the values of $$$x$$$ and $$$y$$$?"" Jose doesn't know the answer, but he believes he has narrowed the possible answers down to $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$. Now, he wants to know the number of unordered pairs formed by two different elements from these $$$n$$$ integers such that the legend could be true if $$$x$$$ and $$$y$$$ were equal to these two values. Note that it is possible that Jose is wrong, and that no pairs could possibly make the legend true.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"593"Alexandra has an even-length array $$$a$$$, consisting of $$$0$$$s and $$$1$$$s. The elements of the array are enumerated from $$$1$$$ to $$$n$$$. She wants to remove at most $$$\frac{n}{2}$$$ elements (where $$$n$$$ — length of array) in the way that alternating sum of the array will be equal $$$0$$$ (i.e. $$$a_1 - a_2 + a_3 - a_4 + \dotsc = 0$$$). In other words, Alexandra wants sum of all elements at the odd positions and sum of all elements at the even positions to become equal. The elements that you remove don't have to be consecutive.For example, if she has $$$a = [1, 0, 1, 0, 0, 0]$$$ and she removes $$$2$$$nd and $$$4$$$th elements, $$$a$$$ will become equal $$$[1, 1, 0, 0]$$$ and its alternating sum is $$$1 - 1 + 0 - 0 = 0$$$.Help her!",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"594"You have a set of dominoes. Each domino is a rectangular tile with a line dividing its face into two square ends. Can you put all dominoes in a line one by one from left to right so that any two dominoes touched with the sides that had the same number of points? You can rotate the dominoes, changing the left and the right side (domino ""1-4"" turns into ""4-1"").",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"595"We are sum for we are manySome NumberThis version of the problem differs from the next one only in the constraint on $$$t$$$. You can make hacks only if both versions of the problem are solved.You are given two positive integers $$$l$$$ and $$$r$$$.Count the number of distinct triplets of integers $$$(i, j, k)$$$ such that $$$l \le i < j < k \le r$$$ and $$$\operatorname{lcm}(i,j,k) \ge i + j + k$$$.Here $$$\operatorname{lcm}(i, j, k)$$$ denotes the least common multiple (LCM) of integers $$$i$$$, $$$j$$$, and $$$k$$$.","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"596"The only difference between easy and hard versions is constraints.Now elections are held in Berland and you want to win them. More precisely, you want everyone to vote for you.There are $$$n$$$ voters, and two ways to convince each of them to vote for you. The first way to convince the $$$i$$$-th voter is to pay him $$$p_i$$$ coins. The second way is to make $$$m_i$$$ other voters vote for you, and the $$$i$$$-th voter will vote for free.Moreover, the process of such voting takes place in several steps. For example, if there are five voters with $$$m_1 = 1$$$, $$$m_2 = 2$$$, $$$m_3 = 2$$$, $$$m_4 = 4$$$, $$$m_5 = 5$$$, then you can buy the vote of the fifth voter, and eventually everyone will vote for you. Set of people voting for you will change as follows: $$${5} \rightarrow {1, 5} \rightarrow {1, 2, 3, 5} \rightarrow {1, 2, 3, 4, 5}$$$.Calculate the minimum number of coins you have to spend so that everyone votes for you.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"597"Programmer Vasya is studying a new programming language &K*. The &K* language resembles the languages of the C family in its syntax. However, it is more powerful, which is why the rules of the actual C-like languages are unapplicable to it. To fully understand the statement, please read the language's description below carefully and follow it and not the similar rules in real programming languages.There is a very powerful system of pointers on &K* — you can add an asterisk to the right of the existing type X — that will result in new type X * . That is called pointer-definition operation. Also, there is the operation that does the opposite — to any type of X, which is a pointer, you can add an ampersand — that will result in a type &X, to which refers X. That is called a dereference operation.The &K* language has only two basic data types — void and errtype. Also, the language has operators typedef and typeof. The operator ""typedef A B"" defines a new data type B, which is equivalent to A. A can have asterisks and ampersands, and B cannot have them. For example, the operator typedef void** ptptvoid will create a new type ptptvoid, that can be used as void**. The operator ""typeof A"" returns type of A, brought to void, that is, returns the type void**...*, equivalent to it with the necessary number of asterisks (the number can possibly be zero). That is, having defined the ptptvoid type, as shown above, the typeof ptptvoid operator will return void**.An attempt of dereferencing of the void type will lead to an error: to a special data type errtype. For errtype the following equation holds true: errtype* = &errtype = errtype. An attempt to use the data type that hasn't been defined before that will also lead to the errtype.Using typedef, we can define one type several times. Of all the definitions only the last one is valid. However, all the types that have been defined earlier using this type do not change.Let us also note that the dereference operation has the lower priority that the pointer operation, in other words &T *  is always equal to T.Note, that the operators are executed consecutively one by one. If we have two operators ""typedef &void a"" and ""typedef a* b"", then at first a becomes errtype, and after that b becomes errtype* = errtype, but not &void* = void (see sample 2).Vasya does not yet fully understand this powerful technology, that's why he asked you to help him. Write a program that analyzes these operators. ",[],"[0, 0, 0, 0, 0, 0, 0, 0]"598"You are playing the game ""Arranging The Sheep"". The goal of this game is to make the sheep line up. The level in the game is described by a string of length $$$n$$$, consisting of the characters '.' (empty space) and '*' (sheep). In one move, you can move any sheep one square to the left or one square to the right, if the corresponding square exists and is empty. The game ends as soon as the sheep are lined up, that is, there should be no empty cells between any sheep.For example, if $$$n=6$$$ and the level is described by the string ""**.*.."", then the following game scenario is possible:   the sheep at the $$$4$$$ position moves to the right, the state of the level: ""**..*."";  the sheep at the $$$2$$$ position moves to the right, the state of the level: ""*.*.*."";  the sheep at the $$$1$$$ position moves to the right, the state of the level: "".**.*."";  the sheep at the $$$3$$$ position moves to the right, the state of the level: "".*.**."";  the sheep at the $$$2$$$ position moves to the right, the state of the level: ""..***."";  the sheep are lined up and the game ends. For a given level, determine the minimum number of moves you need to make to complete the level.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"599"Lunar rover finally reached planet X. After landing, he met an obstacle, that contains permutation $$$p$$$ of length $$$n$$$. Scientists found out, that to overcome an obstacle, the robot should make $$$p$$$ an identity permutation (make $$$p_i = i$$$ for all $$$i$$$).Unfortunately, scientists can't control the robot. Thus the only way to make $$$p$$$ an identity permutation is applying the following operation to $$$p$$$ multiple times:   Select two indices $$$i$$$ and $$$j$$$ ($$$i \neq j$$$), such that $$$p_j = i$$$ and swap the values of $$$p_i$$$ and $$$p_j$$$. It takes robot $$$(j - i)^2$$$ seconds to do this operation.  Positions $$$i$$$ and $$$j$$$ are selected by the robot (scientists can't control it). He will apply this operation while $$$p$$$ isn't an identity permutation. We can show that the robot will make no more than $$$n$$$ operations regardless of the choice of $$$i$$$ and $$$j$$$ on each operation.Scientists asked you to find out the maximum possible time it will take the robot to finish making $$$p$$$ an identity permutation (i. e. worst-case scenario), so they can decide whether they should construct a new lunar rover or just rest and wait. They won't believe you without proof, so you should build an example of $$$p$$$ and robot's operations that maximizes the answer.For a better understanding of the statement, read the sample description.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"600"Alexey is travelling on a train. Unfortunately, due to the bad weather, the train moves slower that it should!Alexey took the train at the railroad terminal. Let's say that the train starts from the terminal at the moment $$$0$$$. Also, let's say that the train will visit $$$n$$$ stations numbered from $$$1$$$ to $$$n$$$ along its way, and that Alexey destination is the station $$$n$$$.Alexey learned from the train schedule $$$n$$$ integer pairs $$$(a_i, b_i)$$$ where $$$a_i$$$ is the expected time of train's arrival at the $$$i$$$-th station and $$$b_i$$$ is the expected time of departure.Also, using all information he has, Alexey was able to calculate $$$n$$$ integers $$$tm_1, tm_2, \dots, tm_n$$$ where $$$tm_i$$$ is the extra time the train need to travel from the station $$$i - 1$$$ to the station $$$i$$$. Formally, the train needs exactly $$$a_i - b_{i-1} + tm_i$$$ time to travel from station $$$i - 1$$$ to station $$$i$$$ (if $$$i = 1$$$ then $$$b_0$$$ is the moment the train leave the terminal, and it's equal to $$$0$$$).The train leaves the station $$$i$$$, if both conditions are met:   it's on the station for at least $$$\left\lceil \frac{b_i - a_i}{2} \right\rceil$$$ units of time (division with ceiling);  current time $$$\ge b_i$$$. Since Alexey spent all his energy on prediction of time delays, help him to calculate the time of arrival at the station $$$n$$$.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"601Shaass has n books. He wants to make a bookshelf for all his books. He wants the bookshelf's dimensions to be as small as possible. The thickness of the i-th book is ti and its pages' width is equal to wi. The thickness of each book is either 1 or 2. All books have the same page heights.  Shaass puts the books on the bookshelf in the following way. First he selects some of the books and put them vertically. Then he puts the rest of the books horizontally above the vertical books. The sum of the widths of the horizontal books must be no more than the total thickness of the vertical books. A sample arrangement of the books is depicted in the figure.  Help Shaass to find the minimum total thickness of the vertical books that we can achieve.,[],"[0, 0, 0, 0, 0, 0, 0, 0]"602"On a random day, Neko found $$$n$$$ treasure chests and $$$m$$$ keys. The $$$i$$$-th chest has an integer $$$a_i$$$ written on it and the $$$j$$$-th key has an integer $$$b_j$$$ on it. Neko knows those chests contain the powerful mysterious green Grapes, thus Neko wants to open as many treasure chests as possible.The $$$j$$$-th key can be used to unlock the $$$i$$$-th chest if and only if the sum of the key number and the chest number is an odd number. Formally, $$$a_i + b_j \equiv 1 \pmod{2}$$$. One key can be used to open at most one chest, and one chest can be opened at most once.Find the maximum number of chests Neko can open.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"603"Mishka wants to buy some food in the nearby shop. Initially, he has $$$s$$$ burles on his card. Mishka can perform the following operation any number of times (possibly, zero): choose some positive integer number $$$1 \le x \le s$$$, buy food that costs exactly $$$x$$$ burles and obtain $$$\lfloor\frac{x}{10}\rfloor$$$ burles as a cashback (in other words, Mishka spends $$$x$$$ burles and obtains $$$\lfloor\frac{x}{10}\rfloor$$$ back). The operation $$$\lfloor\frac{a}{b}\rfloor$$$ means $$$a$$$ divided by $$$b$$$ rounded down.It is guaranteed that you can always buy some food that costs $$$x$$$ for any possible value of $$$x$$$.Your task is to say the maximum number of burles Mishka can spend if he buys food optimally.For example, if Mishka has $$$s=19$$$ burles then the maximum number of burles he can spend is $$$21$$$. Firstly, he can spend $$$x=10$$$ burles, obtain $$$1$$$ burle as a cashback. Now he has $$$s=10$$$ burles, so can spend $$$x=10$$$ burles, obtain $$$1$$$ burle as a cashback and spend it too.You have to answer $$$t$$$ independent test cases.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"604"Oleg the client and Igor the analyst are good friends. However, sometimes they argue over little things. Recently, they started a new company, but they are having trouble finding a name for the company.To settle this problem, they've decided to play a game. The company name will consist of n letters. Oleg and Igor each have a set of n letters (which might contain multiple copies of the same letter, the sets can be different). Initially, the company name is denoted by n question marks. Oleg and Igor takes turns to play the game, Oleg moves first. In each turn, a player can choose one of the letters c in his set and replace any of the question marks with c. Then, a copy of the letter c is removed from his set. The game ends when all the question marks has been replaced by some letter.For example, suppose Oleg has the set of letters {i, o, i} and Igor has the set of letters {i, m, o}. One possible game is as follows :Initially, the company name is ???.Oleg replaces the second question mark with 'i'. The company name becomes ?i?. The set of letters Oleg have now is {i, o}.Igor replaces the third question mark with 'o'. The company name becomes ?io. The set of letters Igor have now is {i, m}.Finally, Oleg replaces the first question mark with 'o'. The company name becomes oio. The set of letters Oleg have now is {i}.In the end, the company name is oio.Oleg wants the company name to be as lexicographically small as possible while Igor wants the company name to be as lexicographically large as possible. What will be the company name if Oleg and Igor always play optimally?A string s = s1s2...sm is called lexicographically smaller than a string t = t1t2...tm (where s ≠ t) if si < ti where i is the smallest index such that si ≠ ti. (so sj = tj for all j < i)",['games'],"[1, 0, 0, 0, 0, 0, 0, 0]"605"During the hypnosis session, Nicholas suddenly remembered a positive integer $$$n$$$, which doesn't contain zeros in decimal notation. Soon, when he returned home, he got curious: what is the maximum number of digits that can be removed from the number so that the number becomes not prime, that is, either composite or equal to one?For some numbers doing so is impossible: for example, for number $$$53$$$ it's impossible to delete some of its digits to obtain a not prime integer. However, for all $$$n$$$ in the test cases of this problem, it's guaranteed that it's possible to delete some of their digits to obtain a not prime number.Note that you cannot remove all the digits from the number.A prime number is a number that has no divisors except one and itself. A composite is a number that has more than two divisors. $$$1$$$ is neither a prime nor a composite number.","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"606"Two people are playing a game with a string $$$s$$$, consisting of lowercase latin letters. On a player's turn, he should choose two consecutive equal letters in the string and delete them. For example, if the string is equal to ""xaax"" than there is only one possible turn: delete ""aa"", so the string will become ""xx"". A player not able to make a turn loses.Your task is to determine which player will win if both play optimally.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"607"Peter decided to wish happy birthday to his friend from Australia and send him a card. To make his present more mysterious, he decided to make a chain. Chain here is such a sequence of envelopes A = {a1,  a2,  ...,  an}, where the width and the height of the i-th envelope is strictly higher than the width and the height of the (i  -  1)-th envelope respectively. Chain size is the number of envelopes in the chain. Peter wants to make the chain of the maximum size from the envelopes he has, the chain should be such, that he'll be able to put a card into it. The card fits into the chain if its width and height is lower than the width and the height of the smallest envelope in the chain respectively. It's forbidden to turn the card and the envelopes. Peter has very many envelopes and very little time, this hard task is entrusted to you.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"608"Recently Max has got himself into popular CCG ""BrainStone"". As ""BrainStone"" is a pretty intellectual game, Max has to solve numerous hard problems during the gameplay. Here is one of them:Max owns n creatures, i-th of them can be described with two numbers — its health hpi and its damage dmgi. Max also has two types of spells in stock:  Doubles health of the creature (hpi := hpi·2);  Assigns value of health of the creature to its damage (dmgi := hpi). Spell of first type can be used no more than a times in total, of the second type — no more than b times in total. Spell can be used on a certain creature multiple times. Spells can be used in arbitrary order. It isn't necessary to use all the spells.Max is really busy preparing for his final exams, so he asks you to determine what is the maximal total damage of all creatures he can achieve if he uses spells in most optimal way.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"609"Kana was just an ordinary high school girl before a talent scout discovered her. Then, she became an idol. But different from the stereotype, she is also a gameholic. One day Kana gets interested in a new adventure game called Dragon Quest. In this game, her quest is to beat a dragon. The dragon has a hit point of $$$x$$$ initially. When its hit point goes to $$$0$$$ or under $$$0$$$, it will be defeated. In order to defeat the dragon, Kana can cast the two following types of spells.  Void Absorption Assume that the dragon's current hit point is $$$h$$$, after casting this spell its hit point will become $$$\left\lfloor \frac{h}{2} \right\rfloor + 10$$$. Here $$$\left\lfloor \frac{h}{2} \right\rfloor$$$ denotes $$$h$$$ divided by two, rounded down. Lightning Strike This spell will decrease the dragon's hit point by $$$10$$$. Assume that the dragon's current hit point is $$$h$$$, after casting this spell its hit point will be lowered to $$$h-10$$$.Due to some reasons Kana can only cast no more than $$$n$$$ Void Absorptions and $$$m$$$ Lightning Strikes. She can cast the spells in any order and doesn't have to cast all the spells. Kana isn't good at math, so you are going to help her to find out whether it is possible to defeat the dragon.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"610"The prison of your city has n prisoners. As the prison can't accommodate all of them, the city mayor has decided to transfer c of the prisoners to a prison located in another city.For this reason, he made the n prisoners to stand in a line, with a number written on their chests. The number is the severity of the crime he/she has committed. The greater the number, the more severe his/her crime was.Then, the mayor told you to choose the c prisoners, who will be transferred to the other prison. He also imposed two conditions. They are,  The chosen c prisoners has to form a contiguous segment of prisoners.  Any of the chosen prisoner's crime level should not be greater then t. Because, that will make the prisoner a severe criminal and the mayor doesn't want to take the risk of his running away during the transfer. Find the number of ways you can choose the c prisoners.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"611"Little Timofey likes integers a lot. Unfortunately, he is very young and can't work with very big integers, so he does all the operations modulo his favorite prime m. Also, Timofey likes to look for arithmetical progressions everywhere.One of his birthday presents was a sequence of distinct integers a1, a2, ..., an. Timofey wants to know whether he can rearrange the elements of the sequence so that is will be an arithmetical progression modulo m, or not.Arithmetical progression modulo m of length n with first element x and difference d is sequence of integers x, x + d, x + 2d, ..., x + (n - 1)·d, each taken modulo m.","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"612"This problem is same as the previous one, but has larger constraints.Shiro's just moved to the new house. She wants to invite all friends of her to the house so they can play monopoly. However, her house is too small, so she can only invite one friend at a time.For each of the $$$n$$$ days since the day Shiro moved to the new house, there will be exactly one cat coming to the Shiro's house. The cat coming in the $$$i$$$-th day has a ribbon with color $$$u_i$$$. Shiro wants to know the largest number $$$x$$$, such that if we consider the streak of the first $$$x$$$ days, it is possible to remove exactly one day from this streak so that every ribbon color that has appeared among the remaining $$$x - 1$$$ will have the same number of occurrences.For example, consider the following sequence of $$$u_i$$$: $$$[2, 2, 1, 1, 5, 4, 4, 5]$$$. Then $$$x = 7$$$ makes a streak, since if we remove the leftmost $$$u_i = 5$$$, each ribbon color will appear exactly twice in the prefix of $$$x - 1$$$ days. Note that $$$x = 8$$$ doesn't form a streak, since you must remove exactly one day. Since Shiro is just a cat, she is not very good at counting and needs your help finding the longest streak.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"613"A country called Berland consists of n cities, numbered with integer numbers from 1 to n. Some of them are connected by bidirectional roads. Each road has some length. There is a path from each city to any other one by these roads. According to some Super Duper Documents, Berland is protected by the Super Duper Missiles. The exact position of the Super Duper Secret Missile Silos is kept secret but Bob managed to get hold of the information. That information says that all silos are located exactly at a distance l from the capital. The capital is located in the city with number s.The documents give the formal definition: the Super Duper Secret Missile Silo is located at some place (which is either city or a point on a road) if and only if the shortest distance from this place to the capital along the roads of the country equals exactly l.Bob wants to know how many missile silos are located in Berland to sell the information then to enemy spies. Help Bob.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"614" Gathering darkness shrouds the woods and the world. The moon sheds its light on the boat and the river.""To curtain off the moonlight should be hardly possible; the shades present its mellow beauty and restful nature."" Intonates Mino.""See? The clouds are coming."" Kanno gazes into the distance.""That can't be better,"" Mino turns to Kanno. The sky can be seen as a one-dimensional axis. The moon is at the origin whose coordinate is $$$0$$$.There are $$$n$$$ clouds floating in the sky. Each cloud has the same length $$$l$$$. The $$$i$$$-th initially covers the range of $$$(x_i, x_i + l)$$$ (endpoints excluded). Initially, it moves at a velocity of $$$v_i$$$, which equals either $$$1$$$ or $$$-1$$$.Furthermore, no pair of clouds intersect initially, that is, for all $$$1 \leq i \lt j \leq n$$$, $$$\lvert x_i - x_j \rvert \geq l$$$.With a wind velocity of $$$w$$$, the velocity of the $$$i$$$-th cloud becomes $$$v_i + w$$$. That is, its coordinate increases by $$$v_i + w$$$ during each unit of time. Note that the wind can be strong and clouds can change their direction.You are to help Mino count the number of pairs $$$(i, j)$$$ ($$$i < j$$$), such that with a proper choice of wind velocity $$$w$$$ not exceeding $$$w_\mathrm{max}$$$ in absolute value (possibly negative and/or fractional), the $$$i$$$-th and $$$j$$$-th clouds both cover the moon at the same future moment. This $$$w$$$ doesn't need to be the same across different pairs.","['geometry', 'math']","[0, 1, 0, 1, 0, 0, 0, 0]"615"Arkady decided to buy roses for his girlfriend.A flower shop has white, orange and red roses, and the total amount of them is n. Arkady thinks that red roses are not good together with white roses, so he won't buy a bouquet containing both red and white roses. Also, Arkady won't buy a bouquet where all roses have the same color. Arkady wants to buy exactly k roses. For each rose in the shop he knows its beauty and color: the beauty of the i-th rose is bi, and its color is ci ('W' for a white rose, 'O' for an orange rose and 'R' for a red rose). Compute the maximum possible total beauty of a bouquet of k roses satisfying the constraints above or determine that it is not possible to make such a bouquet.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"616"In the evening, after the contest Ilya was bored, and he really felt like maximizing. He remembered that he had a set of n sticks and an instrument. Each stick is characterized by its length li.Ilya decided to make a rectangle from the sticks. And due to his whim, he decided to make rectangles in such a way that maximizes their total area. Each stick is used in making at most one rectangle, it is possible that some of sticks remain unused. Bending sticks is not allowed.Sticks with lengths a1, a2, a3 and a4 can make a rectangle if the following properties are observed:  a1 ≤ a2 ≤ a3 ≤ a4  a1 = a2  a3 = a4 A rectangle can be made of sticks with lengths of, for example, 3 3 3 3 or 2 2 4 4. A rectangle cannot be made of, for example, sticks 5 5 5 7.Ilya also has an instrument which can reduce the length of the sticks. The sticks are made of a special material, so the length of each stick can be reduced by at most one. For example, a stick with length 5 can either stay at this length or be transformed into a stick of length 4.You have to answer the question — what maximum total area of the rectangles can Ilya get with a file if makes rectangles from the available sticks?",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"617"A permutation of length n is an array containing each integer from 1 to n exactly once. For example, q = [4, 5, 1, 2, 3] is a permutation. For the permutation q the square of permutation is the permutation p that p[i] = q[q[i]] for each i = 1... n. For example, the square of q = [4, 5, 1, 2, 3] is p = q2 = [2, 3, 4, 5, 1].This problem is about the inverse operation: given the permutation p you task is to find such permutation q that q2 = p. If there are several such q find any of them.","['graphs', 'math']","[0, 0, 1, 1, 0, 0, 0, 0]"618"Your program fails again. This time it gets ""Wrong answer on test 233"".This is the easier version of the problem. In this version $$$1 \le n \le 2000$$$. You can hack this problem only if you solve and lock both problems.The problem is about a test containing $$$n$$$ one-choice-questions. Each of the questions contains $$$k$$$ options, and only one of them is correct. The answer to the $$$i$$$-th question is $$$h_{i}$$$, and if your answer of the question $$$i$$$ is $$$h_{i}$$$, you earn $$$1$$$ point, otherwise, you earn $$$0$$$ points for this question. The values $$$h_1, h_2, \dots, h_n$$$ are known to you in this problem.However, you have a mistake in your program. It moves the answer clockwise! Consider all the $$$n$$$ answers are written in a circle. Due to the mistake in your program, they are shifted by one cyclically.Formally, the mistake moves the answer for the question $$$i$$$ to the question $$$i \bmod n + 1$$$. So it moves the answer for the question $$$1$$$ to question $$$2$$$, the answer for the question $$$2$$$ to the question $$$3$$$, ..., the answer for the question $$$n$$$ to the question $$$1$$$.We call all the $$$n$$$ answers together an answer suit. There are $$$k^n$$$ possible answer suits in total.You're wondering, how many answer suits satisfy the following condition: after moving clockwise by $$$1$$$, the total number of points of the new answer suit is strictly larger than the number of points of the old one. You need to find the answer modulo $$$998\,244\,353$$$.For example, if $$$n = 5$$$, and your answer suit is $$$a=[1,2,3,4,5]$$$, it will submitted as $$$a'=[5,1,2,3,4]$$$ because of a mistake. If the correct answer suit is $$$h=[5,2,2,3,4]$$$, the answer suit $$$a$$$ earns $$$1$$$ point and the answer suite $$$a'$$$ earns $$$4$$$ points. Since $$$4 > 1$$$, the answer suit $$$a=[1,2,3,4,5]$$$ should be counted.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"619"When he's not training for IOI, Little Alawn enjoys playing with puzzles of various types to stimulate his brain. Today, he's playing with a puzzle that consists of a $$$2 \times n$$$ grid where each row is a permutation of the numbers $$$1,2,3,\ldots,n$$$.The goal of Little Alawn's puzzle is to make sure no numbers on the same column or row are the same (we'll call this state of the puzzle as solved), and to achieve this he is able to swap the numbers in any column. However, after solving the puzzle many times, Little Alawn got bored and began wondering about the number of possible solved configurations of the puzzle he could achieve from an initial solved configuration only by swapping numbers in a column.Unfortunately, Little Alawn got stuck while trying to solve this harder problem, so he was wondering if you could help him with it. Find the answer modulo $$$10^9+7$$$.","['graphs', 'math']","[0, 0, 1, 1, 0, 0, 0, 0]"620"It has finally been decided to build a roof over the football field in School 179. Its construction will require placing $$$n$$$ consecutive vertical pillars. Furthermore, the headmaster wants the heights of all the pillars to form a permutation $$$p$$$ of integers from $$$0$$$ to $$$n - 1$$$, where $$$p_i$$$ is the height of the $$$i$$$-th pillar from the left $$$(1 \le i \le n)$$$.As the chief, you know that the cost of construction of consecutive pillars is equal to the maximum value of the bitwise XOR of heights of all pairs of adjacent pillars. In other words, the cost of construction is equal to $$$\max\limits_{1 \le i \le n - 1}{p_i \oplus p_{i + 1}}$$$, where $$$\oplus$$$ denotes the bitwise XOR operation.Find any sequence of pillar heights $$$p$$$ of length $$$n$$$ with the smallest construction cost.In this problem, a permutation is an array consisting of $$$n$$$ distinct integers from $$$0$$$ to $$$n - 1$$$ in arbitrary order. For example, $$$[2,3,1,0,4]$$$ is a permutation, but $$$[1,0,1]$$$ is not a permutation ($$$1$$$ appears twice in the array) and $$$[1,0,3]$$$ is also not a permutation ($$$n=3$$$, but $$$3$$$ is in the array).",[],"[0, 0, 0, 0, 0, 0, 0, 0]"621"Piegirl got bored with binary, decimal and other integer based counting systems. Recently she discovered some interesting properties about number , in particular that q2 = q + 1, and she thinks it would make a good base for her new unique system. She called it ""golden system"". In golden system the number is a non-empty string containing 0's and 1's as digits. The decimal value of expression a0a1...an equals to .Soon Piegirl found out that this system doesn't have same properties that integer base systems do and some operations can not be performed on it. She wasn't able to come up with a fast way of comparing two numbers. She is asking for your help.Given two numbers written in golden system notation, determine which of them has larger decimal value.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"622"One day shooshuns found a sequence of n integers, written on a blackboard. The shooshuns can perform one operation with it, the operation consists of two steps:  Find the number that goes k-th in the current sequence and add the same number to the end of the sequence;  Delete the first number of the current sequence. The shooshuns wonder after how many operations all numbers on the board will be the same and whether all numbers will ever be the same.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"623"You are given the array of integer numbers a0, a1, ..., an - 1. For each element find the distance to the nearest zero (to the element which equals to zero). There is at least one zero element in the given array.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"624"You are given an array $$$a$$$ consisting of $$$n$$$ positive integers.You are allowed to perform this operation any number of times (possibly, zero):   choose an index $$$i$$$ ($$$2 \le i \le n$$$), and change $$$a_i$$$ to $$$a_i - a_{i-1}$$$. Is it possible to make $$$a_i=0$$$ for all $$$2\le i\le n$$$?",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"625"Vasya wants to turn on Christmas lights consisting of m bulbs. Initially, all bulbs are turned off. There are n buttons, each of them is connected to some set of bulbs. Vasya can press any of these buttons. When the button is pressed, it turns on all the bulbs it's connected to. Can Vasya light up all the bulbs?If Vasya presses the button such that some bulbs connected to it are already turned on, they do not change their state, i.e. remain turned on.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"626"Polycarp thinks about the meaning of life very often. He does this constantly, even when typing in the editor. Every time he starts brooding he can no longer fully concentrate and repeatedly presses the keys that need to be pressed only once. For example, instead of the phrase ""how are you"" he can type ""hhoow aaaare yyoouu"". Polycarp decided to automate the process of correcting such errors. He decided to write a plug-in to the text editor that will remove pairs of identical consecutive letters (if there are any in the text). Of course, this is not exactly what Polycarp needs, but he's got to start from something! Help Polycarp and write the main plug-in module. Your program should remove from a string all pairs of identical letters, which are consecutive. If after the removal there appear new pairs, the program should remove them as well. Technically, its work should be equivalent to the following: while the string contains a pair of consecutive identical letters, the pair should be deleted. Note that deleting of the consecutive identical letters can be done in any order, as any order leads to the same result. ",[],"[0, 0, 0, 0, 0, 0, 0, 0]"627"It can be shown that any positive integer x can be uniquely represented as x = 1 + 2 + 4 + ... + 2k - 1 + r, where k and r are integers, k ≥ 0, 0 < r ≤ 2k. Let's call that representation prairie partition of x.For example, the prairie partitions of 12, 17, 7 and 1 are:  12 = 1 + 2 + 4 + 5,17 = 1 + 2 + 4 + 8 + 2,7 = 1 + 2 + 4,1 = 1. Alice took a sequence of positive integers (possibly with repeating elements), replaced every element with the sequence of summands in its prairie partition, arranged the resulting numbers in non-decreasing order and gave them to Borys. Now Borys wonders how many elements Alice's original sequence could contain. Find all possible options!",[],"[0, 0, 0, 0, 0, 0, 0, 0]"628"A number $$$a_2$$$ is said to be the arithmetic mean of two numbers $$$a_1$$$ and $$$a_3$$$, if the following condition holds: $$$a_1 + a_3 = 2\cdot a_2$$$. We define an arithmetic mean deviation of three numbers $$$a_1$$$, $$$a_2$$$ and $$$a_3$$$ as follows: $$$d(a_1, a_2, a_3) = |a_1 + a_3 - 2 \cdot a_2|$$$.Arithmetic means a lot to Jeevan. He has three numbers $$$a_1$$$, $$$a_2$$$ and $$$a_3$$$ and he wants to minimize the arithmetic mean deviation $$$d(a_1, a_2, a_3)$$$. To do so, he can perform the following operation any number of times (possibly zero):  Choose $$$i, j$$$ from $$$\{1, 2, 3\}$$$ such that $$$i \ne j$$$ and increment $$$a_i$$$ by $$$1$$$ and decrement $$$a_j$$$ by $$$1$$$ Help Jeevan find out the minimum value of $$$d(a_1, a_2, a_3)$$$ that can be obtained after applying the operation any number of times.","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"629"The Smart Beaver from ABBYY got hooked on square matrices. Now he is busy studying an n × n size matrix, where n is odd. The Smart Beaver considers the following matrix elements good:    Elements of the main diagonal.   Elements of the secondary diagonal.   Elements of the ""middle"" row — the row which has exactly  rows above it and the same number of rows below it.   Elements of the ""middle"" column — the column that has exactly  columns to the left of it and the same number of columns to the right of it.    The figure shows a 5 × 5 matrix.   The good elements are marked with green.  Help the Smart Beaver count the sum of good elements of the given matrix.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"630"You are playing a strange game with Li Chen. You have a tree with $$$n$$$ nodes drawn on a piece of paper. All nodes are unlabeled and distinguishable. Each of you independently labeled the vertices from $$$1$$$ to $$$n$$$. Neither of you know the other's labelling of the tree.You and Li Chen each chose a subtree (i.e., a connected subgraph) in that tree. Your subtree consists of the vertices labeled $$$x_1, x_2, \ldots, x_{k_1}$$$ in your labeling, Li Chen's subtree consists of the vertices labeled $$$y_1, y_2, \ldots, y_{k_2}$$$ in his labeling. The values of $$$x_1, x_2, \ldots, x_{k_1}$$$ and $$$y_1, y_2, \ldots, y_{k_2}$$$ are known to both of you.   The picture shows two labelings of a possible tree: yours on the left and Li Chen's on the right. The selected trees are highlighted. There are two common nodes. You want to determine whether your subtrees have at least one common vertex. Luckily, your friend Andrew knows both labelings of the tree. You can ask Andrew at most $$$5$$$ questions, each of which is in one of the following two forms:   A x: Andrew will look at vertex $$$x$$$ in your labeling and tell you the number of this vertex in Li Chen's labeling.  B y: Andrew will look at vertex $$$y$$$ in Li Chen's labeling and tell you the number of this vertex in your labeling. Determine whether the two subtrees have at least one common vertex after asking some questions. If there is at least one common vertex, determine one of your labels for any of the common vertices.",['trees'],"[0, 0, 0, 0, 0, 0, 0, 1]"631"$$$n$$$ robots have escaped from your laboratory! You have to find them as soon as possible, because these robots are experimental, and their behavior is not tested yet, so they may be really dangerous!Fortunately, even though your robots have escaped, you still have some control over them. First of all, you know the location of each robot: the world you live in can be modeled as an infinite coordinate plane, and the $$$i$$$-th robot is currently located at the point having coordinates ($$$x_i$$$, $$$y_i$$$). Furthermore, you may send exactly one command to all of the robots. The command should contain two integer numbers $$$X$$$ and $$$Y$$$, and when each robot receives this command, it starts moving towards the point having coordinates ($$$X$$$, $$$Y$$$). The robot stops its movement in two cases:  either it reaches ($$$X$$$, $$$Y$$$);  or it cannot get any closer to ($$$X$$$, $$$Y$$$). Normally, all robots should be able to get from any point of the coordinate plane to any other point. Each robot usually can perform four actions to move. Let's denote the current coordinates of the robot as ($$$x_c$$$, $$$y_c$$$). Then the movement system allows it to move to any of the four adjacent points:  the first action allows it to move from ($$$x_c$$$, $$$y_c$$$) to ($$$x_c - 1$$$, $$$y_c$$$);  the second action allows it to move from ($$$x_c$$$, $$$y_c$$$) to ($$$x_c$$$, $$$y_c + 1$$$);  the third action allows it to move from ($$$x_c$$$, $$$y_c$$$) to ($$$x_c + 1$$$, $$$y_c$$$);  the fourth action allows it to move from ($$$x_c$$$, $$$y_c$$$) to ($$$x_c$$$, $$$y_c - 1$$$). Unfortunately, it seems that some movement systems of some robots are malfunctioning. For each robot you know which actions it can perform, and which it cannot perform.You want to send a command so all robots gather at the same point. To do so, you have to choose a pair of integer numbers $$$X$$$ and $$$Y$$$ so that each robot can reach the point ($$$X$$$, $$$Y$$$). Is it possible to find such a point?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"632"Consider a game with $$$n$$$ cards ($$$n$$$ is even). Each card has a number written on it, between $$$1$$$ and $$$n$$$. All numbers on the cards are different. We say that a card with number $$$x$$$ is stronger than a card with number $$$y$$$ if $$$x > y$$$.Two players, Alex and Boris, play this game. In the beginning, each of them receives exactly $$$\frac{n}{2}$$$ cards, so each card belongs to exactly one player. Then, they take turns. Alex goes first, then Boris, then Alex again, and so on.On a player's turn, he must play exactly one of his cards. Then, if the opponent doesn't have any cards stronger than the card played, the opponent loses, and the game ends. Otherwise, the opponent has to play a stronger card (exactly one card as well). These two cards are removed from the game, and the turn ends. If there are no cards left, the game ends in a draw; otherwise it's the opponent's turn.Consider all possible ways to distribute the cards between two players, so that each of them receives exactly half of the cards. You have to calculate three numbers:  the number of ways to distribute the cards so that Alex wins;  the number of ways to distribute the cards so that Boris wins;  the number of ways to distribute the cards so that the game ends in a draw. You may assume that both players play optimally (i. e. if a player can win no matter how his opponent plays, he wins). Two ways to distribute the cards are different if there is at least one card such that, in one of these ways, it is given to Alex, and in the other way, it is given to Boris.For example, suppose $$$n = 4$$$, Alex receives the cards $$$[2, 3]$$$, and Boris receives the cards $$$[1, 4]$$$. Then the game may go as follows:  if Alex plays the card $$$2$$$, then Boris has to respond with the card $$$4$$$. Then, Alex's turn ends, and Boris' turn starts. Boris has only one card left, which is $$$1$$$; he plays it, and Alex responds with the card $$$3$$$. So, the game ends in a draw;  if Alex plays the card $$$3$$$, then Boris has to respond with the card $$$4$$$. Then, Alex's turn ends, and Boris' turn starts. Boris has only one card left, which is $$$1$$$; he plays it, and Alex responds with the card $$$2$$$. So, the game ends in a draw. So, in this case, the game ends in a draw.",['games'],"[1, 0, 0, 0, 0, 0, 0, 0]"633"There are $$$n$$$ warriors in a row. The power of the $$$i$$$-th warrior is $$$a_i$$$. All powers are pairwise distinct.You have two types of spells which you may cast:   Fireball: you spend $$$x$$$ mana and destroy exactly $$$k$$$ consecutive warriors;  Berserk: you spend $$$y$$$ mana, choose two consecutive warriors, and the warrior with greater power destroys the warrior with smaller power. For example, let the powers of warriors be $$$[2, 3, 7, 8, 11, 5, 4]$$$, and $$$k = 3$$$. If you cast Berserk on warriors with powers $$$8$$$ and $$$11$$$, the resulting sequence of powers becomes $$$[2, 3, 7, 11, 5, 4]$$$. Then, for example, if you cast Fireball on consecutive warriors with powers $$$[7, 11, 5]$$$, the resulting sequence of powers becomes $$$[2, 3, 4]$$$.You want to turn the current sequence of warriors powers $$$a_1, a_2, \dots, a_n$$$ into $$$b_1, b_2, \dots, b_m$$$. Calculate the minimum amount of mana you need to spend on it.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"634"Gosha's universe is a table consisting of n rows and m columns. Both the rows and columns are numbered with consecutive integers starting with 1. We will use (r, c) to denote a cell located in the row r and column c.Gosha is often invited somewhere. Every time he gets an invitation, he first calculates the number of ways to get to this place, and only then he goes. Gosha's house is located in the cell (1, 1).At any moment of time, Gosha moves from the cell he is currently located in to a cell adjacent to it (two cells are adjacent if they share a common side). Of course, the movement is possible only if such a cell exists, i.e. Gosha will not go beyond the boundaries of the table. Thus, from the cell (r, c) he is able to make a move to one of the cells (r - 1, c), (r, c - 1), (r + 1, c), (r, c + 1). Also, Ghosha can skip a move and stay in the current cell (r, c).Besides the love of strange calculations, Gosha is allergic to cats, so he never goes to the cell that has a cat in it. Gosha knows exactly where and when he will be invited and the schedule of cats travelling along the table. Formally, he has q records, the i-th of them has one of the following forms:   1, xi, yi, ti — Gosha is invited to come to cell (xi, yi) at the moment of time ti. It is guaranteed that there is no cat inside cell (xi, yi) at this moment of time.  2, xi, yi, ti — at the moment ti a cat appears in cell (xi, yi). It is guaranteed that no other cat is located in this cell (xi, yi) at that moment of time.  3, xi, yi, ti — at the moment ti a cat leaves cell (xi, yi). It is guaranteed that there is cat located in the cell (xi, yi). Gosha plans to accept only one invitation, but he has not yet decided, which particular one. In order to make this decision, he asks you to calculate for each of the invitations i the number of ways to get to the cell (xi, yi) at the moment ti. For every invitation, assume that Gosha he starts moving from cell (1, 1) at the moment 1.Moving between two neighboring cells takes Gosha exactly one unit of tim. In particular, this means that Gosha can come into the cell only if a cat sitting in it leaves the moment when Gosha begins his movement from the neighboring cell, and if none of the cats comes to the cell at the time when Gosha is in it.Two ways to go from cell (1, 1) to cell (x, y) at time t are considered distinct if for at least one moment of time from 1 to t Gosha's positions are distinct for the two ways at this moment. Note, that during this travel Gosha is allowed to visit both (1, 1) and (x, y) multiple times. Since the number of ways can be quite large, print it modulo 109 + 7.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"635"A positive integer number n is written on a blackboard. It consists of not more than 105 digits. You have to transform it into a beautiful number by erasing some of the digits, and you want to erase as few digits as possible.The number is called beautiful if it consists of at least one digit, doesn't have leading zeroes and is a multiple of 3. For example, 0, 99, 10110 are beautiful numbers, and 00, 03, 122 are not.Write a program which for the given n will find a beautiful number such that n can be transformed into this number by erasing as few digits as possible. You can erase an arbitraty set of digits. For example, they don't have to go one after another in the number n.If it's impossible to obtain a beautiful number, print -1. If there are multiple answers, print any of them.","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"636"In the army, it isn't easy to form a group of soldiers that will be effective on the battlefield. The communication is crucial and thus no two soldiers should share a name (what would happen if they got an order that Bob is a scouter, if there are two Bobs?).A group of soldiers is effective if and only if their names are different. For example, a group (John, Bob, Limak) would be effective, while groups (Gary, Bob, Gary) and (Alice, Alice) wouldn't.You are a spy in the enemy's camp. You noticed n soldiers standing in a row, numbered 1 through n. The general wants to choose a group of k consecutive soldiers. For every k consecutive soldiers, the general wrote down whether they would be an effective group or not.You managed to steal the general's notes, with n - k + 1 strings s1, s2, ..., sn - k + 1, each either ""YES"" or ""NO"".   The string s1 describes a group of soldiers 1 through k (""YES"" if the group is effective, and ""NO"" otherwise).  The string s2 describes a group of soldiers 2 through k + 1.  And so on, till the string sn - k + 1 that describes a group of soldiers n - k + 1 through n. Your task is to find possible names of n soldiers. Names should match the stolen notes. Each name should be a string that consists of between 1 and 10 English letters, inclusive. The first letter should be uppercase, and all other letters should be lowercase. Names don't have to be existing names — it's allowed to print ""Xyzzzdj"" or ""T"" for example.Find and print any solution. It can be proved that there always exists at least one solution.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"637"Along the railroad there are stations indexed from $$$1$$$ to $$$10^9$$$. An express train always travels along a route consisting of $$$n$$$ stations with indices $$$u_1, u_2, \dots, u_n$$$, where ($$$1 \le u_i \le 10^9$$$). The train travels along the route from left to right. It starts at station $$$u_1$$$, then stops at station $$$u_2$$$, then at $$$u_3$$$, and so on. Station $$$u_n$$$ — the terminus.It is possible that the train will visit the same station more than once. That is, there may be duplicates among the values $$$u_1, u_2, \dots, u_n$$$.You are given $$$k$$$ queries, each containing two different integers $$$a_j$$$ and $$$b_j$$$ ($$$1 \le a_j, b_j \le 10^9$$$). For each query, determine whether it is possible to travel by train from the station with index $$$a_j$$$ to the station with index $$$b_j$$$.For example, let the train route consist of $$$6$$$ of stations with indices [$$$3, 7, 1, 5, 1, 4$$$] and give $$$3$$$ of the following queries:   $$$a_1 = 3$$$, $$$b_1 = 5$$$It is possible to travel from station $$$3$$$ to station $$$5$$$ by taking a section of the route consisting of stations [$$$3, 7, 1, 5$$$]. Answer: YES.  $$$a_2 = 1$$$, $$$b_2 = 7$$$You cannot travel from station $$$1$$$ to station $$$7$$$ because the train cannot travel in the opposite direction. Answer: NO.  $$$a_3 = 3$$$, $$$b_3 = 10$$$It is not possible to travel from station $$$3$$$ to station $$$10$$$ because station $$$10$$$ is not part of the train's route. Answer: NO. ",[],"[0, 0, 0, 0, 0, 0, 0, 0]"638"It turns out that you are a great fan of rock band AC/PE. Peter learned that and started the following game: he plays the first song of the list of n songs of the group, and you have to find out the name of the song. After you tell the song name, Peter immediately plays the following song in order, and so on.The i-th song of AC/PE has its recognizability pi. This means that if the song has not yet been recognized by you, you listen to it for exactly one more second and with probability of pi percent you recognize it and tell it's name. Otherwise you continue listening it. Note that you can only try to guess it only when it is integer number of seconds after the moment the song starts playing.In all AC/PE songs the first words of chorus are the same as the title, so when you've heard the first ti seconds of i-th song and its chorus starts, you immediately guess its name for sure.For example, in the song Highway To Red the chorus sounds pretty late, but the song has high recognizability. In the song Back In Blue, on the other hand, the words from the title sound close to the beginning of the song, but it's hard to name it before hearing those words. You can name both of these songs during a few more first seconds.Determine the expected number songs of you will recognize if the game lasts for exactly T seconds (i. e. you can make the last guess on the second T, after that the game stops).If all songs are recognized faster than in T seconds, the game stops after the last song is recognized.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"639"The mayor of the Central Town wants to modernize Central Street, represented in this problem by the $$$(Ox)$$$ axis.On this street, there are $$$n$$$ antennas, numbered from $$$1$$$ to $$$n$$$. The $$$i$$$-th antenna lies on the position $$$x_i$$$ and has an initial scope of $$$s_i$$$: it covers all integer positions inside the interval $$$[x_i - s_i; x_i + s_i]$$$.It is possible to increment the scope of any antenna by $$$1$$$, this operation costs $$$1$$$ coin. We can do this operation as much as we want (multiple times on the same antenna if we want).To modernize the street, we need to make all integer positions from $$$1$$$ to $$$m$$$ inclusive covered by at least one antenna. Note that it is authorized to cover positions outside $$$[1; m]$$$, even if it's not required.What is the minimum amount of coins needed to achieve this modernization?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"640"You have an array $$$a$$$ consisting of $$$n$$$ distinct positive integers, numbered from $$$1$$$ to $$$n$$$. Define $$$p_k$$$ as $$$$$$p_k = \sum_{1 \le i, j \le k} a_i \bmod a_j,$$$$$$ where $$$x \bmod y$$$ denotes the remainder when $$$x$$$ is divided by $$$y$$$. You have to find and print $$$p_1, p_2, \ldots, p_n$$$. ",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"641"The only difference between the easy and the hard versions is the maximum value of $$$k$$$.You are given an infinite sequence of form ""112123123412345$$$\dots$$$"" which consist of blocks of all consecutive positive integers written one after another. The first block consists of all numbers from $$$1$$$ to $$$1$$$, the second one — from $$$1$$$ to $$$2$$$, the third one — from $$$1$$$ to $$$3$$$, $$$\dots$$$, the $$$i$$$-th block consists of all numbers from $$$1$$$ to $$$i$$$. So the first $$$56$$$ elements of the sequence are ""11212312341234512345612345671234567812345678912345678910"". Elements of the sequence are numbered from one. For example, the $$$1$$$-st element of the sequence is $$$1$$$, the $$$3$$$-rd element of the sequence is $$$2$$$, the $$$20$$$-th element of the sequence is $$$5$$$, the $$$38$$$-th element is $$$2$$$, the $$$56$$$-th element of the sequence is $$$0$$$.Your task is to answer $$$q$$$ independent queries. In the $$$i$$$-th query you are given one integer $$$k_i$$$. Calculate the digit at the position $$$k_i$$$ of the sequence.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"642"A company of $$$n$$$ friends wants to order exactly two pizzas. It is known that in total there are $$$9$$$ pizza ingredients in nature, which are denoted by integers from $$$1$$$ to $$$9$$$.Each of the $$$n$$$ friends has one or more favorite ingredients: the $$$i$$$-th of friends has the number of favorite ingredients equal to $$$f_i$$$ ($$$1 \le f_i \le 9$$$) and your favorite ingredients form the sequence $$$b_{i1}, b_{i2}, \dots, b_{if_i}$$$ ($$$1 \le b_{it} \le 9$$$).The website of CodePizza restaurant has exactly $$$m$$$ ($$$m \ge 2$$$) pizzas. Each pizza is characterized by a set of $$$r_j$$$ ingredients $$$a_{j1}, a_{j2}, \dots, a_{jr_j}$$$ ($$$1 \le r_j \le 9$$$, $$$1 \le a_{jt} \le 9$$$) , which are included in it, and its price is $$$c_j$$$.Help your friends choose exactly two pizzas in such a way as to please the maximum number of people in the company. It is known that a person is pleased with the choice if each of his/her favorite ingredients is in at least one ordered pizza. If there are several ways to choose two pizzas so as to please the maximum number of friends, then choose the one that minimizes the total price of two pizzas.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"643"You have a string $$$s$$$ consisting of digits from $$$0$$$ to $$$9$$$ inclusive. You can perform the following operation any (possibly zero) number of times:   You can choose a position $$$i$$$ and delete a digit $$$d$$$ on the $$$i$$$-th position. Then insert the digit $$$\min{(d + 1, 9)}$$$ on any position (at the beginning, at the end or in between any two adjacent digits). What is the lexicographically smallest string you can get by performing these operations?A string $$$a$$$ is lexicographically smaller than a string $$$b$$$ of the same length if and only if the following holds:   in the first position where $$$a$$$ and $$$b$$$ differ, the string $$$a$$$ has a smaller digit than the corresponding digit in $$$b$$$. ",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"644"You are given $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$. Is it possible to arrange them on a circle so that each number is strictly greater than both its neighbors or strictly smaller than both its neighbors?In other words, check if there exists a rearrangement $$$b_1, b_2, \ldots, b_n$$$ of the integers $$$a_1, a_2, \ldots, a_n$$$ such that for each $$$i$$$ from $$$1$$$ to $$$n$$$ at least one of the following conditions holds: $$$b_{i-1} < b_i > b_{i+1}$$$ $$$b_{i-1} > b_i < b_{i+1}$$$To make sense of the previous formulas for $$$i=1$$$ and $$$i=n$$$, one shall define $$$b_0=b_n$$$ and $$$b_{n+1}=b_1$$$.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"645"The only difference between easy and hard versions is on constraints. In this version constraints are higher. You can make hacks only if all versions of the problem are solved.Koa the Koala is at the beach!The beach consists (from left to right) of a shore, $$$n+1$$$ meters of sea and an island at $$$n+1$$$ meters from the shore.She measured the depth of the sea at $$$1, 2, \dots, n$$$ meters from the shore and saved them in array $$$d$$$. $$$d_i$$$ denotes the depth of the sea at $$$i$$$ meters from the shore for $$$1 \le i \le n$$$.Like any beach this one has tide, the intensity of the tide is measured by parameter $$$k$$$ and affects all depths from the beginning at time $$$t=0$$$ in the following way:  For a total of $$$k$$$ seconds, each second, tide increases all depths by $$$1$$$.  Then, for a total of $$$k$$$ seconds, each second, tide decreases all depths by $$$1$$$.  This process repeats again and again (ie. depths increase for $$$k$$$ seconds then decrease for $$$k$$$ seconds and so on ...).Formally, let's define $$$0$$$-indexed array $$$p = [0, 1, 2, \ldots, k - 2, k - 1, k, k - 1, k - 2, \ldots, 2, 1]$$$ of length $$$2k$$$. At time $$$t$$$ ($$$0 \le t$$$) depth at $$$i$$$ meters from the shore equals $$$d_i + p[t \bmod 2k]$$$ ($$$t \bmod 2k$$$ denotes the remainder of the division of $$$t$$$ by $$$2k$$$). Note that the changes occur instantaneously after each second, see the notes for better understanding. At time $$$t=0$$$ Koa is standing at the shore and wants to get to the island. Suppose that at some time $$$t$$$ ($$$0 \le t$$$) she is at $$$x$$$ ($$$0 \le x \le n$$$) meters from the shore:  In one second Koa can swim $$$1$$$ meter further from the shore ($$$x$$$ changes to $$$x+1$$$) or not swim at all ($$$x$$$ stays the same), in both cases $$$t$$$ changes to $$$t+1$$$.  As Koa is a bad swimmer, the depth of the sea at the point where she is can't exceed $$$l$$$ at integer points of time (or she will drown). More formally, if Koa is at $$$x$$$ ($$$1 \le x \le n$$$) meters from the shore at the moment $$$t$$$ (for some integer $$$t\ge 0$$$), the depth of the sea at this point  — $$$d_x + p[t \bmod 2k]$$$  — can't exceed $$$l$$$. In other words, $$$d_x + p[t \bmod 2k] \le l$$$ must hold always.  Once Koa reaches the island at $$$n+1$$$ meters from the shore, she stops and can rest.Note that while Koa swims tide doesn't have effect on her (ie. she can't drown while swimming). Note that Koa can choose to stay on the shore for as long as she needs and neither the shore or the island are affected by the tide (they are solid ground and she won't drown there). Koa wants to know whether she can go from the shore to the island. Help her!",[],"[0, 0, 0, 0, 0, 0, 0, 0]"646"You are given a weighted directed graph, consisting of n vertices and m edges. You should answer q queries of two types:  1 v — find the length of shortest path from vertex 1 to vertex v.  2 c l1 l2 ... lc — add 1 to weights of edges with indices l1, l2, ..., lc. ",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"647"Amugae has a sentence consisting of $$$n$$$ words. He want to compress this sentence into one word. Amugae doesn't like repetitions, so when he merges two words into one word, he removes the longest prefix of the second word that coincides with a suffix of the first word. For example, he merges ""sample"" and ""please"" into ""samplease"".Amugae will merge his sentence left to right (i.e. first merge the first two words, then merge the result with the third word and so on). Write a program that prints the compressed word after the merging process ends.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"648"Sereja has got an array, consisting of n integers, a1, a2, ..., an. Sereja is an active boy, so he is now going to complete m operations. Each operation will have one of the three forms:  Make vi-th array element equal to xi. In other words, perform the assignment avi = xi.  Increase each array element by yi. In other words, perform n assignments ai = ai + yi (1 ≤ i ≤ n).  Take a piece of paper and write out the qi-th array element. That is, the element aqi. Help Sereja, complete all his operations.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"649"Little Nastya has a hobby, she likes to remove some letters from word, to obtain another word. But it turns out to be pretty hard for her, because she is too young. Therefore, her brother Sergey always helps her.Sergey gives Nastya the word t and wants to get the word p out of it. Nastya removes letters in a certain order (one after another, in this order strictly), which is specified by permutation of letters' indices of the word t: a1... a|t|. We denote the length of word x as |x|. Note that after removing one letter, the indices of other letters don't change. For example, if t = ""nastya"" and a = [4, 1, 5, 3, 2, 6] then removals make the following sequence of words ""nastya""  ""nastya""  ""nastya""  ""nastya""  ""nastya""  ""nastya""  ""nastya"".Sergey knows this permutation. His goal is to stop his sister at some point and continue removing by himself to get the word p. Since Nastya likes this activity, Sergey wants to stop her as late as possible. Your task is to determine, how many letters Nastya can remove before she will be stopped by Sergey.It is guaranteed that the word p can be obtained by removing the letters from word t.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"650"Iahub recently has learned Bubble Sort, an algorithm that is used to sort a permutation with n elements a1, a2, ..., an in ascending order. He is bored of this so simple algorithm, so he invents his own graph. The graph (let's call it G) initially has n vertices and 0 edges. During Bubble Sort execution, edges appear as described in the following algorithm (pseudocode). procedure bubbleSortGraph()    build a graph G with n vertices and 0 edges    repeat        swapped = false        for i = 1 to n - 1 inclusive do:            if a[i] > a[i + 1] then                add an undirected edge in G between a[i] and a[i + 1]                swap( a[i], a[i + 1] )                swapped = true            end if        end for    until not swapped     /* repeat the algorithm as long as swapped value is true. */ end procedureFor a graph, an independent set is a set of vertices in a graph, no two of which are adjacent (so there are no edges between vertices of an independent set). A maximum independent set is an independent set which has maximum cardinality. Given the permutation, find the size of the maximum independent set of graph G, if we use such permutation as the premutation a in procedure bubbleSortGraph.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"651" You are given an array $$$a_1, a_2, \dots, a_n$$$ where all $$$a_i$$$ are integers and greater than $$$0$$$. In one operation, you can choose two different indices $$$i$$$ and $$$j$$$ ($$$1 \le i, j \le n$$$). If $$$gcd(a_i, a_j)$$$ is equal to the minimum element of the whole array $$$a$$$, you can swap $$$a_i$$$ and $$$a_j$$$. $$$gcd(x, y)$$$ denotes the greatest common divisor (GCD) of integers $$$x$$$ and $$$y$$$. Now you'd like to make $$$a$$$ non-decreasing using the operation any number of times (possibly zero). Determine if you can do this. An array $$$a$$$ is non-decreasing if and only if $$$a_1 \le a_2 \le \ldots \le a_n$$$.","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"652"There are $$$n$$$ people participating in some contest, they start participating in $$$x$$$ minutes intervals. That means the first participant starts at time $$$0$$$, the second participant starts at time $$$x$$$, the third — at time $$$2 \cdot x$$$, and so on.Duration of contest is $$$t$$$ minutes for each participant, so the first participant finishes the contest at time $$$t$$$, the second — at time $$$t + x$$$, and so on. When a participant finishes the contest, their dissatisfaction equals to the number of participants that started the contest (or starting it now), but haven't yet finished it.Determine the sum of dissatisfaction of all participants.","['geometry', 'math']","[0, 1, 0, 1, 0, 0, 0, 0]"653"This is the harder version of the problem. In this version, $$$1 \le n \le 10^6$$$ and $$$0 \leq a_i \leq 10^6$$$. You can hack this problem if you locked it. But you can hack the previous problem only if you locked both problemsChristmas is coming, and our protagonist, Bob, is preparing a spectacular present for his long-time best friend Alice. This year, he decides to prepare $$$n$$$ boxes of chocolate, numbered from $$$1$$$ to $$$n$$$. Initially, the $$$i$$$-th box contains $$$a_i$$$ chocolate pieces.Since Bob is a typical nice guy, he will not send Alice $$$n$$$ empty boxes. In other words, at least one of $$$a_1, a_2, \ldots, a_n$$$ is positive. Since Alice dislikes coprime sets, she will be happy only if there exists some integer $$$k > 1$$$ such that the number of pieces in each box is divisible by $$$k$$$. Note that Alice won't mind if there exists some empty boxes. Charlie, Alice's boyfriend, also is Bob's second best friend, so he decides to help Bob by rearranging the chocolate pieces. In one second, Charlie can pick up a piece in box $$$i$$$ and put it into either box $$$i-1$$$ or box $$$i+1$$$ (if such boxes exist). Of course, he wants to help his friend as quickly as possible. Therefore, he asks you to calculate the minimum number of seconds he would need to make Alice happy.","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"654"The Dogeforces company has $$$k$$$ employees. Each employee, except for lower-level employees, has at least $$$2$$$ subordinates. Lower-level employees have no subordinates. Each employee, except for the head of the company, has exactly one direct supervisor. The head of the company is a direct or indirect supervisor of all employees. It is known that in Dogeforces, each supervisor receives a salary strictly more than all his subordinates.The full structure of the company is a secret, but you know the number of lower-level employees and for each pair of lower-level employees, the salary of their common supervisor is known (if there are several such supervisors, then the supervisor with the minimum salary). You have to restore the structure of the company.",['trees'],"[0, 0, 0, 0, 0, 0, 0, 1]"655"We say an infinite sequence $$$a_{0}, a_{1}, a_2, \ldots$$$ is non-increasing if and only if for all $$$i\ge 0$$$, $$$a_i \ge a_{i+1}$$$.There is an infinite right and down grid. The upper-left cell has coordinates $$$(0,0)$$$. Rows are numbered $$$0$$$ to infinity from top to bottom, columns are numbered from $$$0$$$ to infinity from left to right.There is also a non-increasing infinite sequence $$$a_{0}, a_{1}, a_2, \ldots$$$. You are given $$$a_0$$$, $$$a_1$$$, $$$\ldots$$$, $$$a_n$$$; for all $$$i>n$$$, $$$a_i=0$$$. For every pair of $$$x$$$, $$$y$$$, the cell with coordinates $$$(x,y)$$$ (which is located at the intersection of $$$x$$$-th row and $$$y$$$-th column) is white if $$$y<a_x$$$ and black otherwise.Initially there is one doll named Jina on $$$(0,0)$$$. You can do the following operation.  Select one doll on $$$(x,y)$$$. Remove it and place a doll on $$$(x,y+1)$$$ and place a doll on $$$(x+1,y)$$$. Note that multiple dolls can be present at a cell at the same time; in one operation, you remove only one. Your goal is to make all white cells contain $$$0$$$ dolls.What's the minimum number of operations needed to achieve the goal? Print the answer modulo $$$10^9+7$$$.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"656"Nikolay lives in a two-storied house. There are $$$n$$$ rooms on each floor, arranged in a row and numbered from one from left to right. So each room can be represented by the number of the floor and the number of the room on this floor (room number is an integer between $$$1$$$ and $$$n$$$). If Nikolay is currently in some room, he can move to any of the neighbouring rooms (if they exist). Rooms with numbers $$$i$$$ and $$$i+1$$$ on each floor are neighbouring, for all $$$1 \leq i \leq n - 1$$$. There may also be staircases that connect two rooms from different floors having the same numbers. If there is a staircase connecting the room $$$x$$$ on the first floor and the room $$$x$$$ on the second floor, then Nikolay can use it to move from one room to another.     The picture illustrates a house with $$$n = 4$$$. There is a staircase between the room $$$2$$$ on the first floor and the room $$$2$$$ on the second floor, and another staircase between the room $$$4$$$ on the first floor and the room $$$4$$$ on the second floor. The arrows denote possible directions in which Nikolay can move. The picture corresponds to the string ""0101"" in the input.  Nikolay wants to move through some rooms in his house. To do this, he firstly chooses any room where he starts. Then Nikolay moves between rooms according to the aforementioned rules. Nikolay never visits the same room twice (he won't enter a room where he has already been). Calculate the maximum number of rooms Nikolay can visit during his tour, if:  he can start in any room on any floor of his choice,  and he won't visit the same room twice. ",[],"[0, 0, 0, 0, 0, 0, 0, 0]"657"You are given a sequence of numbers a1, a2, ..., an, and a number m.Check if it is possible to choose a non-empty subsequence aij such that the sum of numbers in this subsequence is divisible by m.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"658"You are given a permutation $$$p_1, p_2, \ldots, p_n$$$ of integers from $$$1$$$ to $$$n$$$ and an integer $$$k$$$, such that $$$1 \leq k \leq n$$$. A permutation means that every number from $$$1$$$ to $$$n$$$ is contained in $$$p$$$ exactly once.Let's consider all partitions of this permutation into $$$k$$$ disjoint segments. Formally, a partition is a set of segments $$$\{[l_1, r_1], [l_2, r_2], \ldots, [l_k, r_k]\}$$$, such that:  $$$1 \leq l_i \leq r_i \leq n$$$ for all $$$1 \leq i \leq k$$$;  For all $$$1 \leq j \leq n$$$ there exists exactly one segment $$$[l_i, r_i]$$$, such that $$$l_i \leq j \leq r_i$$$. Two partitions are different if there exists a segment that lies in one partition but not the other.Let's calculate the partition value, defined as $$$\sum\limits_{i=1}^{k} {\max\limits_{l_i \leq j \leq r_i} {p_j}}$$$, for all possible partitions of the permutation into $$$k$$$ disjoint segments. Find the maximum possible partition value over all such partitions, and the number of partitions with this value. As the second value can be very large, you should find its remainder when divided by $$$998\,244\,353$$$.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"659"You are given a sequence of $$$n$$$ integers $$$a_1, \, a_2, \, \dots, \, a_n$$$.Does there exist a sequence of $$$n$$$ integers $$$b_1, \, b_2, \, \dots, \, b_n$$$ such that the following property holds?  For each $$$1 \le i \le n$$$, there exist two (not necessarily distinct) indices $$$j$$$ and $$$k$$$ ($$$1 \le j, \, k \le n$$$) such that $$$a_i = b_j - b_k$$$. ","['graphs', 'math']","[0, 0, 1, 1, 0, 0, 0, 0]"660"People do many crazy things to stand out in a crowd. Some of them dance, some learn by heart rules of Russian language, some try to become an outstanding competitive programmers, while others collect funny math objects.Alis is among these collectors. Right now she wants to get one of k-special tables. In case you forget, the table n × n is called k-special if the following three conditions are satisfied:  every integer from 1 to n2 appears in the table exactly once;  in each row numbers are situated in increasing order;  the sum of numbers in the k-th column is maximum possible. Your goal is to help Alice and find at least one k-special table of size n × n. Both rows and columns are numbered from 1 to n, with rows numbered from top to bottom and columns numbered from left to right.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"661"Recently a dog was bought for Polycarp. The dog's name is Cormen. Now Polycarp has a lot of troubles. For example, Cormen likes going for a walk. Empirically Polycarp learned that the dog needs at least k walks for any two consecutive days in order to feel good. For example, if k = 5 and yesterday Polycarp went for a walk with Cormen 2 times, today he has to go for a walk at least 3 times. Polycarp analysed all his affairs over the next n days and made a sequence of n integers a1, a2, ..., an, where ai is the number of times Polycarp will walk with the dog on the i-th day while doing all his affairs (for example, he has to go to a shop, throw out the trash, etc.).Help Polycarp determine the minimum number of walks he needs to do additionaly in the next n days so that Cormen will feel good during all the n days. You can assume that on the day before the first day and on the day after the n-th day Polycarp will go for a walk with Cormen exactly k times. Write a program that will find the minumum number of additional walks and the appropriate schedule — the sequence of integers b1, b2, ..., bn (bi ≥ ai), where bi means the total number of walks with the dog on the i-th day.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"662"There is a chessboard of size $$$n$$$ by $$$n$$$. The square in the $$$i$$$-th row from top and $$$j$$$-th column from the left is labelled $$$(i,j)$$$.Currently, Gregor has some pawns in the $$$n$$$-th row. There are also enemy pawns in the $$$1$$$-st row. On one turn, Gregor moves one of his pawns. A pawn can move one square up (from $$$(i,j)$$$ to $$$(i-1,j)$$$) if there is no pawn in the destination square. Additionally, a pawn can move one square diagonally up (from $$$(i,j)$$$ to either $$$(i-1,j-1)$$$ or $$$(i-1,j+1)$$$) if and only if there is an enemy pawn in that square. The enemy pawn is also removed.Gregor wants to know what is the maximum number of his pawns that can reach row $$$1$$$?Note that only Gregor takes turns in this game, and the enemy pawns never move. Also, when Gregor's pawn reaches row $$$1$$$, it is stuck and cannot make any further moves.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"663"Polycarp likes to play with numbers. He takes some integer number $$$x$$$, writes it down on the board, and then performs with it $$$n - 1$$$ operations of the two kinds:   divide the number $$$x$$$ by $$$3$$$ ($$$x$$$ must be divisible by $$$3$$$);  multiply the number $$$x$$$ by $$$2$$$. After each operation, Polycarp writes down the result on the board and replaces $$$x$$$ by the result. So there will be $$$n$$$ numbers on the board after all.You are given a sequence of length $$$n$$$ — the numbers that Polycarp wrote down. This sequence is given in arbitrary order, i.e. the order of the sequence can mismatch the order of the numbers written on the board.Your problem is to rearrange (reorder) elements of this sequence in such a way that it can match possible Polycarp's game in the order of the numbers written on the board. I.e. each next number will be exactly two times of the previous number or exactly one third of previous number.It is guaranteed that the answer exists.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"664"You are given a sequence $$$a_1, a_2, \dots, a_n$$$ consisting of $$$n$$$ integers.You may perform the following operation on this sequence: choose any element and either increase or decrease it by one.Calculate the minimum possible difference between the maximum element and the minimum element in the sequence, if you can perform the aforementioned operation no more than $$$k$$$ times.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"665"Let's call a string good if its length is at least $$$2$$$ and all of its characters are $$$\texttt{A}$$$ except for the last character which is $$$\texttt{B}$$$. The good strings are $$$\texttt{AB},\texttt{AAB},\texttt{AAAB},\ldots$$$. Note that $$$\texttt{B}$$$ is not a good string.You are given an initially empty string $$$s_1$$$.You can perform the following operation any number of times:   Choose any position of $$$s_1$$$ and insert some good string in that position. Given a string $$$s_2$$$, can we turn $$$s_1$$$ into $$$s_2$$$ after some number of operations?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"666"Phoenix is picking berries in his backyard. There are $$$n$$$ shrubs, and each shrub has $$$a_i$$$ red berries and $$$b_i$$$ blue berries.Each basket can contain $$$k$$$ berries. But, Phoenix has decided that each basket may only contain berries from the same shrub or berries of the same color (red or blue). In other words, all berries in a basket must be from the same shrub or/and have the same color.For example, if there are two shrubs with $$$5$$$ red and $$$2$$$ blue berries in the first shrub and $$$2$$$ red and $$$1$$$ blue berries in the second shrub then Phoenix can fill $$$2$$$ baskets of capacity $$$4$$$ completely:   the first basket will contain $$$3$$$ red and $$$1$$$ blue berries from the first shrub;  the second basket will contain the $$$2$$$ remaining red berries from the first shrub and $$$2$$$ red berries from the second shrub. Help Phoenix determine the maximum number of baskets he can fill completely!",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"667"Can the greatest common divisor and bitwise operations have anything in common? It is time to answer this question.Suppose you are given a positive integer $$$a$$$. You want to choose some integer $$$b$$$ from $$$1$$$ to $$$a - 1$$$ inclusive in such a way that the greatest common divisor (GCD) of integers $$$a \oplus b$$$ and $$$a \> \& \> b$$$ is as large as possible. In other words, you'd like to compute the following function:$$$$$$f(a) = \max_{0 < b < a}{gcd(a \oplus b, a \> \& \> b)}.$$$$$$Here $$$\oplus$$$ denotes the bitwise XOR operation, and $$$\&$$$ denotes the bitwise AND operation.The greatest common divisor of two integers $$$x$$$ and $$$y$$$ is the largest integer $$$g$$$ such that both $$$x$$$ and $$$y$$$ are divided by $$$g$$$ without remainder.You are given $$$q$$$ integers $$$a_1, a_2, \ldots, a_q$$$. For each of these integers compute the largest possible value of the greatest common divisor (when $$$b$$$ is chosen optimally). ","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"668"Hilbert's Hotel is a very unusual hotel since the number of rooms is infinite! In fact, there is exactly one room for every integer, including zero and negative integers. Even stranger, the hotel is currently at full capacity, meaning there is exactly one guest in every room. The hotel's manager, David Hilbert himself, decides he wants to shuffle the guests around because he thinks this will create a vacancy (a room without a guest).For any integer $$$k$$$ and positive integer $$$n$$$, let $$$k\bmod n$$$ denote the remainder when $$$k$$$ is divided by $$$n$$$. More formally, $$$r=k\bmod n$$$ is the smallest non-negative integer such that $$$k-r$$$ is divisible by $$$n$$$. It always holds that $$$0\le k\bmod n\le n-1$$$. For example, $$$100\bmod 12=4$$$ and $$$(-1337)\bmod 3=1$$$.Then the shuffling works as follows. There is an array of $$$n$$$ integers $$$a_0,a_1,\ldots,a_{n-1}$$$. Then for each integer $$$k$$$, the guest in room $$$k$$$ is moved to room number $$$k+a_{k\bmod n}$$$.After this shuffling process, determine if there is still exactly one guest assigned to each room. That is, there are no vacancies or rooms with multiple guests.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"669"You are given a non-empty string s consisting of lowercase letters. Find the number of pairs of non-overlapping palindromic substrings of this string.In a more formal way, you have to find the quantity of tuples (a, b, x, y) such that 1 ≤ a ≤ b < x ≤ y ≤ |s| and substrings s[a... b], s[x... y] are palindromes.A palindrome is a string that can be read the same way from left to right and from right to left. For example, ""abacaba"", ""z"", ""abba"" are palindromes.A substring s[i... j] (1 ≤ i ≤ j ≤ |s|) of string s = s1s2... s|s| is a string sisi + 1... sj. For example, substring s[2...4] of string s = ""abacaba"" equals ""bac"".",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"670"You are given a set of n elements indexed from 1 to n. The weight of i-th element is wi. The weight of some subset of a given set is denoted as . The weight of some partition R of a given set into k subsets is  (recall that a partition of a given set is a set of its subsets such that every element of the given set belongs to exactly one subset in partition).Calculate the sum of weights of all partitions of a given set into exactly k non-empty subsets, and print it modulo 109 + 7. Two partitions are considered different iff there exist two elements x and y such that they belong to the same set in one of the partitions, and to different sets in another partition.","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"671"We often go to supermarkets to buy some fruits or vegetables, and on the tag there prints the price for a kilo. But in some supermarkets, when asked how much the items are, the clerk will say that $$$a$$$ yuan for $$$b$$$ kilos (You don't need to care about what ""yuan"" is), the same as $$$a/b$$$ yuan for a kilo.Now imagine you'd like to buy $$$m$$$ kilos of apples. You've asked $$$n$$$ supermarkets and got the prices. Find the minimum cost for those apples.You can assume that there are enough apples in all supermarkets.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"672"You are given an array $$$a$$$ of $$$n$$$ integers, and another integer $$$k$$$ such that $$$2k \le n$$$.You have to perform exactly $$$k$$$ operations with this array. In one operation, you have to choose two elements of the array (let them be $$$a_i$$$ and $$$a_j$$$; they can be equal or different, but their positions in the array must not be the same), remove them from the array, and add $$$\lfloor \frac{a_i}{a_j} \rfloor$$$ to your score, where $$$\lfloor \frac{x}{y} \rfloor$$$ is the maximum integer not exceeding $$$\frac{x}{y}$$$.Initially, your score is $$$0$$$. After you perform exactly $$$k$$$ operations, you add all the remaining elements of the array to the score.Calculate the minimum possible score you can get.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"673"According to a new ISO standard, a flag of every country should have a chequered field n × m, each square should be of one of 10 colours, and the flag should be «striped»: each horizontal row of the flag should contain squares of the same colour, and the colours of adjacent horizontal rows should be different. Berland's government asked you to find out whether their flag meets the new ISO standard.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"674"This is an interactive problem.Natasha is going to fly to Mars. Finally, Natasha sat in the rocket. She flies, flies... but gets bored. She wishes to arrive to Mars already! So she decides to find something to occupy herself. She couldn't think of anything better to do than to calculate the distance to the red planet.Let's define $$$x$$$ as the distance to Mars. Unfortunately, Natasha does not know $$$x$$$. But it is known that $$$1 \le x \le m$$$, where Natasha knows the number $$$m$$$. Besides, $$$x$$$ and $$$m$$$ are positive integers.Natasha can ask the rocket questions. Every question is an integer $$$y$$$ ($$$1 \le y \le m$$$). The correct answer to the question is $$$-1$$$, if $$$x<y$$$, $$$0$$$, if $$$x=y$$$, and $$$1$$$, if $$$x>y$$$. But the rocket is broken — it does not always answer correctly. Precisely: let the correct answer to the current question be equal to $$$t$$$, then, if the rocket answers this question correctly, then it will answer $$$t$$$, otherwise it will answer $$$-t$$$.In addition, the rocket has a sequence $$$p$$$ of length $$$n$$$. Each element of the sequence is either $$$0$$$ or $$$1$$$. The rocket processes this sequence in the cyclic order, that is $$$1$$$-st element, $$$2$$$-nd, $$$3$$$-rd, $$$\ldots$$$, $$$(n-1)$$$-th, $$$n$$$-th, $$$1$$$-st, $$$2$$$-nd, $$$3$$$-rd, $$$\ldots$$$, $$$(n-1)$$$-th, $$$n$$$-th, $$$\ldots$$$. If the current element is $$$1$$$, the rocket answers correctly, if $$$0$$$ — lies. Natasha doesn't know the sequence $$$p$$$, but she knows its length — $$$n$$$.You can ask the rocket no more than $$$60$$$ questions.Help Natasha find the distance to Mars. Assume, that the distance to Mars does not change while Natasha is asking questions.Your solution will not be accepted, if it does not receive an answer $$$0$$$ from the rocket (even if the distance to Mars is uniquely determined by the already received rocket's answers).",[],"[0, 0, 0, 0, 0, 0, 0, 0]"675"For a collection of integers $$$S$$$, define $$$\operatorname{mex}(S)$$$ as the smallest non-negative integer that does not appear in $$$S$$$.NIT, the cleaver, decides to destroy the universe. He is not so powerful as Thanos, so he can only destroy the universe by snapping his fingers several times.The universe can be represented as a 1-indexed array $$$a$$$ of length $$$n$$$. When NIT snaps his fingers, he does the following operation on the array:  He selects positive integers $$$l$$$ and $$$r$$$ such that $$$1\le l\le r\le n$$$. Let $$$w=\operatorname{mex}(\{a_l,a_{l+1},\dots,a_r\})$$$. Then, for all $$$l\le i\le r$$$, set $$$a_i$$$ to $$$w$$$. We say the universe is destroyed if and only if for all $$$1\le i\le n$$$, $$$a_i=0$$$ holds.Find the minimum number of times NIT needs to snap his fingers to destroy the universe. That is, find the minimum number of operations NIT needs to perform to make all elements in the array equal to $$$0$$$.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"676"You are given an undirected unweighted graph consisting of $$$n$$$ vertices and $$$m$$$ edges (which represents the map of Bertown) and the array of prices $$$p$$$ of length $$$m$$$. It is guaranteed that there is a path between each pair of vertices (districts).Mike has planned a trip from the vertex (district) $$$a$$$ to the vertex (district) $$$b$$$ and then from the vertex (district) $$$b$$$ to the vertex (district) $$$c$$$. He can visit the same district twice or more. But there is one issue: authorities of the city want to set a price for using the road so if someone goes along the road then he should pay the price corresponding to this road (he pays each time he goes along the road). The list of prices that will be used $$$p$$$ is ready and they just want to distribute it between all roads in the town in such a way that each price from the array corresponds to exactly one road.You are a good friend of Mike (and suddenly a mayor of Bertown) and want to help him to make his trip as cheap as possible. So, your task is to distribute prices between roads in such a way that if Mike chooses the optimal path then the price of the trip is the minimum possible. Note that you cannot rearrange prices after the start of the trip.You have to answer $$$t$$$ independent test cases.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"677"This is the easy version of the problem. The difference between the versions is that the easy version has no swap operations. You can make hacks only if all versions of the problem are solved.Pikachu is a cute and friendly pokémon living in the wild pikachu herd.But it has become known recently that infamous team R wanted to steal all these pokémon! Pokémon trainer Andrew decided to help Pikachu to build a pokémon army to resist.First, Andrew counted all the pokémon — there were exactly $$$n$$$ pikachu. The strength of the $$$i$$$-th pokémon is equal to $$$a_i$$$, and all these numbers are distinct.As an army, Andrew can choose any non-empty subsequence of pokemons. In other words, Andrew chooses some array $$$b$$$ from $$$k$$$ indices such that $$$1 \le b_1 < b_2 < \dots < b_k \le n$$$, and his army will consist of pokémons with forces $$$a_{b_1}, a_{b_2}, \dots, a_{b_k}$$$.The strength of the army is equal to the alternating sum of elements of the subsequence; that is, $$$a_{b_1} - a_{b_2} + a_{b_3} - a_{b_4} + \dots$$$.Andrew is experimenting with pokémon order. He performs $$$q$$$ operations. In $$$i$$$-th operation Andrew swaps $$$l_i$$$-th and $$$r_i$$$-th pokémon.Note: $$$q=0$$$ in this version of the task.Andrew wants to know the maximal stregth of the army he can achieve with the initial pokémon placement. He also needs to know the maximal strength after each operation.Help Andrew and the pokémon, or team R will realize their tricky plan!",[],"[0, 0, 0, 0, 0, 0, 0, 0]"678"One must train much to do well on wizardry contests. So, there are numerous wizardry schools and magic fees.One of such magic schools consists of n tours. A winner of each tour gets a huge prize. The school is organised quite far away, so one will have to take all the prizes home in one go. And the bags that you've brought with you have space for no more than k huge prizes.Besides the fact that you want to take all the prizes home, you also want to perform well. You will consider your performance good if you win at least l tours.In fact, years of organizing contests proved to the organizers that transporting huge prizes is an issue for the participants. Alas, no one has ever invented a spell that would shrink the prizes... So, here's the solution: for some tours the winner gets a bag instead of a huge prize. Each bag is characterized by number ai — the number of huge prizes that will fit into it.You already know the subject of all tours, so you can estimate the probability pi of winning the i-th tour. You cannot skip the tour under any circumstances.Find the probability that you will perform well on the contest and will be able to take all won prizes home (that is, that you will be able to fit all the huge prizes that you won into the bags that you either won or brought from home).","['math', 'probabilities']","[0, 0, 0, 1, 0, 1, 0, 0]"679"You are given a tree $$$G$$$ with $$$n$$$ vertices and an integer $$$k$$$. The vertices of the tree are numbered from $$$1$$$ to $$$n$$$.For a vertex $$$r$$$ and a subset $$$S$$$ of vertices of $$$G$$$, such that $$$|S| = k$$$, we define $$$f(r, S)$$$ as the size of the smallest rooted subtree containing all vertices in $$$S$$$ when the tree is rooted at $$$r$$$. A set of vertices $$$T$$$ is called a rooted subtree, if all the vertices in $$$T$$$ are connected, and for each vertex in $$$T$$$, all its descendants belong to $$$T$$$.You need to calculate the sum of $$$f(r, S)$$$ over all possible distinct combinations of vertices $$$r$$$ and subsets $$$S$$$, where $$$|S| = k$$$. Formally, compute the following: $$$$$$\sum_{r \in V} \sum_{S \subseteq V, |S| = k} f(r, S),$$$$$$ where $$$V$$$ is the set of vertices in $$$G$$$.Output the answer modulo $$$10^9 + 7$$$.","['math', 'trees']","[0, 0, 0, 1, 0, 0, 0, 1]"680"You are given a table $$$a$$$ of size $$$2 \times n$$$ (i.e. two rows and $$$n$$$ columns) consisting of integers from $$$1$$$ to $$$n$$$.In one move, you can choose some column $$$j$$$ ($$$1 \le j \le n$$$) and swap values $$$a_{1, j}$$$ and $$$a_{2, j}$$$ in it. Each column can be chosen no more than once.Your task is to find the minimum number of moves required to obtain permutations of size $$$n$$$ in both first and second rows of the table or determine if it is impossible to do that.You have to answer $$$t$$$ independent test cases.Recall that the permutation of size $$$n$$$ is such an array of size $$$n$$$ that contains each integer from $$$1$$$ to $$$n$$$ exactly once (the order of elements doesn't matter).",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"681"There are some ambiguities when one writes Berland names with the letters of the Latin alphabet.For example, the Berland sound u can be written in the Latin alphabet as ""u"", and can be written as ""oo"". For this reason, two words ""ulyana"" and ""oolyana"" denote the same name.The second ambiguity is about the Berland sound h: one can use both ""h"" and ""kh"" to write it. For example, the words ""mihail"" and ""mikhail"" denote the same name.There are n users registered on the Polycarp's website. Each of them indicated a name represented by the Latin letters. How many distinct names are there among them, if two ambiguities described above are taken into account?Formally, we assume that two words denote the same name, if using the replacements ""u""  ""oo"" and ""h""  ""kh"", you can make the words equal. One can make replacements in both directions, in any of the two words an arbitrary number of times. A letter that resulted from the previous replacement can participate in the next replacements.For example, the following pairs of words denote the same name:  ""koouper"" and ""kuooper"". Making the replacements described above, you can make both words to be equal: ""koouper""  ""kuuper"" and ""kuooper""  ""kuuper"".  ""khun"" and ""kkkhoon"". With the replacements described above you can make both words to be equal: ""khun""  ""khoon"" and ""kkkhoon""  ""kkhoon""  ""khoon"". For a given list of words, find the minimal number of groups where the words in each group denote the same name.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"682"Little girl Margarita is a big fan of competitive programming. She especially loves problems about arrays and queries on them.Recently, she was presented with an array $$$a$$$ of the size of $$$10^9$$$ elements that is filled as follows:   $$$a_1 = -1$$$  $$$a_2 = 2$$$  $$$a_3 = -3$$$  $$$a_4 = 4$$$  $$$a_5 = -5$$$  And so on ... That is, the value of the $$$i$$$-th element of the array $$$a$$$ is calculated using the formula $$$a_i = i \cdot (-1)^i$$$.She immediately came up with $$$q$$$ queries on this array. Each query is described with two numbers: $$$l$$$ and $$$r$$$. The answer to a query is the sum of all the elements of the array at positions from $$$l$$$ to $$$r$$$ inclusive.Margarita really wants to know the answer to each of the requests. She doesn't want to count all this manually, but unfortunately, she couldn't write the program that solves the problem either. She has turned to you — the best programmer.Help her find the answers!",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"683"Once Vasya and Petya assembled a figure of m cubes, each of them is associated with a number between 0 and m - 1 (inclusive, each number appeared exactly once). Let's consider a coordinate system such that the OX is the ground, and the OY is directed upwards. Each cube is associated with the coordinates of its lower left corner, these coordinates are integers for each cube.The figure turned out to be stable. This means that for any cube that is not on the ground, there is at least one cube under it such that those two cubes touch by a side or a corner. More formally, this means that for the cube with coordinates (x, y) either y = 0, or there is a cube with coordinates (x - 1, y - 1), (x, y - 1) or (x + 1, y - 1).Now the boys want to disassemble the figure and put all the cubes in a row. In one step the cube is removed from the figure and being put to the right of the blocks that have already been laid. The guys remove the cubes in such order that the figure remains stable. To make the process more interesting, the guys decided to play the following game. The guys take out the cubes from the figure in turns. It is easy to see that after the figure is disassembled, the integers written on the cubes form a number, written in the m-ary positional numerical system (possibly, with a leading zero). Vasya wants the resulting number to be maximum possible, and Petya, on the contrary, tries to make it as small as possible. Vasya starts the game.Your task is to determine what number is formed after the figure is disassembled, if the boys play optimally. Determine the remainder of the answer modulo 109 + 9.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"684Monocarp wrote down two numbers on a whiteboard. Both numbers follow a specific format: a positive integer $$$x$$$ with $$$p$$$ zeros appended to its end.Now Monocarp asks you to compare these two numbers. Can you help him?,['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"685"This is the hard version of the problem. The only difference is that in this version there are remove queries.Initially you have a set containing one element — $$$0$$$. You need to handle $$$q$$$ queries of the following types:+ $$$x$$$ — add the integer $$$x$$$ to the set. It is guaranteed that this integer is not contained in the set; - $$$x$$$ — remove the integer $$$x$$$ from the set. It is guaranteed that this integer is contained in the set; ? $$$k$$$ — find the $$$k\text{-mex}$$$ of the set. In our problem, we define the $$$k\text{-mex}$$$ of a set of integers as the smallest non-negative integer $$$x$$$ that is divisible by $$$k$$$ and which is not contained in the set.",['number theory'],"[0, 0, 0, 0, 1, 0, 0, 0]"686"You have an array $$$a$$$ of size $$$n$$$ consisting only of zeroes and ones. You can do the following operation:  choose two indices $$$1 \le i , j \le n$$$, $$$i \ne j$$$,  add $$$a_{i}$$$ to $$$a_{j}$$$,  remove $$$a_{i}$$$ from $$$a$$$. Note that elements of $$$a$$$ can become bigger than $$$1$$$ after performing some operations. Also note that $$$n$$$ becomes $$$1$$$ less after the operation.What is the minimum number of operations needed to make $$$a$$$ non-decreasing, i. e. that each element is not less than the previous element?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"687"You are given two matrices $$$A$$$ and $$$B$$$. Each matrix contains exactly $$$n$$$ rows and $$$m$$$ columns. Each element of $$$A$$$ is either $$$0$$$ or $$$1$$$; each element of $$$B$$$ is initially $$$0$$$.You may perform some operations with matrix $$$B$$$. During each operation, you choose any submatrix of $$$B$$$ having size $$$2 \times 2$$$, and replace every element in the chosen submatrix with $$$1$$$. In other words, you choose two integers $$$x$$$ and $$$y$$$ such that $$$1 \le x < n$$$ and $$$1 \le y < m$$$, and then set $$$B_{x, y}$$$, $$$B_{x, y + 1}$$$, $$$B_{x + 1, y}$$$ and $$$B_{x + 1, y + 1}$$$ to $$$1$$$.Your goal is to make matrix $$$B$$$ equal to matrix $$$A$$$. Two matrices $$$A$$$ and $$$B$$$ are equal if and only if every element of matrix $$$A$$$ is equal to the corresponding element of matrix $$$B$$$.Is it possible to make these matrices equal? If it is, you have to come up with a sequence of operations that makes $$$B$$$ equal to $$$A$$$. Note that you don't have to minimize the number of operations.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"688"You are given a sequence $$$a$$$, initially consisting of $$$n$$$ integers.You want to transform this sequence so that all elements in it are equal (i. e. it contains several occurrences of the same element).To achieve this, you choose some integer $$$x$$$ that occurs at least once in $$$a$$$, and then perform the following operation any number of times (possibly zero): choose some segment $$$[l, r]$$$ of the sequence and remove it. But there is one exception: you are not allowed to choose a segment that contains $$$x$$$. More formally, you choose some contiguous subsequence $$$[a_l, a_{l + 1}, \dots, a_r]$$$ such that $$$a_i \ne x$$$ if $$$l \le i \le r$$$, and remove it. After removal, the numbering of elements to the right of the removed segment changes: the element that was the $$$(r+1)$$$-th is now $$$l$$$-th, the element that was $$$(r+2)$$$-th is now $$$(l+1)$$$-th, and so on (i. e. the remaining sequence just collapses).Note that you can not change $$$x$$$ after you chose it.For example, suppose $$$n = 6$$$, $$$a = [1, 3, 2, 4, 1, 2]$$$. Then one of the ways to transform it in two operations is to choose $$$x = 1$$$, then:  choose $$$l = 2$$$, $$$r = 4$$$, so the resulting sequence is $$$a = [1, 1, 2]$$$;  choose $$$l = 3$$$, $$$r = 3$$$, so the resulting sequence is $$$a = [1, 1]$$$. Note that choosing $$$x$$$ is not an operation. Also, note that you can not remove any occurrence of $$$x$$$.Your task is to find the minimum number of operations required to transform the sequence in a way described above.You have to answer $$$t$$$ independent test cases.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"689"This is an easy version of the problem. In this version, all numbers in the given array are distinct and the constraints on $$$n$$$ are less than in the hard version of the problem.You are given an array $$$a$$$ of $$$n$$$ integers (there are no equals elements in the array). You can perform the following operations on array elements:  choose any index $$$i$$$ ($$$1 \le i \le n$$$) and move the element $$$a[i]$$$ to the begin of the array;  choose any index $$$i$$$ ($$$1 \le i \le n$$$) and move the element $$$a[i]$$$ to the end of the array. For example, if $$$n = 5$$$, $$$a = [4, 7, 2, 3, 9]$$$, then the following sequence of operations can be performed:   after performing the operation of the first type to the second element, the array $$$a$$$ will become $$$[7, 4, 2, 3, 9]$$$;  after performing the operation of the second type to the second element, the array $$$a$$$ will become $$$[7, 2, 3, 9, 4]$$$. You can perform operations of any type any number of times in any order.Find the minimum total number of operations of the first and second type that will make the $$$a$$$ array sorted in non-decreasing order. In other words, what is the minimum number of operations that must be performed so the array satisfies the inequalities $$$a[1] \le a[2] \le \ldots \le a[n]$$$.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"690"Let's call a binary string $$$T$$$ of length $$$m$$$ indexed from $$$1$$$ to $$$m$$$ paranoid if we can obtain a string of length $$$1$$$ by performing the following two kinds of operations $$$m-1$$$ times in any order : Select any substring of $$$T$$$ that is equal to 01, and then replace it with 1. Select any substring of $$$T$$$ that is equal to 10, and then replace it with 0.For example, if $$$T = $$$ 001, we can select the substring $$$[T_2T_3]$$$ and perform the first operation. So we obtain $$$T = $$$ 01.You are given a binary string $$$S$$$ of length $$$n$$$ indexed from $$$1$$$ to $$$n$$$. Find the number of pairs of integers $$$(l, r)$$$ $$$1 \le l \le r \le n$$$ such that $$$S[l \ldots r]$$$ (the substring of $$$S$$$ from $$$l$$$ to $$$r$$$) is a paranoid string. ",[],"[0, 0, 0, 0, 0, 0, 0, 0]"691"A penguin Rocher has $$$n$$$ sticks. He has exactly one stick with length $$$i$$$ for all $$$1 \le i \le n$$$.He can connect some sticks. If he connects two sticks that have lengths $$$a$$$ and $$$b$$$, he gets one stick with length $$$a + b$$$. Two sticks, that were used in the operation disappear from his set and the new connected stick appears in his set and can be used for the next connections.He wants to create the maximum number of sticks that have the same length. It is not necessary to make all sticks have the same length, some sticks can have the other length. How many sticks with the equal length he can create?",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"692"This is an interactive problem.Vasya and Petya are going to play the following game: Petya has some positive integer number $$$a$$$. After that Vasya should guess this number using the following questions. He can say a pair of non-negative integer numbers $$$(x, y)$$$. Petya will answer him:   ""x"", if $$$(x \bmod a) \geq (y \bmod a)$$$.  ""y"", if $$$(x \bmod a) < (y \bmod a)$$$. We define $$$(x \bmod a)$$$ as a remainder of division $$$x$$$ by $$$a$$$.Vasya should guess the number $$$a$$$ using no more, than 60 questions.It's guaranteed that Petya has a number, that satisfies the inequality $$$1 \leq a \leq 10^9$$$.Help Vasya playing this game and write a program, that will guess the number $$$a$$$.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"693"This is an interactive problem.After getting AC after 13 Time Limit Exceeded verdicts on a geometry problem, Kuroni went to an Italian restaurant to celebrate this holy achievement. Unfortunately, the excess sauce disoriented him, and he's now lost!The United States of America can be modeled as a tree (why though) with $$$n$$$ vertices. The tree is rooted at vertex $$$r$$$, wherein lies Kuroni's hotel.Kuroni has a phone app designed to help him in such emergency cases. To use the app, he has to input two vertices $$$u$$$ and $$$v$$$, and it'll return a vertex $$$w$$$, which is the lowest common ancestor of those two vertices.However, since the phone's battery has been almost drained out from live-streaming Kuroni's celebration party, he could only use the app at most $$$\lfloor \frac{n}{2} \rfloor$$$ times. After that, the phone would die and there will be nothing left to help our dear friend! :(As the night is cold and dark, Kuroni needs to get back, so that he can reunite with his comfy bed and pillow(s). Can you help him figure out his hotel's location?",['trees'],"[0, 0, 0, 0, 0, 0, 0, 1]"694"Cowboy Beblop is a funny little boy who likes sitting at his computer. He somehow obtained two elastic hoops in the shape of 2D polygons, which are not necessarily convex. Since there's no gravity on his spaceship, the hoops are standing still in the air. Since the hoops are very elastic, Cowboy Beblop can stretch, rotate, translate or shorten their edges as much as he wants.For both hoops, you are given the number of their vertices, as well as the position of each vertex, defined by the X , Y and Z coordinates. The vertices are given in the order they're connected: the 1st vertex is connected to the 2nd, which is connected to the 3rd, etc., and the last vertex is connected to the first one. Two hoops are connected if it's impossible to pull them to infinity in different directions by manipulating their edges, without having their edges or vertices intersect at any point – just like when two links of a chain are connected. The polygons' edges do not intersect or overlap. To make things easier, we say that two polygons are well-connected, if the edges of one polygon cross the area of the other polygon in two different directions (from the upper and lower sides of the plane defined by that polygon) a different number of times.Cowboy Beblop is fascinated with the hoops he has obtained and he would like to know whether they are well-connected or not. Since he’s busy playing with his dog, Zwei, he’d like you to figure it out for him. He promised you some sweets if you help him! ",['geometry'],"[0, 1, 0, 0, 0, 0, 0, 0]"695"You are given an array $$$a$$$ consisting of $$$n$$$ integers.Your task is to say the number of such positive integers $$$x$$$ such that $$$x$$$ divides each number from the array. In other words, you have to find the number of common divisors of all elements in the array.For example, if the array $$$a$$$ will be $$$[2, 4, 6, 2, 10]$$$, then $$$1$$$ and $$$2$$$ divide each number from the array (so the answer for this test is $$$2$$$).",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"696"The classic programming language of Bitland is Bit++. This language is so peculiar and complicated.The language is that peculiar as it has exactly one variable, called x. Also, there are two operations:  Operation ++ increases the value of variable x by 1.  Operation -- decreases the value of variable x by 1. A statement in language Bit++ is a sequence, consisting of exactly one operation and one variable x. The statement is written without spaces, that is, it can only contain characters ""+"", ""-"", ""X"". Executing a statement means applying the operation it contains.A programme in Bit++ is a sequence of statements, each of them needs to be executed. Executing a programme means executing all the statements it contains.You're given a programme in language Bit++. The initial value of x is 0. Execute the programme and find its final value (the value of the variable when this programme is executed).",[],"[0, 0, 0, 0, 0, 0, 0, 0]"697"Vasya has started watching football games. He has learned that for some fouls the players receive yellow cards, and for some fouls they receive red cards. A player who receives the second yellow card automatically receives a red card.Vasya is watching a recorded football match now and makes notes of all the fouls that he would give a card for. Help Vasya determine all the moments in time when players would be given red cards if Vasya were the judge. For each player, Vasya wants to know only the first moment of time when he would receive a red card from Vasya.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"698"When you play the game of thrones, you win, or you die. There is no middle ground.Cersei Lannister, A Game of Thrones by George R. R. MartinThere are $$$n$$$ nobles, numbered from $$$1$$$ to $$$n$$$. Noble $$$i$$$ has a power of $$$i$$$. There are also $$$m$$$ ""friendships"". A friendship between nobles $$$a$$$ and $$$b$$$ is always mutual.A noble is defined to be vulnerable if both of the following conditions are satisfied:   the noble has at least one friend, and  all of that noble's friends have a higher power. You will have to process the following three types of queries.   Add a friendship between nobles $$$u$$$ and $$$v$$$.  Remove a friendship between nobles $$$u$$$ and $$$v$$$.  Calculate the answer to the following process. The process: all vulnerable nobles are simultaneously killed, and all their friendships end. Then, it is possible that new nobles become vulnerable. The process repeats itself until no nobles are vulnerable. It can be proven that the process will end in finite time. After the process is complete, you need to calculate the number of remaining nobles.Note that the results of the process are not carried over between queries, that is, every process starts with all nobles being alive!",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"699"Everybody knows that the capital of Berland is connected to Bercouver (the Olympic capital) by a direct road. To improve the road's traffic capacity, there was placed just one traffic sign, limiting the maximum speed. Traffic signs in Berland are a bit peculiar, because they limit the speed only at that point on the road where they are placed. Right after passing the sign it is allowed to drive at any speed.It is known that the car of an average Berland citizen has the acceleration (deceleration) speed of a km/h2, and has maximum speed of v km/h. The road has the length of l km, and the speed sign, limiting the speed to w km/h, is placed d km (1 ≤ d < l) away from the capital of Berland. The car has a zero speed at the beginning of the journey. Find the minimum time that an average Berland citizen will need to get from the capital to Bercouver, if he drives at the optimal speed.The car can enter Bercouver at any speed.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"700"While searching for the pizza, baby Hosssam came across two permutations $$$a$$$ and $$$b$$$ of length $$$n$$$.Recall that a permutation is an array consisting of $$$n$$$ distinct integers from $$$1$$$ to $$$n$$$ in arbitrary order. For example, $$$[2,3,1,5,4]$$$ is a permutation, but $$$[1,2,2]$$$ is not a permutation ($$$2$$$ appears twice in the array) and $$$[1,3,4]$$$ is also not a permutation ($$$n=3$$$ but there is $$$4$$$ in the array).Baby Hosssam forgot about the pizza and started playing around with the two permutations. While he was playing with them, some elements of the first permutation got mixed up with some elements of the second permutation, and to his surprise those elements also formed a permutation of size $$$n$$$.Specifically, he mixed up the permutations to form a new array $$$c$$$ in the following way.  For each $$$i$$$ ($$$1\le i\le n$$$), he either made $$$c_i=a_i$$$ or $$$c_i=b_i$$$.  The array $$$c$$$ is a permutation. You know permutations $$$a$$$, $$$b$$$, and values at some positions in $$$c$$$. Please count the number different permutations $$$c$$$ that are consistent with the described process and the given values. Since the answer can be large, print it modulo $$$10^9+7$$$.It is guaranteed that there exists at least one permutation $$$c$$$ that satisfies all the requirements.","['graphs', 'math']","[0, 0, 1, 1, 0, 0, 0, 0]"701"Ashish has a tree consisting of $$$n$$$ nodes numbered $$$1$$$ to $$$n$$$ rooted at node $$$1$$$. The $$$i$$$-th node in the tree has a cost $$$a_i$$$, and binary digit $$$b_i$$$ is written in it. He wants to have binary digit $$$c_i$$$ written in the $$$i$$$-th node in the end.To achieve this, he can perform the following operation any number of times:   Select any $$$k$$$ nodes from the subtree of any node $$$u$$$, and shuffle the digits in these nodes as he wishes, incurring a cost of $$$k \cdot a_u$$$. Here, he can choose $$$k$$$ ranging from $$$1$$$ to the size of the subtree of $$$u$$$. He wants to perform the operations in such a way that every node finally has the digit corresponding to its target.Help him find the minimum total cost he needs to spend so that after all the operations, every node $$$u$$$ has digit $$$c_u$$$ written in it, or determine that it is impossible.",['trees'],"[0, 0, 0, 0, 0, 0, 0, 1]"702"Mad scientist Mike has just finished constructing a new device to search for extraterrestrial intelligence! He was in such a hurry to launch it for the first time that he plugged in the power wires without giving it a proper glance and started experimenting right away. After a while Mike observed that the wires ended up entangled and now have to be untangled again.The device is powered by two wires ""plus"" and ""minus"". The wires run along the floor from the wall (on the left) to the device (on the right). Both the wall and the device have two contacts in them on the same level, into which the wires are plugged in some order. The wires are considered entangled if there are one or more places where one wire runs above the other one. For example, the picture below has four such places (top view):  Mike knows the sequence in which the wires run above each other. Mike also noticed that on the left side, the ""plus"" wire is always plugged into the top contact (as seen on the picture). He would like to untangle the wires without unplugging them and without moving the device. Determine if it is possible to do that. A wire can be freely moved and stretched on the floor, but cannot be cut.To understand the problem better please read the notes to the test samples.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"703"Everyone knows that 2010 FIFA World Cup is being held in South Africa now. By the decision of BFA (Berland's Football Association) next World Cup will be held in Berland. BFA took the decision to change some World Cup regulations:  the final tournament features n teams (n is always even)  the first n / 2 teams (according to the standings) come through to the knockout stage  the standings are made on the following principle: for a victory a team gets 3 points, for a draw — 1 point, for a defeat — 0 points. In the first place, teams are ordered in the standings in decreasing order of their points; in the second place — in decreasing order of the difference between scored and missed goals; in the third place — in the decreasing order of scored goals  it's written in Berland's Constitution that the previous regulation helps to order the teams without ambiguity. You are asked to write a program that, by the given list of the competing teams and the results of all the matches, will find the list of teams that managed to get through to the knockout stage.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"704"You are given an array consisting of $$$n$$$ integers $$$a_1, a_2, \dots , a_n$$$ and an integer $$$x$$$. It is guaranteed that for every $$$i$$$, $$$1 \le a_i \le x$$$.Let's denote a function $$$f(l, r)$$$ which erases all values such that $$$l \le a_i \le r$$$ from the array $$$a$$$ and returns the resulting array. For example, if $$$a = [4, 1, 1, 4, 5, 2, 4, 3]$$$, then $$$f(2, 4) = [1, 1, 5]$$$.Your task is to calculate the number of pairs $$$(l, r)$$$ such that $$$1 \le l \le r \le x$$$ and $$$f(l, r)$$$ is sorted in non-descending order. Note that the empty array is also considered sorted.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"705"You will receive 5 points for solving this problem.Manao has invented a new operation on strings that is called folding. Each fold happens between a pair of consecutive letters and places the second part of the string above first part, running in the opposite direction and aligned to the position of the fold. Using this operation, Manao converts the string into a structure that has one more level than there were fold operations performed. See the following examples for clarity.We will denote the positions of folds with '|' characters. For example, the word ""ABRACADABRA"" written as ""AB|RACA|DAB|RA"" indicates that it has been folded three times: first, between the leftmost pair of 'B' and 'R' letters; second, between 'A' and 'D'; and third, between the rightmost pair of 'B' and 'R' letters. Here are several examples of folded strings:""ABCDEF|GHIJK"" |  ""A|BCDEFGHIJK"" |  ""AB|RACA|DAB|RA"" |  ""X|XXXXX|X|X|XXXXXX""               |                 |                   |       XXXXXX    KJIHG      |   KJIHGFEDCB    |      AR           |       X   ABCDEF      |            A    |     DAB           |       X               |                 |     ACAR          |       XXXXX               |                 |       AB          |           XOne last example for ""ABCD|EFGH|IJ|K"":  KIJHGFEABCDManao noticed that each folded string can be viewed as several piles of letters. For instance, in the previous example, there are four piles, which can be read as ""AHI"", ""BGJK"", ""CF"", and ""DE"" from bottom to top. Manao wonders what is the highest pile of identical letters he can build using fold operations on a given word. Note that the pile should not contain gaps and should start at the bottom level. For example, in the rightmost of the four examples above, none of the piles would be considered valid since each of them has gaps, starts above the bottom level, or both.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"706"You are living on an infinite plane with the Cartesian coordinate system on it. In one move you can go to any of the four adjacent points (left, right, up, down).More formally, if you are standing at the point $$$(x, y)$$$, you can:   go left, and move to $$$(x - 1, y)$$$, or  go right, and move to $$$(x + 1, y)$$$, or  go up, and move to $$$(x, y + 1)$$$, or  go down, and move to $$$(x, y - 1)$$$. There are $$$n$$$ boxes on this plane. The $$$i$$$-th box has coordinates $$$(x_i,y_i)$$$. It is guaranteed that the boxes are either on the $$$x$$$-axis or the $$$y$$$-axis. That is, either $$$x_i=0$$$ or $$$y_i=0$$$.You can collect a box if you and the box are at the same point. Find the minimum number of moves you have to perform to collect all of these boxes if you have to start and finish at the point $$$(0,0)$$$.",['geometry'],"[0, 1, 0, 0, 0, 0, 0, 0]"707"Vasily has a deck of cards consisting of n cards. There is an integer on each of the cards, this integer is between 1 and 100 000, inclusive. It is possible that some cards have the same integers on them.Vasily decided to sort the cards. To do this, he repeatedly takes the top card from the deck, and if the number on it equals the minimum number written on the cards in the deck, then he places the card away. Otherwise, he puts it under the deck and takes the next card from the top, and so on. The process ends as soon as there are no cards in the deck. You can assume that Vasily always knows the minimum number written on some card in the remaining deck, but doesn't know where this card (or these cards) is.You are to determine the total number of times Vasily takes the top card from the deck.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"708"A number is ternary if it contains only digits $$$0$$$, $$$1$$$ and $$$2$$$. For example, the following numbers are ternary: $$$1022$$$, $$$11$$$, $$$21$$$, $$$2002$$$.You are given a long ternary number $$$x$$$. The first (leftmost) digit of $$$x$$$ is guaranteed to be $$$2$$$, the other digits of $$$x$$$ can be $$$0$$$, $$$1$$$ or $$$2$$$.Let's define the ternary XOR operation $$$\odot$$$ of two ternary numbers $$$a$$$ and $$$b$$$ (both of length $$$n$$$) as a number $$$c = a \odot b$$$ of length $$$n$$$, where $$$c_i = (a_i + b_i) \% 3$$$ (where $$$\%$$$ is modulo operation). In other words, add the corresponding digits and take the remainders of the sums when divided by $$$3$$$. For example, $$$10222 \odot 11021 = 21210$$$.Your task is to find such ternary numbers $$$a$$$ and $$$b$$$ both of length $$$n$$$ and both without leading zeros that $$$a \odot b = x$$$ and $$$max(a, b)$$$ is the minimum possible.You have to answer $$$t$$$ independent test cases.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"709"Vasya's bicycle chain drive consists of two parts: n stars are attached to the pedal axle, m stars are attached to the rear wheel axle. The chain helps to rotate the rear wheel by transmitting the pedal rotation.We know that the i-th star on the pedal axle has ai (0 < a1 < a2 < ... < an) teeth, and the j-th star on the rear wheel axle has bj (0 < b1 < b2 < ... < bm) teeth. Any pair (i, j) (1 ≤ i ≤ n; 1 ≤ j ≤ m) is called a gear and sets the indexes of stars to which the chain is currently attached. Gear (i, j) has a gear ratio, equal to the value .Since Vasya likes integers, he wants to find such gears (i, j), that their ratios are integers. On the other hand, Vasya likes fast driving, so among all ""integer"" gears (i, j) he wants to choose a gear with the maximum ratio. Help him to find the number of such gears.In the problem, fraction  denotes division in real numbers, that is, no rounding is performed.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"710"A sequence a0, a1, ... is called a recurrent binary sequence, if each term ai (i = 0, 1, ...) is equal to 0 or 1 and there exist coefficients  such that an = c1·an - 1 + c2·an - 2 + ... + ck·an - k (mod 2),  for all n ≥ k. Assume that not all of ci are zeros.Note that such a sequence can be uniquely recovered from any k-tuple {as, as + 1, ..., as + k - 1} and so it is periodic. Moreover, if a k-tuple contains only zeros, then the sequence contains only zeros, so this case is not very interesting. Otherwise the minimal period of the sequence is not greater than 2k - 1, as k-tuple determines next element, and there are 2k - 1 non-zero k-tuples. Let us call a sequence long if its minimal period is exactly 2k - 1. Your task is to find a long sequence for a given k, if there is any.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"711"Igor has been into chess for a long time and now he is sick of the game by the ordinary rules. He is going to think of new rules of the game and become world famous.Igor's chessboard is a square of size n × n cells. Igor decided that simple rules guarantee success, that's why his game will have only one type of pieces. Besides, all pieces in his game are of the same color. The possible moves of a piece are described by a set of shift vectors. The next passage contains a formal description of available moves.Let the rows of the board be numbered from top to bottom and the columns be numbered from left to right from 1 to n. Let's assign to each square a pair of integers (x, y) — the number of the corresponding column and row. Each of the possible moves of the piece is defined by a pair of integers (dx, dy); using this move, the piece moves from the field (x, y) to the field (x + dx, y + dy). You can perform the move if the cell (x + dx, y + dy) is within the boundaries of the board and doesn't contain another piece. Pieces that stand on the cells other than (x, y) and (x + dx, y + dy) are not important when considering the possibility of making the given move (for example, like when a knight moves in usual chess).Igor offers you to find out what moves his chess piece can make. He placed several pieces on the board and for each unoccupied square he told you whether it is attacked by any present piece (i.e. whether some of the pieces on the field can move to that cell). Restore a possible set of shift vectors of the piece, or else determine that Igor has made a mistake and such situation is impossible for any set of shift vectors.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"712"The popular improv website Interpretation Impetus hosts regular improv contests and maintains a rating of the best performers. However, since improv can often go horribly wrong, the website is notorious for declaring improv contests unrated. It now holds a wager before each improv contest where the participants try to predict whether it will be rated or unrated, and they are now more popular than the improv itself.Izzy and $$$n$$$ other participants take part in each wager. First, they each make their prediction, expressed as 1 (""rated"") or 0 (""unrated""). Izzy always goes last, so she knows the predictions of the other participants when making her own. Then, the actual competition takes place and it is declared either rated or unrated.You need to write a program that will interactively play as Izzy. There will be $$$m$$$ wagers held in 2021, and Izzy's goal is to have at most $$$1.3\cdot b + 100$$$ wrong predictions after all those wagers, where $$$b$$$ is the smallest number of wrong predictions that any other wager participant will have after all those wagers. The number $$$b$$$ is not known in advance. Izzy also knows nothing about the other participants — they might somehow always guess correctly, or their predictions might be correlated. Izzy's predictions, though, do not affect the predictions of the other participants and the decision on the contest being rated or not — in other words, in each test case, your program always receives the same inputs, no matter what it outputs.","['math', 'probabilities']","[0, 0, 0, 1, 0, 1, 0, 0]"713"A Russian space traveller Alisa Selezneva, like any other schoolgirl of the late 21 century, is interested in science. She has recently visited the MIT (Moscow Institute of Time), where its chairman and the co-inventor of the time machine academician Petrov told her about the construction of a time machine.During the demonstration of the time machine performance Alisa noticed that the machine does not have high speed and the girl got interested in the reason for such disadvantage. As it turns out on closer examination, one of the problems that should be solved for the time machine isn't solved by an optimal algorithm. If you find a way to solve this problem optimally, the time machine will run faster and use less energy.A task that none of the staff can solve optimally is as follows. There exists a matrix a, which is filled by the following rule:The cells are consecutive positive integers, starting with one. Besides, ai, j < at, k (i, j, t, k ≥ 1), if:  max(i, j) < max(t, k);  max(i, j) = max(t, k) and j < k;  max(i, j) = max(t, k), j = k and i > t. So, after the first 36 numbers are inserted, matrix a will look as follows:  To solve the problem, you should learn to find rather quickly for the given values of x1, y1, x2 and y2 (x1 ≤ x2, y1 ≤ y2) the meaning of expression:As the meaning of this expression can be large enough, it is sufficient to know only the last 10 digits of the sought value.So, no one in MTI can solve the given task. Alice was brave enough to use the time machine and travel the past to help you.Your task is to write a program that uses the given values x1, y1, x2 and y2 finds the last 10 digits of the given expression.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"714"Something happened in Uzhlyandia again... There are riots on the streets... Famous Uzhlyandian superheroes Shean the Sheep and Stas the Giraffe were called in order to save the situation. Upon the arriving, they found that citizens are worried about maximum values of the Main Uzhlyandian Function f, which is defined as follows:In the above formula, 1 ≤ l < r ≤ n must hold, where n is the size of the Main Uzhlyandian Array a, and |x| means absolute value of x. But the heroes skipped their math lessons in school, so they asked you for help. Help them calculate the maximum value of f among all possible values of l and r for the given array a.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"715"Two neighbours, Alan and Bob, live in the city, where there are three buildings only: a cinema, a shop and the house, where they live. The rest is a big asphalt square. Once they went to the cinema, and the film impressed them so deeply, that when they left the cinema, they did not want to stop discussing it.Bob wants to get home, but Alan has to go to the shop first, and only then go home. So, they agreed to cover some distance together discussing the film (their common path might pass through the shop, or they might walk circles around the cinema together), and then to part each other's company and go each his own way. After they part, they will start thinking about their daily pursuits; and even if they meet again, they won't be able to go on with the discussion. Thus, Bob's path will be a continuous curve, having the cinema and the house as its ends. Alan's path — a continuous curve, going through the shop, and having the cinema and the house as its ends.The film ended late, that's why the whole distance covered by Alan should not differ from the shortest one by more than t1, and the distance covered by Bob should not differ from the shortest one by more than t2.Find the maximum distance that Alan and Bob will cover together, discussing the film.",['geometry'],"[0, 1, 0, 0, 0, 0, 0, 0]"716"Yes, that's another problem with definition of ""beautiful"" numbers.Let's call a positive integer x beautiful if its decimal representation without leading zeroes contains even number of digits, and there exists a permutation of this representation which is palindromic. For example, 4242 is a beautiful number, since it contains 4 digits, and there exists a palindromic permutation 2442.Given a positive integer s, find the largest beautiful number which is less than s.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"717"The School №0 of the capital of Berland has n children studying in it. All the children in this school are gifted: some of them are good at programming, some are good at maths, others are good at PE (Physical Education). Hence, for each child we know value ti:  ti = 1, if the i-th child is good at programming,  ti = 2, if the i-th child is good at maths,  ti = 3, if the i-th child is good at PE Each child happens to be good at exactly one of these three subjects.The Team Scientific Decathlon Olympias requires teams of three students. The school teachers decided that the teams will be composed of three children that are good at different subjects. That is, each team must have one mathematician, one programmer and one sportsman. Of course, each child can be a member of no more than one team.What is the maximum number of teams that the school will be able to present at the Olympiad? How should the teams be formed for that?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"718"For years, the Day of city N was held in the most rainy day of summer. New mayor decided to break this tradition and select a not-so-rainy day for the celebration. The mayor knows the weather forecast for the $$$n$$$ days of summer. On the $$$i$$$-th day, $$$a_i$$$ millimeters of rain will fall. All values $$$a_i$$$ are distinct.The mayor knows that citizens will watch the weather $$$x$$$ days before the celebration and $$$y$$$ days after. Because of that, he says that a day $$$d$$$ is not-so-rainy if $$$a_d$$$ is smaller than rain amounts at each of $$$x$$$ days before day $$$d$$$ and and each of $$$y$$$ days after day $$$d$$$. In other words, $$$a_d < a_j$$$ should hold for all $$$d - x \le j < d$$$ and $$$d < j \le d + y$$$. Citizens only watch the weather during summer, so we only consider such $$$j$$$ that $$$1 \le j \le n$$$.Help mayor find the earliest not-so-rainy day of summer.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"719"Recently Vasya got interested in finding extra-terrestrial intelligence. He made a simple extra-terrestrial signals’ receiver and was keeping a record of the signals for n days in a row. Each of those n days Vasya wrote a 1 in his notebook if he had received a signal that day and a 0 if he hadn’t. Vasya thinks that he has found extra-terrestrial intelligence if there is a system in the way the signals has been received, i.e. if all the intervals between successive signals are equal. Otherwise, Vasya thinks that the signals were sent by some stupid aliens no one cares about. Help Vasya to deduce from the information given by the receiver if he has found extra-terrestrial intelligence or not.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"720"Vasya has got two number: a and b. However, Vasya finds number a too short. So he decided to repeat the operation of lengthening number a n times.One operation of lengthening a number means adding exactly one digit to the number (in the decimal notation) to the right provided that the resulting number is divisible by Vasya's number b. If it is impossible to obtain the number which is divisible by b, then the lengthening operation cannot be performed.Your task is to help Vasya and print the number he can get after applying the lengthening operation to number a n times.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"721Pashmak decided to give Parmida a pair of flowers from the garden. There are n flowers in the garden and the i-th of them has a beauty number bi. Parmida is a very strange girl so she doesn't want to have the two most beautiful flowers necessarily. She wants to have those pairs of flowers that their beauty difference is maximal possible!Your task is to write a program which calculates two things:  The maximum beauty difference of flowers that Pashmak can give to Parmida.  The number of ways that Pashmak can pick the flowers. Two ways are considered different if and only if there is at least one flower that is chosen in the first way and not chosen in the second way. ,[],"[0, 0, 0, 0, 0, 0, 0, 0]"722"Acacius is studying strings theory. Today he came with the following problem.You are given a string $$$s$$$ of length $$$n$$$ consisting of lowercase English letters and question marks. It is possible to replace question marks with lowercase English letters in such a way that a string ""abacaba"" occurs as a substring in a resulting string exactly once?Each question mark should be replaced with exactly one lowercase English letter. For example, string ""a?b?c"" can be transformed into strings ""aabbc"" and ""azbzc"", but can't be transformed into strings ""aabc"", ""a?bbc"" and ""babbc"".Occurrence of a string $$$t$$$ of length $$$m$$$ in the string $$$s$$$ of length $$$n$$$ as a substring is a index $$$i$$$ ($$$1 \leq i \leq n - m + 1$$$) such that string $$$s[i..i+m-1]$$$ consisting of $$$m$$$ consecutive symbols of $$$s$$$ starting from $$$i$$$-th equals to string $$$t$$$. For example string ""ababa"" has two occurrences of a string ""aba"" as a substring with $$$i = 1$$$ and $$$i = 3$$$, but there are no occurrences of a string ""aba"" in the string ""acba"" as a substring.Please help Acacius to check if it is possible to replace all question marks with lowercase English letters in such a way that a string ""abacaba"" occurs as a substring in a resulting string exactly once.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"723"The string $$$t_1t_2 \dots t_k$$$ is good if each letter of this string belongs to at least one palindrome of length greater than 1.A palindrome is a string that reads the same backward as forward. For example, the strings A, BAB, ABBA, BAABBBAAB are palindromes, but the strings AB, ABBBAA, BBBA are not.Here are some examples of good strings:   $$$t$$$ = AABBB (letters $$$t_1$$$, $$$t_2$$$ belong to palindrome $$$t_1 \dots t_2$$$ and letters $$$t_3$$$, $$$t_4$$$, $$$t_5$$$ belong to palindrome $$$t_3 \dots t_5$$$);  $$$t$$$ = ABAA (letters $$$t_1$$$, $$$t_2$$$, $$$t_3$$$ belong to palindrome $$$t_1 \dots t_3$$$ and letter $$$t_4$$$ belongs to palindrome $$$t_3 \dots t_4$$$);  $$$t$$$ = AAAAA (all letters belong to palindrome $$$t_1 \dots t_5$$$); You are given a string $$$s$$$ of length $$$n$$$, consisting of only letters A and B.You have to calculate the number of good substrings of string $$$s$$$.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"724"Recently n students from city S moved to city P to attend a programming camp.They moved there by train. In the evening, all students in the train decided that they want to drink some tea. Of course, no two people can use the same teapot simultaneously, so the students had to form a queue to get their tea.i-th student comes to the end of the queue at the beginning of li-th second. If there are multiple students coming to the queue in the same moment, then the student with greater index comes after the student with lesser index. Students in the queue behave as follows: if there is nobody in the queue before the student, then he uses the teapot for exactly one second and leaves the queue with his tea; otherwise the student waits for the people before him to get their tea. If at the beginning of ri-th second student i still cannot get his tea (there is someone before him in the queue), then he leaves the queue without getting any tea. For each student determine the second he will use the teapot and get his tea (if he actually gets it).",[],"[0, 0, 0, 0, 0, 0, 0, 0]"725"You are given an integer $$$n$$$.Let's define $$$s(n)$$$ as the string ""BAN"" concatenated $$$n$$$ times. For example, $$$s(1)$$$ = ""BAN"", $$$s(3)$$$ = ""BANBANBAN"". Note that the length of the string $$$s(n)$$$ is equal to $$$3n$$$.Consider $$$s(n)$$$. You can perform the following operation on $$$s(n)$$$ any number of times (possibly zero): Select any two distinct indices $$$i$$$ and $$$j$$$ $$$(1 \leq i, j \leq 3n, i \ne j)$$$. Then, swap $$$s(n)_i$$$ and $$$s(n)_j$$$. You want the string ""BAN"" to not appear in $$$s(n)$$$ as a subsequence. What's the smallest number of operations you have to do to achieve this? Also, find one such shortest sequence of operations.A string $$$a$$$ is a subsequence of a string $$$b$$$ if $$$a$$$ can be obtained from $$$b$$$ by deletion of several (possibly, zero or all) characters.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"726"Recently, Valery have come across an entirely new programming language. Most of all the language attracted him with template functions and procedures. Let us remind you that templates are tools of a language, designed to encode generic algorithms, without reference to some parameters (e.g., data types, buffer sizes, default values).Valery decided to examine template procedures in this language in more detail. The description of a template procedure consists of the procedure name and the list of its parameter types. The generic type T parameters can be used as parameters of template procedures.A procedure call consists of a procedure name and a list of variable parameters. Let's call a procedure suitable for this call if the following conditions are fulfilled:   its name equals to the name of the called procedure;  the number of its parameters equals to the number of parameters of the procedure call;  the types of variables in the procedure call match the corresponding types of its parameters. The variable type matches the type of a parameter if the parameter has a generic type T or the type of the variable and the parameter are the same. You are given a description of some set of template procedures. You are also given a list of variables used in the program, as well as direct procedure calls that use the described variables. For each call you need to count the number of procedures that are suitable for this call.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"727"For some reason in many American cartoons anvils fall from time to time onto heroes' heads. Of course, safes, wardrobes, cruisers, planes fall sometimes too... But anvils do so most of all.Anvils come in different sizes and shapes. Quite often they get the hero stuck deep in the ground. But have you ever thought who throws anvils from the sky? From what height? We are sure that such questions have never troubled you!It turns out that throwing an anvil properly is not an easy task at all. Let's describe one of the most popular anvil throwing models.Let the height p of the potential victim vary in the range [0;a] and the direction of the wind q vary in the range [ - b;b]. p and q could be any real (floating) numbers. Then we can assume that the anvil will fit the toon's head perfectly only if the following equation has at least one real root:   Determine the probability with which an aim can be successfully hit by an anvil.You can assume that the p and q coefficients are chosen equiprobably and independently in their ranges.","['math', 'probabilities']","[0, 0, 0, 1, 0, 1, 0, 0]"728You are given an array of $$$n$$$ integers. You need to split all integers into two groups so that the GCD of all integers in the first group is equal to one and the GCD of all integers in the second group is equal to one.The GCD of a group of integers is the largest non-negative integer that divides all the integers in the group.Both groups have to be non-empty.,"['number theory', 'probabilities']","[0, 0, 0, 0, 1, 1, 0, 0]"729You are given an undirected unweighted connected graph consisting of $$$n$$$ vertices and $$$m$$$ edges. It is guaranteed that there are no self-loops or multiple edges in the given graph.Your task is to find any spanning tree of this graph such that the degree of the first vertex (vertex with label $$$1$$$ on it) is equal to $$$D$$$ (or say that there are no such spanning trees). Recall that the degree of a vertex is the number of edges incident to it.,['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"730"INSPION FullBand Master - INSPION INSPION - IOLITE-SUNSTONEOn another floor of the A.R.C. Markland-N, the young man Simon ""Xenon"" Jackson, takes a break after finishing his project early (as always). Having a lot of free time, he decides to put on his legendary hacker ""X"" instinct and fight against the gangs of the cyber world.His target is a network of $$$n$$$ small gangs. This network contains exactly $$$n - 1$$$ direct links, each of them connecting two gangs together. The links are placed in such a way that every pair of gangs is connected through a sequence of direct links.By mining data, Xenon figured out that the gangs used a form of cross-encryption to avoid being busted: every link was assigned an integer from $$$0$$$ to $$$n - 2$$$ such that all assigned integers are distinct and every integer was assigned to some link. If an intruder tries to access the encrypted data, they will have to surpass $$$S$$$ password layers, with $$$S$$$ being defined by the following formula:$$$$$$S = \sum_{1 \leq u < v \leq n} mex(u, v)$$$$$$Here, $$$mex(u, v)$$$ denotes the smallest non-negative integer that does not appear on any link on the unique simple path from gang $$$u$$$ to gang $$$v$$$.Xenon doesn't know the way the integers are assigned, but it's not a problem. He decides to let his AI's instances try all the passwords on his behalf, but before that, he needs to know the maximum possible value of $$$S$$$, so that the AIs can be deployed efficiently.Now, Xenon is out to write the AI scripts, and he is expected to finish them in two hours. Can you find the maximum possible $$$S$$$ before he returns?",['trees'],"[0, 0, 0, 0, 0, 0, 0, 1]"731"Nastya came to her informatics lesson, and her teacher who is, by the way, a little bit famous here gave her the following task.Two matrices $$$A$$$ and $$$B$$$ are given, each of them has size $$$n \times m$$$. Nastya can perform the following operation to matrix $$$A$$$ unlimited number of times:   take any square square submatrix of $$$A$$$ and transpose it (i.e. the element of the submatrix which was in the $$$i$$$-th row and $$$j$$$-th column of the submatrix will be in the $$$j$$$-th row and $$$i$$$-th column after transposing, and the transposed submatrix itself will keep its place in the matrix $$$A$$$). Nastya's task is to check whether it is possible to transform the matrix $$$A$$$ to the matrix $$$B$$$.    Example of the operation As it may require a lot of operations, you are asked to answer this question for Nastya.A square submatrix of matrix $$$M$$$ is a matrix which consist of all elements which comes from one of the rows with indeces $$$x, x+1, \dots, x+k-1$$$ of matrix $$$M$$$ and comes from one of the columns with indeces $$$y, y+1, \dots, y+k-1$$$ of matrix $$$M$$$. $$$k$$$ is the size of square submatrix. In other words, square submatrix is the set of elements of source matrix which form a solid square (i.e. without holes).",[],"[0, 0, 0, 0, 0, 0, 0, 0]"732"Some time ago Leonid have known about idempotent functions. Idempotent function defined on a set {1, 2, ..., n} is such function , that for any  the formula g(g(x)) = g(x) holds.Let's denote as f(k)(x) the function f applied k times to the value x. More formally, f(1)(x) = f(x), f(k)(x) = f(f(k - 1)(x)) for each k > 1.You are given some function . Your task is to find minimum positive integer k such that function f(k)(x) is idempotent.","['graphs', 'math']","[0, 0, 1, 1, 0, 0, 0, 0]"733"You have a string $$$s_1 s_2 \ldots s_n$$$ and you stand on the left of the string looking right. You want to choose an index $$$k$$$ ($$$1 \le k \le n$$$) and place a mirror after the $$$k$$$-th letter, so that what you see is $$$s_1 s_2 \ldots s_k s_k s_{k - 1} \ldots s_1$$$. What is the lexicographically smallest string you can see?A string $$$a$$$ is lexicographically smaller than a string $$$b$$$ if and only if one of the following holds:   $$$a$$$ is a prefix of $$$b$$$, but $$$a \ne b$$$;  in the first position where $$$a$$$ and $$$b$$$ differ, the string $$$a$$$ has a letter that appears earlier in the alphabet than the corresponding letter in $$$b$$$. ",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"734"Vus the Cossack has $$$n$$$ real numbers $$$a_i$$$. It is known that the sum of all numbers is equal to $$$0$$$. He wants to choose a sequence $$$b$$$ the size of which is $$$n$$$ such that the sum of all numbers is $$$0$$$ and each $$$b_i$$$ is either $$$\lfloor a_i \rfloor$$$ or $$$\lceil a_i \rceil$$$. In other words, $$$b_i$$$ equals $$$a_i$$$ rounded up or down. It is not necessary to round to the nearest integer.For example, if $$$a = [4.58413, 1.22491, -2.10517, -3.70387]$$$, then $$$b$$$ can be equal, for example, to $$$[4, 2, -2, -4]$$$. Note that if $$$a_i$$$ is an integer, then there is no difference between $$$\lfloor a_i \rfloor$$$ and $$$\lceil a_i \rceil$$$, $$$b_i$$$ will always be equal to $$$a_i$$$.Help Vus the Cossack find such sequence!",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"735"Two friends are travelling through Bubble galaxy. They say ""Hello!"" via signals to each other if their distance is smaller or equal than $$$d_1$$$ and    it's the first time they speak to each other or   at some point in time after their last talk their distance was greater than $$$d_2$$$. We need to calculate how many times friends said ""Hello!"" to each other. For $$$N$$$ moments, you'll have an array of points for each friend representing their positions at that moment. A person can stay in the same position between two moments in time, but if a person made a move we assume this movement as movement with constant speed in constant direction.",['geometry'],"[0, 1, 0, 0, 0, 0, 0, 0]"736"You've got string s, consisting of small English letters. Some of the English letters are good, the rest are bad.A substring s[l...r] (1 ≤ l ≤ r ≤ |s|) of string s  =  s1s2...s|s| (where |s| is the length of string s) is string  slsl + 1...sr.The substring s[l...r] is good, if among the letters  sl, sl + 1, ..., sr there are at most k bad ones (look at the sample's explanation to understand it more clear).Your task is to find the number of distinct good substrings of the given string s. Two substrings s[x...y] and s[p...q] are considered distinct if their content is different, i.e. s[x...y] ≠ s[p...q].",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"737"You are given $$$n$$$ words of equal length $$$m$$$, consisting of lowercase Latin alphabet letters. The $$$i$$$-th word is denoted $$$s_i$$$.In one move you can choose any position in any single word and change the letter at that position to the previous or next letter in alphabetical order. For example:  you can change 'e' to 'd' or to 'f';  'a' can only be changed to 'b';  'z' can only be changed to 'y'. The difference between two words is the minimum number of moves required to make them equal. For example, the difference between ""best"" and ""cost"" is $$$1 + 10 + 0 + 0 = 11$$$.Find the minimum difference of $$$s_i$$$ and $$$s_j$$$ such that $$$(i < j)$$$. In other words, find the minimum difference over all possible pairs of the $$$n$$$ words.","['math', 'strings']","[0, 0, 0, 1, 0, 0, 1, 0]"738"You are given a rooted tree with n vertices. The vertices are numbered from 1 to n, the root is the vertex number 1.Each vertex has a color, let's denote the color of vertex v by cv. Initially cv = 0.You have to color the tree into the given colors using the smallest possible number of steps. On each step you can choose a vertex v and a color x, and then color all vectices in the subtree of v (including v itself) in color x. In other words, for every vertex u, such that the path from root to u passes through v, set cu = x.It is guaranteed that you have to color each vertex in a color different from 0.You can learn what a rooted tree is using the link: https://en.wikipedia.org/wiki/Tree_(graph_theory).",[],"[0, 0, 0, 0, 0, 0, 0, 0]"739"ВКонтакте открыла второй штаб в Санкт-Петербурге! Вы не преминули возможностью сменить обстановку и решили переехать из офиса в Доме Зингера в офис на Красном мосту.Для комфортной работы вам потребуются два монитора с одинаковой высотой, чтобы изображение на них выглядело единым целым. На складе офиса на Красном мосту есть $$$n$$$ мониторов, $$$i$$$-й из них имеет ширину $$$w_i$$$ и высоту $$$h_i$$$. Любой монитор можно повернуть на 90 градусов, и тогда он будет иметь ширину $$$h_i$$$ и высоту $$$w_i$$$.Назовём неупорядоченную пару из двух различных мониторов подходящей, если можно их повернуть так, чтобы они имели одинаковую высоту. Любой из мониторов в паре можно как повернуть относительно исходной ориентации, так и не поворачивать.Подсчитайте подходящие пары мониторов.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"740"Pinkie Pie has bought a bag of patty-cakes with different fillings! But it appeared that not all patty-cakes differ from one another with filling. In other words, the bag contains some patty-cakes with the same filling.Pinkie Pie eats the patty-cakes one-by-one. She likes having fun so she decided not to simply eat the patty-cakes but to try not to eat the patty-cakes with the same filling way too often. To achieve this she wants the minimum distance between the eaten with the same filling to be the largest possible. Herein Pinkie Pie called the distance between two patty-cakes the number of eaten patty-cakes strictly between them.Pinkie Pie can eat the patty-cakes in any order. She is impatient about eating all the patty-cakes up so she asks you to help her to count the greatest minimum distance between the eaten patty-cakes with the same filling amongst all possible orders of eating!Pinkie Pie is going to buy more bags of patty-cakes so she asks you to solve this problem for several bags!",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"741"You are given a positive integer $$$x$$$. Check whether the number $$$x$$$ is representable as the sum of the cubes of two positive integers.Formally, you need to check if there are two integers $$$a$$$ and $$$b$$$ ($$$1 \le a, b$$$) such that $$$a^3+b^3=x$$$.For example, if $$$x = 35$$$, then the numbers $$$a=2$$$ and $$$b=3$$$ are suitable ($$$2^3+3^3=8+27=35$$$). If $$$x=4$$$, then no pair of numbers $$$a$$$ and $$$b$$$ is suitable.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"742"Vasya and Petya take part in a Codeforces round. The round lasts for two hours and contains five problems.For this round the dynamic problem scoring is used. If you were lucky not to participate in any Codeforces round with dynamic problem scoring, here is what it means. The maximum point value of the problem depends on the ratio of the number of participants who solved the problem to the total number of round participants. Everyone who made at least one submission is considered to be participating in the round.Pay attention to the range bounds. For example, if 40 people are taking part in the round, and 10 of them solve a particular problem, then the solvers fraction is equal to 1 / 4, and the problem's maximum point value is equal to 1500.If the problem's maximum point value is equal to x, then for each whole minute passed from the beginning of the contest to the moment of the participant's correct submission, the participant loses x / 250 points. For example, if the problem's maximum point value is 2000, and the participant submits a correct solution to it 40 minutes into the round, this participant will be awarded with 2000·(1 - 40 / 250) = 1680 points for this problem.There are n participants in the round, including Vasya and Petya. For each participant and each problem, the number of minutes which passed between the beginning of the contest and the submission of this participant to this problem is known. It's also possible that this participant made no submissions to this problem.With two seconds until the end of the round, all participants' submissions have passed pretests, and not a single hack attempt has been made. Vasya believes that no more submissions or hack attempts will be made in the remaining two seconds, and every submission will pass the system testing.Unfortunately, Vasya is a cheater. He has registered 109 + 7 new accounts for the round. Now Vasya can submit any of his solutions from these new accounts in order to change the maximum point values of the problems. Vasya can also submit any wrong solutions to any problems. Note that Vasya can not submit correct solutions to the problems he hasn't solved.Vasya seeks to score strictly more points than Petya in the current round. Vasya has already prepared the scripts which allow to obfuscate his solutions and submit them into the system from any of the new accounts in just fractions of seconds. However, Vasya doesn't want to make his cheating too obvious, so he wants to achieve his goal while making submissions from the smallest possible number of new accounts.Find the smallest number of new accounts Vasya needs in order to beat Petya (provided that Vasya's assumptions are correct), or report that Vasya can't achieve his goal.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"743"Alice and Bob are playing a game on a line with $$$n$$$ cells. There are $$$n$$$ cells labeled from $$$1$$$ through $$$n$$$. For each $$$i$$$ from $$$1$$$ to $$$n-1$$$, cells $$$i$$$ and $$$i+1$$$ are adjacent.Alice initially has a token on some cell on the line, and Bob tries to guess where it is. Bob guesses a sequence of line cell numbers $$$x_1, x_2, \ldots, x_k$$$ in order. In the $$$i$$$-th question, Bob asks Alice if her token is currently on cell $$$x_i$$$. That is, Alice can answer either ""YES"" or ""NO"" to each Bob's question.At most one time in this process, before or after answering a question, Alice is allowed to move her token from her current cell to some adjacent cell. Alice acted in such a way that she was able to answer ""NO"" to all of Bob's questions.Note that Alice can even move her token before answering the first question or after answering the last question. Alice can also choose to not move at all.You are given $$$n$$$ and Bob's questions $$$x_1, \ldots, x_k$$$. You would like to count the number of scenarios that let Alice answer ""NO"" to all of Bob's questions. Let $$$(a,b)$$$ denote a scenario where Alice starts at cell $$$a$$$ and ends at cell $$$b$$$. Two scenarios $$$(a_i, b_i)$$$ and $$$(a_j, b_j)$$$ are different if $$$a_i \neq a_j$$$ or $$$b_i \neq b_j$$$.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"744"Ivan wants to play a game with you. He picked some string $$$s$$$ of length $$$n$$$ consisting only of lowercase Latin letters. You don't know this string. Ivan has informed you about all its improper prefixes and suffixes (i.e. prefixes and suffixes of lengths from $$$1$$$ to $$$n-1$$$), but he didn't tell you which strings are prefixes and which are suffixes.Ivan wants you to guess which of the given $$$2n-2$$$ strings are prefixes of the given string and which are suffixes. It may be impossible to guess the string Ivan picked (since multiple strings may give the same set of suffixes and prefixes), but Ivan will accept your answer if there is at least one string that is consistent with it. Let the game begin!",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"745"Bob is a competitive programmer. He wants to become red, and for that he needs a strict training regime. He went to the annual meeting of grandmasters and asked $$$n$$$ of them how much effort they needed to reach red.""Oh, I just spent $$$x_i$$$ hours solving problems"", said the $$$i$$$-th of them. Bob wants to train his math skills, so for each answer he wrote down the number of minutes ($$$60 \cdot x_i$$$), thanked the grandmasters and went home. Bob could write numbers with leading zeroes — for example, if some grandmaster answered that he had spent $$$2$$$ hours, Bob could write $$$000120$$$ instead of $$$120$$$.Alice wanted to tease Bob and so she took the numbers Bob wrote down, and for each of them she did one of the following independently:   rearranged its digits, or  wrote a random number. This way, Alice generated $$$n$$$ numbers, denoted $$$y_1$$$, ..., $$$y_n$$$.For each of the numbers, help Bob determine whether $$$y_i$$$ can be a permutation of a number divisible by $$$60$$$ (possibly with leading zeroes).",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"746"Joe has been hurt on the Internet. Now he is storming around the house, destroying everything in his path.Joe's house has n floors, each floor is a segment of m cells. Each cell either contains nothing (it is an empty cell), or has a brick or a concrete wall (always something one of three). It is believed that each floor is surrounded by a concrete wall on the left and on the right.Now Joe is on the n-th floor and in the first cell, counting from left to right. At each moment of time, Joe has the direction of his gaze, to the right or to the left (always one direction of the two). Initially, Joe looks to the right.Joe moves by a particular algorithm. Every second he makes one of the following actions:   If the cell directly under Joe is empty, then Joe falls down. That is, he moves to this cell, the gaze direction is preserved.  Otherwise consider the next cell in the current direction of the gaze.   If the cell is empty, then Joe moves into it, the gaze direction is preserved.  If this cell has bricks, then Joe breaks them with his forehead (the cell becomes empty), and changes the direction of his gaze to the opposite.  If this cell has a concrete wall, then Joe just changes the direction of his gaze to the opposite (concrete can withstand any number of forehead hits).  Joe calms down as soon as he reaches any cell of the first floor.The figure below shows an example Joe's movements around the house.  Determine how many seconds Joe will need to calm down.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"747"The Hedgehog likes to give presents to his friend, but no less he likes to receive them.Having received another present today, the Hedgehog suddenly understood that he has no place to put it as there was no room left on the special shelf in the cupboard. He will have to choose another shelf, but which one should he choose, how large should it be?In order to get to know this, the Hedgehog asks you to write him a program that will count the estimated number of presents that he will receive during the following N days. Besides, he is guided by the principle:   on each holiday day the Hedgehog will necessarily receive a present,  he receives presents at least every K days (i.e., if he received a present on the i-th day, he will receive the next present no later than on the i + K-th day).  For the given N and K, as well as the list of holidays among the following N days count the minimal number of presents that could be given to the Hedgehog. The number of today's day is zero, and you should regard today's present as already given (i.e., you shouldn't count it in the answer).",[],"[0, 0, 0, 0, 0, 0, 0, 0]"748"To satisfy his love of matching socks, Phoenix has brought his $$$n$$$ socks ($$$n$$$ is even) to the sock store. Each of his socks has a color $$$c_i$$$ and is either a left sock or right sock. Phoenix can pay one dollar to the sock store to either:   recolor a sock to any color $$$c'$$$ $$$(1 \le c' \le n)$$$  turn a left sock into a right sock  turn a right sock into a left sock  The sock store may perform each of these changes any number of times. Note that the color of a left sock doesn't change when it turns into a right sock, and vice versa. A matching pair of socks is a left and right sock with the same color. What is the minimum cost for Phoenix to make $$$n/2$$$ matching pairs? Each sock must be included in exactly one matching pair.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"749"Vasya has written some permutation $$$p_1, p_2, \ldots, p_n$$$ of integers from $$$1$$$ to $$$n$$$, so for all $$$1 \leq i \leq n$$$ it is true that $$$1 \leq p_i \leq n$$$ and all $$$p_1, p_2, \ldots, p_n$$$ are different. After that he wrote $$$n$$$ numbers $$$next_1, next_2, \ldots, next_n$$$. The number $$$next_i$$$ is equal to the minimal index $$$i < j \leq n$$$, such that $$$p_j > p_i$$$. If there is no such $$$j$$$ let's let's define as $$$next_i = n + 1$$$.In the evening Vasya went home from school and due to rain, his notebook got wet. Now it is impossible to read some written numbers. Permutation and some values $$$next_i$$$ are completely lost! If for some $$$i$$$ the value $$$next_i$$$ is lost, let's say that $$$next_i = -1$$$.You are given numbers $$$next_1, next_2, \ldots, next_n$$$ (maybe some of them are equal to $$$-1$$$). Help Vasya to find such permutation $$$p_1, p_2, \ldots, p_n$$$ of integers from $$$1$$$ to $$$n$$$, that he can write it to the notebook and all numbers $$$next_i$$$, which are not equal to $$$-1$$$, will be correct. ","['graphs', 'math']","[0, 0, 1, 1, 0, 0, 0, 0]"750"A rectangle with its opposite corners in $$$(0, 0)$$$ and $$$(w, h)$$$ and sides parallel to the axes is drawn on a plane.You are given a list of lattice points such that each point lies on a side of a rectangle but not in its corner. Also, there are at least two points on every side of a rectangle.Your task is to choose three points in such a way that:   exactly two of them belong to the same side of a rectangle;  the area of a triangle formed by them is maximum possible. Print the doubled area of this triangle. It can be shown that the doubled area of any triangle formed by lattice points is always an integer.","['geometry', 'math']","[0, 1, 0, 1, 0, 0, 0, 0]"751"Sereja has an array a, consisting of n integers a1, a2, ..., an. The boy cannot sit and do nothing, he decided to study an array. Sereja took a piece of paper and wrote out m integers l1, l2, ..., lm (1 ≤ li ≤ n). For each number li he wants to know how many distinct numbers are staying on the positions li, li + 1, ..., n. Formally, he want to find the number of distinct numbers among ali, ali + 1, ..., an.?Sereja wrote out the necessary array elements but the array was so large and the boy was so pressed for time. Help him, find the answer for the described question for each li.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"752"You and your friends live in $$$n$$$ houses. Each house is located on a 2D plane, in a point with integer coordinates. There might be different houses located in the same point. The mayor of the city is asking you for places for the building of the Eastern exhibition. You have to find the number of places (points with integer coordinates), so that the summary distance from all the houses to the exhibition is minimal. The exhibition can be built in the same point as some house. The distance between two points $$$(x_1, y_1)$$$ and $$$(x_2, y_2)$$$ is $$$|x_1 - x_2| + |y_1 - y_2|$$$, where $$$|x|$$$ is the absolute value of $$$x$$$. ",['geometry'],"[0, 1, 0, 0, 0, 0, 0, 0]"753"After Fox Ciel got off a bus, she found that the bus she was on was a wrong bus and she lost her way in a strange town. However, she fortunately met her friend Beaver Taro and asked which way to go to her castle. Taro's response to her was a string s, and she tried to remember the string s correctly.However, Ciel feels n strings b1, b2, ... , bn are really boring, and unfortunately she dislikes to remember a string that contains a boring substring. To make the thing worse, what she can remember is only the contiguous substring of s.Determine the longest contiguous substring of s that does not contain any boring string, so that she can remember the longest part of Taro's response.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"754"Sometime the classic solution are not powerful enough and we have to design our own. For the purpose of this problem you have to implement the part of the system of task scheduling.Each task should be executed at some particular moments of time. In our system you may set the exact value for the second, minute, hour, day of the week, day and month, when the task should be executed. Moreover, one can set a special value -1 that means any value of this parameter is valid.For example, if the parameter string is -1 59 23 -1 -1 -1, the problem will be executed every day at 23:59:00, 23:59:01, 23:59:02, ..., 23:59:59 (60 times in total).Seconds, minutes and hours are numbered starting from zero, while day, months and days of the week are numbered starting from one. The first day of the week is Monday.There is one special case that is treated separately. If both day of the week and day are given (i.e. differ from -1) to execute the task only one of these two (at least one, if both match this is fine too) parameters should match the current time (of course, all other parameters should match too). For example, the string of parameters 0 0 12 6 3 7 means that the task will be executed both on Saturday, July 2nd, 2016 and on Sunday, July 3rd, 2016 at noon.One should not forget about the existence of the leap years. The year is leap if it's number is divisible by 400, or is not divisible by 100, but is divisible by 4. Each leap year has 366 days instead of usual 365, by extending February to 29 days rather than the common 28.The current time is represented as the number of seconds passed after 00:00:00 January 1st, 1970 (Thursday).You are given the string of six parameters, describing the moments of time the task should be executed. You are also given a number of moments of time. For each of them you have to find the first moment of time strictly greater than the current when the task will be executed.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"755"Masha really loves algebra. On the last lesson, her strict teacher Dvastan gave she new exercise.You are given geometric progression b defined by two integers b1 and q. Remind that a geometric progression is a sequence of integers b1, b2, b3, ..., where for each i > 1 the respective term satisfies the condition bi = bi - 1·q, where q is called the common ratio of the progression. Progressions in Uzhlyandia are unusual: both b1 and q can equal 0. Also, Dvastan gave Masha m ""bad"" integers a1, a2, ..., am, and an integer l.Masha writes all progression terms one by one onto the board (including repetitive) while condition |bi| ≤ l is satisfied (|x| means absolute value of x). There is an exception: if a term equals one of the ""bad"" integers, Masha skips it (doesn't write onto the board) and moves forward to the next term.But the lesson is going to end soon, so Masha has to calculate how many integers will be written on the board. In order not to get into depression, Masha asked you for help: help her calculate how many numbers she will write, or print ""inf"" in case she needs to write infinitely many integers.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"756"The Bitlandians are quite weird people. They do everything differently. They have a different alphabet so they have a different definition for a string.A Bitlandish string is a string made only of characters ""0"" and ""1"".BitHaval (the mayor of Bitland) loves to play with Bitlandish strings. He takes some Bitlandish string a, and applies several (possibly zero) operations to it. In one operation the mayor may take any two adjacent characters of a string, define one of them as x and the other one as y. Then he calculates two values p and q: p = x xor y, q = x or y. Then he replaces one of the two taken characters by p and the other one by q.The xor operation means the bitwise excluding OR operation. The or operation is the bitwise OR operation.So for example one operation can transform string 11 to string 10 or to string 01. String 1 cannot be transformed into any other string.You've got two Bitlandish strings a and b. Your task is to check if it is possible for BitHaval to transform string a to string b in several (possibly zero) described operations.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"757"There is a square grid of size $$$n \times n$$$. Some cells are colored in black, all others are colored in white. In one operation you can select some rectangle and color all its cells in white. It costs $$$\min(h, w)$$$ to color a rectangle of size $$$h \times w$$$. You are to make all cells white for minimum total cost.The square is large, so we give it to you in a compressed way. The set of black cells is the union of $$$m$$$ rectangles.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"758"Julia's $$$n$$$ friends want to organize a startup in a new country they moved to. They assigned each other numbers from 1 to $$$n$$$ according to the jobs they have, from the most front-end tasks to the most back-end ones. They also estimated a matrix $$$c$$$, where $$$c_{ij} = c_{ji}$$$ is the average number of messages per month between people doing jobs $$$i$$$ and $$$j$$$.Now they want to make a hierarchy tree. It will be a binary tree with each node containing one member of the team. Some member will be selected as a leader of the team and will be contained in the root node. In order for the leader to be able to easily reach any subordinate, for each node $$$v$$$ of the tree, the following should apply: all members in its left subtree must have smaller numbers than $$$v$$$, and all members in its right subtree must have larger numbers than $$$v$$$.After the hierarchy tree is settled, people doing jobs $$$i$$$ and $$$j$$$ will be communicating via the shortest path in the tree between their nodes. Let's denote the length of this path as $$$d_{ij}$$$. Thus, the cost of their communication is $$$c_{ij} \cdot d_{ij}$$$.Your task is to find a hierarchy tree that minimizes the total cost of communication over all pairs: $$$\sum_{1 \le i < j \le n} c_{ij} \cdot d_{ij}$$$.",['trees'],"[0, 0, 0, 0, 0, 0, 0, 1]"759"This is the hard version of the problem. The only difference is that here $$$2\leq k\leq 100$$$. You can make hacks only if both the versions of the problem are solved.This is an interactive problem!Every decimal number has a base $$$k$$$ equivalent. The individual digits of a base $$$k$$$ number are called $$$k$$$-its. Let's define the $$$k$$$-itwise XOR of two $$$k$$$-its $$$a$$$ and $$$b$$$ as $$$(a + b)\bmod k$$$.The $$$k$$$-itwise XOR of two base $$$k$$$ numbers is equal to the new number formed by taking the $$$k$$$-itwise XOR of their corresponding $$$k$$$-its. The $$$k$$$-itwise XOR of two decimal numbers $$$a$$$ and $$$b$$$ is denoted by $$$a\oplus_{k} b$$$ and is equal to the decimal representation of the $$$k$$$-itwise XOR of the base $$$k$$$ representations of $$$a$$$ and $$$b$$$. All further numbers used in the statement below are in decimal unless specified.You have hacked the criminal database of Rockport Police Department (RPD), also known as the Rap Sheet. But in order to access it, you require a password. You don't know it, but you are quite sure that it lies between $$$0$$$ and $$$n-1$$$ inclusive. So, you have decided to guess it. Luckily, you can try at most $$$n$$$ times without being blocked by the system. But the system is adaptive. Each time you make an incorrect guess, it changes the password. Specifically, if the password before the guess was $$$x$$$, and you guess a different number $$$y$$$, then the system changes the password to a number $$$z$$$ such that $$$x\oplus_{k} z=y$$$. Guess the password and break into the system.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"760"Frog Gorf is traveling through Swamp kingdom. Unfortunately, after a poor jump, he fell into a well of $$$n$$$ meters depth. Now Gorf is on the bottom of the well and has a long way up.The surface of the well's walls vary in quality: somewhere they are slippery, but somewhere have convenient ledges. In other words, if Gorf is on $$$x$$$ meters below ground level, then in one jump he can go up on any integer distance from $$$0$$$ to $$$a_x$$$ meters inclusive. (Note that Gorf can't jump down, only up).Unfortunately, Gorf has to take a break after each jump (including jump on $$$0$$$ meters). And after jumping up to position $$$x$$$ meters below ground level, he'll slip exactly $$$b_x$$$ meters down while resting.Calculate the minimum number of jumps Gorf needs to reach ground level.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"761"There are n cities and m two-way roads in Berland, each road connects two cities. It is known that there is no more than one road connecting each pair of cities, and there is no road which connects the city with itself. It is possible that there is no way to get from one city to some other city using only these roads.The road minister decided to make a reform in Berland and to orient all roads in the country, i.e. to make each road one-way. The minister wants to maximize the number of cities, for which the number of roads that begins in the city equals to the number of roads that ends in it.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"762"AquaMoon and Cirno are playing an interesting game with arrays. Cirno has prepared two arrays $$$a$$$ and $$$b$$$, both consist of $$$n$$$ non-negative integers. AquaMoon can perform the following operation an arbitrary number of times (possibly zero):  She chooses two indices $$$i$$$ and $$$j$$$ ($$$1 \le i, j \le n$$$), then decreases the $$$i$$$-th element of array $$$a$$$ by $$$1$$$, and increases the $$$j$$$-th element of array $$$a$$$ by $$$1$$$. The resulting values at $$$i$$$-th and $$$j$$$-th index of array $$$a$$$ are $$$a_i - 1$$$ and $$$a_j + 1$$$, respectively. Each element of array $$$a$$$ must be non-negative after each operation. If $$$i = j$$$ this operation doesn't change the array $$$a$$$. AquaMoon wants to make some operations to make arrays $$$a$$$ and $$$b$$$ equal. Two arrays $$$a$$$ and $$$b$$$ are considered equal if and only if $$$a_i = b_i$$$ for all $$$1 \leq i \leq n$$$.Help AquaMoon to find a sequence of operations that will solve her problem or find, that it is impossible to make arrays $$$a$$$ and $$$b$$$ equal.Please note, that you don't have to minimize the number of operations.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"763"You are given a tree consisting of $$$n$$$ vertices, and $$$q$$$ triples $$$(x_i, y_i, s_i)$$$, where $$$x_i$$$ and $$$y_i$$$ are integers from $$$1$$$ to $$$n$$$, and $$$s_i$$$ is a string with length equal to the number of vertices on the simple path from $$$x_i$$$ to $$$y_i$$$.You want to write a lowercase Latin letter on each vertex in such a way that, for each of $$$q$$$ given triples, at least one of the following conditions holds:  if you write out the letters on the vertices on the simple path from $$$x_i$$$ to $$$y_i$$$ in the order they appear on this path, you get the string $$$s_i$$$;  if you write out the letters on the vertices on the simple path from $$$y_i$$$ to $$$x_i$$$ in the order they appear on this path, you get the string $$$s_i$$$. Find any possible way to write a letter on each vertex to meet these constraints, or report that it is impossible.","['graphs', 'trees']","[0, 0, 1, 0, 0, 0, 0, 1]"764"Let's define a balanced multiset the following way. Write down the sum of all elements of the multiset in its decimal representation. For each position of that number check if the multiset includes at least one element such that the digit of the element and the digit of the sum at that position are the same. If that holds for every position, then the multiset is balanced. Otherwise it's unbalanced.For example, multiset $$$\{20, 300, 10001\}$$$ is balanced and multiset $$$\{20, 310, 10001\}$$$ is unbalanced:   The red digits mark the elements and the positions for which these elements have the same digit as the sum. The sum of the first multiset is $$$10321$$$, every position has the digit required. The sum of the second multiset is $$$10331$$$ and the second-to-last digit doesn't appear in any number, thus making the multiset unbalanced.You are given an array $$$a_1, a_2, \dots, a_n$$$, consisting of $$$n$$$ integers.You are asked to perform some queries on it. The queries can be of two types:  $$$1~i~x$$$ — replace $$$a_i$$$ with the value $$$x$$$;  $$$2~l~r$$$ — find the unbalanced subset of the multiset of the numbers $$$a_l, a_{l + 1}, \dots, a_r$$$ with the minimum sum, or report that no unbalanced subset exists. Note that the empty multiset is balanced.For each query of the second type print the lowest sum of the unbalanced subset. Print -1 if no unbalanced subset exists.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"765"Berland has n cities, the capital is located in city s, and the historic home town of the President is in city t (s ≠ t). The cities are connected by one-way roads, the travel time for each of the road is a positive integer.Once a year the President visited his historic home town t, for which his motorcade passes along some path from s to t (he always returns on a personal plane). Since the president is a very busy man, he always chooses the path from s to t, along which he will travel the fastest.The ministry of Roads and Railways wants to learn for each of the road: whether the President will definitely pass through it during his travels, and if not, whether it is possible to repair it so that it would definitely be included in the shortest path from the capital to the historic home town of the President. Obviously, the road can not be repaired so that the travel time on it was less than one. The ministry of Berland, like any other, is interested in maintaining the budget, so it wants to know the minimum cost of repairing the road. Also, it is very fond of accuracy, so it repairs the roads so that the travel time on them is always a positive integer.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"766"Demonstrative competitions will be held in the run-up to the $$$20NN$$$ Berlatov Olympic Games. Today is the day for the running competition!Berlatov team consists of $$$2n$$$ runners which are placed on two running tracks; $$$n$$$ runners are placed on each track. The runners are numbered from $$$1$$$ to $$$n$$$ on each track. The runner with number $$$i$$$ runs through the entire track in $$$i$$$ seconds.The competition is held as follows: first runners on both tracks start running at the same time; when the slower of them arrives at the end of the track, second runners on both tracks start running, and everyone waits until the slower of them finishes running, and so on, until all $$$n$$$ pairs run through the track.The organizers want the run to be as long as possible, but if it lasts for more than $$$k$$$ seconds, the crowd will get bored. As the coach of the team, you may choose any order in which the runners are arranged on each track (but you can't change the number of runners on each track or swap runners between different tracks).You have to choose the order of runners on each track so that the duration of the competition is as long as possible, but does not exceed $$$k$$$ seconds.Formally, you want to find two permutations $$$p$$$ and $$$q$$$ (both consisting of $$$n$$$ elements) such that $$$sum = \sum\limits_{i=1}^{n} max(p_i, q_i)$$$ is maximum possible, but does not exceed $$$k$$$. If there is no such pair, report about it.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"767"Monocarp has just learned a new card trick, and can't wait to present it to you. He shows you the entire deck of $$$n$$$ cards. You see that the values of cards from the topmost to the bottommost are integers $$$a_1, a_2, \dots, a_n$$$, and all values are different.Then he asks you to shuffle the deck $$$m$$$ times. With the $$$j$$$-th shuffle, you should take $$$b_j$$$ topmost cards and move them under the remaining $$$(n - b_j)$$$ cards without changing the order.And then, using some magic, Monocarp tells you the topmost card of the deck. However, you are not really buying that magic. You tell him that you know the topmost card yourself. Can you surprise Monocarp and tell him the topmost card before he shows it?",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"768"One day, liouzhou_101 got a chat record of Freda and Rainbow. Out of curiosity, he wanted to know which sentences were said by Freda, and which were said by Rainbow. According to his experience, he thought that Freda always said ""lala."" at the end of her sentences, while Rainbow always said ""miao."" at the beginning of his sentences. For each sentence in the chat record, help liouzhou_101 find whose sentence it is. ",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"769"You have written on a piece of paper an array of n positive integers a[1], a[2], ..., a[n] and m good pairs of integers (i1, j1), (i2, j2), ..., (im, jm). Each good pair (ik, jk) meets the following conditions: ik + jk is an odd number and 1 ≤ ik < jk ≤ n.In one operation you can perform a sequence of actions:   take one of the good pairs (ik, jk) and some integer v (v > 1), which divides both numbers a[ik] and a[jk];  divide both numbers by v, i. e. perform the assignments:  and . Determine the maximum number of operations you can sequentially perform on the given array. Note that one pair may be used several times in the described operations.",['number theory'],"[0, 0, 0, 0, 1, 0, 0, 0]"770"Iahub wants to enhance his multitasking abilities. In order to do this, he wants to sort n arrays simultaneously, each array consisting of m integers.Iahub can choose a pair of distinct indices i and j (1 ≤ i, j ≤ m, i ≠ j). Then in each array the values at positions i and j are swapped only if the value at position i is strictly greater than the value at position j.Iahub wants to find an array of pairs of distinct indices that, chosen in order, sort all of the n arrays in ascending or descending order (the particular order is given in input). The size of the array can be at most  (at most  pairs). Help Iahub, find any suitable array.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"771"Carousel Boutique is busy again! Rarity has decided to visit the pony ball and she surely needs a new dress, because going out in the same dress several times is a sign of bad manners. First of all, she needs a dress pattern, which she is going to cut out from the rectangular piece of the multicolored fabric.The piece of the multicolored fabric consists of $$$n \times m$$$ separate square scraps. Since Rarity likes dresses in style, a dress pattern must only include scraps sharing the same color. A dress pattern must be the square, and since Rarity is fond of rhombuses, the sides of a pattern must form a $$$45^{\circ}$$$ angle with sides of a piece of fabric (that way it will be resembling the traditional picture of a rhombus).Examples of proper dress patterns:  Examples of improper dress patterns:  The first one consists of multi-colored scraps, the second one goes beyond the bounds of the piece of fabric, the third one is not a square with sides forming a $$$45^{\circ}$$$ angle with sides of the piece of fabric.Rarity wonders how many ways to cut out a dress pattern that satisfies all the conditions that do exist. Please help her and satisfy her curiosity so she can continue working on her new masterpiece!",[],"[0, 0, 0, 0, 0, 0, 0, 0]"772"Now you can take online courses in the Berland State University! Polycarp needs to pass k main online courses of his specialty to get a diploma. In total n courses are availiable for the passage.The situation is complicated by the dependence of online courses, for each course there is a list of those that must be passed before starting this online course (the list can be empty, it means that there is no limitation).Help Polycarp to pass the least number of courses in total to get the specialty (it means to pass all main and necessary courses). Write a program which prints the order of courses. Polycarp passes courses consistently, he starts the next course when he finishes the previous one. Each course can't be passed more than once. ",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"773"Polycarp has built his own web service. Being a modern web service it includes login feature. And that always implies password security problems.Polycarp decided to store the hash of the password, generated by the following algorithm:  take the password $$$p$$$, consisting of lowercase Latin letters, and shuffle the letters randomly in it to obtain $$$p'$$$ ($$$p'$$$ can still be equal to $$$p$$$);  generate two random strings, consisting of lowercase Latin letters, $$$s_1$$$ and $$$s_2$$$ (any of these strings can be empty);  the resulting hash $$$h = s_1 + p' + s_2$$$, where addition is string concatenation. For example, let the password $$$p =$$$ ""abacaba"". Then $$$p'$$$ can be equal to ""aabcaab"". Random strings $$$s1 =$$$ ""zyx"" and $$$s2 =$$$ ""kjh"". Then $$$h =$$$ ""zyxaabcaabkjh"".Note that no letters could be deleted or added to $$$p$$$ to obtain $$$p'$$$, only the order could be changed.Now Polycarp asks you to help him to implement the password check module. Given the password $$$p$$$ and the hash $$$h$$$, check that $$$h$$$ can be the hash for the password $$$p$$$.Your program should answer $$$t$$$ independent test cases.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"774"As you know, all the kids in Berland love playing with cubes. Little Petya has n towers consisting of cubes of the same size. Tower with number i consists of ai cubes stacked one on top of the other. Petya defines the instability of a set of towers as a value equal to the difference between the heights of the highest and the lowest of the towers. For example, if Petya built five cube towers with heights (8, 3, 2, 6, 3), the instability of this set is equal to 6 (the highest tower has height 8, the lowest one has height 2). The boy wants the instability of his set of towers to be as low as possible. All he can do is to perform the following operation several times: take the top cube from some tower and put it on top of some other tower of his set. Please note that Petya would never put the cube on the same tower from which it was removed because he thinks it's a waste of time. Before going to school, the boy will have time to perform no more than k such operations. Petya does not want to be late for class, so you have to help him accomplish this task.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"775"Luckily, Serval got onto the right bus, and he came to the kindergarten on time. After coming to kindergarten, he found the toy bricks very funny.He has a special interest to create difficult problems for others to solve. This time, with many $$$1 \times 1 \times 1$$$ toy bricks, he builds up a 3-dimensional object. We can describe this object with a $$$n \times m$$$ matrix, such that in each cell $$$(i,j)$$$, there are $$$h_{i,j}$$$ bricks standing on the top of each other.However, Serval doesn't give you any $$$h_{i,j}$$$, and just give you the front view, left view, and the top view of this object, and he is now asking you to restore the object. Note that in the front view, there are $$$m$$$ columns, and in the $$$i$$$-th of them, the height is the maximum of $$$h_{1,i},h_{2,i},\dots,h_{n,i}$$$. It is similar for the left view, where there are $$$n$$$ columns. And in the top view, there is an $$$n \times m$$$ matrix $$$t_{i,j}$$$, where $$$t_{i,j}$$$ is $$$0$$$ or $$$1$$$. If $$$t_{i,j}$$$ equals $$$1$$$, that means $$$h_{i,j}>0$$$, otherwise, $$$h_{i,j}=0$$$.However, Serval is very lonely because others are bored about his unsolvable problems before, and refused to solve this one, although this time he promises there will be at least one object satisfying all the views. As his best friend, can you have a try?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"776"Filya just learned new geometry object — rectangle. He is given a field consisting of n × n unit cells. Rows are numbered from bottom to top with integer from 1 to n. Columns are numbered from left to right with integers from 1 to n. Cell, located at the intersection of the row r and column c is denoted as (r, c). Filya has painted two rectangles, such that their sides are parallel to coordinate axes and each cell lies fully inside or fully outside each of them. Moreover, no cell lies in both rectangles.Later, hedgehog Filya became interested in the location of his rectangles but was unable to find the sheet of paper they were painted on. They were taken by Sonya and now she wants to play a little game with Filya. He tells her a query rectangle and she replies with the number of initial rectangles that lie fully inside the given query rectangle. The query rectangle should match the same conditions as initial rectangles. Rectangle lies fully inside the query if each o its cells lies inside the query.Filya knows Sonya really well, so is sure that if he asks more than 200 questions she will stop to reply.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"777"Find the minimum area of a square land on which you can place two identical rectangular $$$a \times b$$$ houses. The sides of the houses should be parallel to the sides of the desired square land.Formally,   You are given two identical rectangles with side lengths $$$a$$$ and $$$b$$$ ($$$1 \le a, b \le 100$$$) — positive integers (you are given just the sizes, but not their positions).  Find the square of the minimum area that contains both given rectangles. Rectangles can be rotated (both or just one), moved, but the sides of the rectangles should be parallel to the sides of the desired square. Two rectangles can touch each other (side or corner), but cannot intersect. Rectangles can also touch the sides of the square but must be completely inside it. You can rotate the rectangles. Take a look at the examples for a better understanding.    The picture shows a square that contains red and green rectangles. ",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"778"One day three best friends Petya, Vasya and Tonya decided to form a team and take part in programming contests. Participants are usually offered several problems during programming contests. Long before the start the friends decided that they will implement a problem if at least two of them are sure about the solution. Otherwise, the friends won't write the problem's solution.This contest offers n problems to the participants. For each problem we know, which friend is sure about the solution. Help the friends find the number of problems for which they will write a solution.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"779"$$$n$$$ people live on the coordinate line, the $$$i$$$-th one lives at the point $$$x_i$$$ ($$$1 \le i \le n$$$). They want to choose a position $$$x_0$$$ to meet. The $$$i$$$-th person will spend $$$|x_i - x_0|$$$ minutes to get to the meeting place. Also, the $$$i$$$-th person needs $$$t_i$$$ minutes to get dressed, so in total he or she needs $$$t_i + |x_i - x_0|$$$ minutes. Here $$$|y|$$$ denotes the absolute value of $$$y$$$.These people ask you to find a position $$$x_0$$$ that minimizes the time in which all $$$n$$$ people can gather at the meeting place. ","['geometry', 'math']","[0, 1, 0, 1, 0, 0, 0, 0]"780"Grigory has $$$n$$$ magic stones, conveniently numbered from $$$1$$$ to $$$n$$$. The charge of the $$$i$$$-th stone is equal to $$$c_i$$$.Sometimes Grigory gets bored and selects some inner stone (that is, some stone with index $$$i$$$, where $$$2 \le i \le n - 1$$$), and after that synchronizes it with neighboring stones. After that, the chosen stone loses its own charge, but acquires the charges from neighboring stones. In other words, its charge $$$c_i$$$ changes to $$$c_i' = c_{i + 1} + c_{i - 1} - c_i$$$.Andrew, Grigory's friend, also has $$$n$$$ stones with charges $$$t_i$$$. Grigory is curious, whether there exists a sequence of zero or more synchronization operations, which transforms charges of Grigory's stones into charges of corresponding Andrew's stones, that is, changes $$$c_i$$$ into $$$t_i$$$ for all $$$i$$$?",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"781"One day, at the ""Russian Code Cup"" event it was decided to play football as an out of competition event. All participants was divided into n teams and played several matches, two teams could not play against each other more than once.The appointed Judge was the most experienced member — Pavel. But since he was the wisest of all, he soon got bored of the game and fell asleep. Waking up, he discovered that the tournament is over and the teams want to know the results of all the matches.Pavel didn't want anyone to discover about him sleeping and not keeping an eye on the results, so he decided to recover the results of all games. To do this, he asked all the teams and learned that the real winner was friendship, that is, each team beat the other teams exactly k times. Help Pavel come up with chronology of the tournir that meets all the conditions, or otherwise report that there is no such table.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"782"Let's define logical OR as an operation on two logical values (i. e. values that belong to the set {0, 1}) that is equal to 1 if either or both of the logical values is set to 1, otherwise it is 0. We can define logical OR of three or more logical values in the same manner: where  is equal to 1 if some ai = 1, otherwise it is equal to 0.Nam has a matrix A consisting of m rows and n columns. The rows are numbered from 1 to m, columns are numbered from 1 to n. Element at row i (1 ≤ i ≤ m) and column j (1 ≤ j ≤ n) is denoted as Aij. All elements of A are either 0 or 1. From matrix A, Nam creates another matrix B of the same size using formula:.(Bij is OR of all elements in row i and column j of matrix A)Nam gives you matrix B and challenges you to guess matrix A. Although Nam is smart, he could probably make a mistake while calculating matrix B, since size of A can be large.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"783"A binary string is a string where each character is either 0 or 1. Two binary strings $$$a$$$ and $$$b$$$ of equal length are similar, if they have the same character in some position (there exists an integer $$$i$$$ such that $$$a_i = b_i$$$). For example:  10010 and 01111 are similar (they have the same character in position $$$4$$$);  10010 and 11111 are similar;  111 and 111 are similar;  0110 and 1001 are not similar. You are given an integer $$$n$$$ and a binary string $$$s$$$ consisting of $$$2n-1$$$ characters. Let's denote $$$s[l..r]$$$ as the contiguous substring of $$$s$$$ starting with $$$l$$$-th character and ending with $$$r$$$-th character (in other words, $$$s[l..r] = s_l s_{l + 1} s_{l + 2} \dots s_r$$$).You have to construct a binary string $$$w$$$ of length $$$n$$$ which is similar to all of the following strings: $$$s[1..n]$$$, $$$s[2..n+1]$$$, $$$s[3..n+2]$$$, ..., $$$s[n..2n-1]$$$.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"784"Petya got an array $$$a$$$ of numbers from $$$1$$$ to $$$n$$$, where $$$a[i]=i$$$.He performed $$$n$$$ operations sequentially. In the end, he received a new state of the $$$a$$$ array.At the $$$i$$$-th operation, Petya chose the first $$$i$$$ elements of the array and cyclically shifted them to the right an arbitrary number of times (elements with indexes $$$i+1$$$ and more remain in their places). One cyclic shift to the right is such a transformation that the array $$$a=[a_1, a_2, \dots, a_n]$$$ becomes equal to the array $$$a = [a_i, a_1, a_2, \dots, a_{i-2}, a_{i-1}, a_{i+1}, a_{i+2}, \dots, a_n]$$$.For example, if $$$a = [5,4,2,1,3]$$$ and $$$i=3$$$ (that is, this is the third operation), then as a result of this operation, he could get any of these three arrays:  $$$a = [5,4,2,1,3]$$$ (makes $$$0$$$ cyclic shifts, or any number that is divisible by $$$3$$$);  $$$a = [2,5,4,1,3]$$$ (makes $$$1$$$ cyclic shift, or any number that has a remainder of $$$1$$$ when divided by $$$3$$$);  $$$a = [4,2,5,1,3]$$$ (makes $$$2$$$ cyclic shifts, or any number that has a remainder of $$$2$$$ when divided by $$$3$$$). Let's look at an example. Let $$$n=6$$$, i.e. initially $$$a=[1,2,3,4,5,6]$$$. A possible scenario is described below.  $$$i=1$$$: no matter how many cyclic shifts Petya makes, the array $$$a$$$ does not change.  $$$i=2$$$: let's say Petya decided to make a $$$1$$$ cyclic shift, then the array will look like $$$a = [\textbf{2}, \textbf{1}, 3, 4, 5, 6]$$$.  $$$i=3$$$: let's say Petya decided to make $$$1$$$ cyclic shift, then the array will look like $$$a = [\textbf{3}, \textbf{2}, \textbf{1}, 4, 5, 6]$$$.  $$$i=4$$$: let's say Petya decided to make $$$2$$$ cyclic shifts, the original array will look like $$$a = [\textbf{1}, \textbf{4}, \textbf{3}, \textbf{2}, 5, 6]$$$.  $$$i=5$$$: let's say Petya decided to make $$$0$$$ cyclic shifts, then the array won't change.  $$$i=6$$$: let's say Petya decided to make $$$4$$$ cyclic shifts, the array will look like $$$a = [\textbf{3}, \textbf{2}, \textbf{5}, \textbf{6}, \textbf{1}, \textbf{4}]$$$. You are given a final array state $$$a$$$ after all $$$n$$$ operations. Determine if there is a way to perform the operation that produces this result. In this case, if an answer exists, print the numbers of cyclical shifts that occurred during each of the $$$n$$$ operations.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"785"Kris works in a large company ""Blake Technologies"". As a best engineer of the company he was assigned a task to develop a printer that will be able to print horizontal and vertical strips. First prototype is already built and Kris wants to tests it. He wants you to implement the program that checks the result of the printing.Printer works with a rectangular sheet of paper of size n × m. Consider the list as a table consisting of n rows and m columns. Rows are numbered from top to bottom with integers from 1 to n, while columns are numbered from left to right with integers from 1 to m. Initially, all cells are painted in color 0.Your program has to support two operations:   Paint all cells in row ri in color ai;  Paint all cells in column ci in color ai. If during some operation i there is a cell that have already been painted, the color of this cell also changes to ai.Your program has to print the resulting table after k operation.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"786"A tree is a connected graph that doesn't contain any cycles.The distance between two vertices of a tree is the length (in edges) of the shortest path between these vertices.You are given a tree with n vertices and a positive number k. Find the number of distinct pairs of the vertices which have a distance of exactly k between them. Note that pairs (v, u) and (u, v) are considered to be the same pair.",['trees'],"[0, 0, 0, 0, 0, 0, 0, 1]"787"Kuro is living in a country called Uberland, consisting of $$$n$$$ towns, numbered from $$$1$$$ to $$$n$$$, and $$$n - 1$$$ bidirectional roads connecting these towns. It is possible to reach each town from any other. Each road connects two towns $$$a$$$ and $$$b$$$. Kuro loves walking and he is planning to take a walking marathon, in which he will choose a pair of towns $$$(u, v)$$$ ($$$u \neq v$$$) and walk from $$$u$$$ using the shortest path to $$$v$$$ (note that $$$(u, v)$$$ is considered to be different from $$$(v, u)$$$).Oddly, there are 2 special towns in Uberland named Flowrisa (denoted with the index $$$x$$$) and Beetopia (denoted with the index $$$y$$$). Flowrisa is a town where there are many strong-scent flowers, and Beetopia is another town where many bees live. In particular, Kuro will avoid any pair of towns $$$(u, v)$$$ if on the path from $$$u$$$ to $$$v$$$, he reaches Beetopia after he reached Flowrisa, since the bees will be attracted with the flower smell on Kuro’s body and sting him.Kuro wants to know how many pair of city $$$(u, v)$$$ he can take as his route. Since he’s not really bright, he asked you to help him with this problem.",['trees'],"[0, 0, 0, 0, 0, 0, 0, 1]"788"The only difference between easy and hard versions is the constraints.Polycarp has to write a coursework. The coursework consists of $$$m$$$ pages.Polycarp also has $$$n$$$ cups of coffee. The coffee in the $$$i$$$-th cup Polycarp has $$$a_i$$$ caffeine in it. Polycarp can drink some cups of coffee (each one no more than once). He can drink cups in any order. Polycarp drinks each cup instantly and completely (i.e. he cannot split any cup into several days).Surely, courseworks are not being written in a single day (in a perfect world of Berland, at least).Let's consider some day of Polycarp's work. Consider Polycarp drinks $$$k$$$ cups of coffee during this day and caffeine dosages of cups Polycarp drink during this day are $$$a_{i_1}, a_{i_2}, \dots, a_{i_k}$$$. Then the first cup he drinks gives him energy to write $$$a_{i_1}$$$ pages of coursework, the second cup gives him energy to write $$$max(0, a_{i_2} - 1)$$$ pages, the third cup gives him energy to write $$$max(0, a_{i_3} - 2)$$$ pages, ..., the $$$k$$$-th cup gives him energy to write $$$max(0, a_{i_k} - k + 1)$$$ pages.If Polycarp doesn't drink coffee during some day, he cannot write coursework at all that day.Polycarp has to finish his coursework as soon as possible (spend the minimum number of days to do it). Your task is to find out this number of days or say that it is impossible.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"789"SmallR is an archer. SmallR is taking a match of archer with Zanoes. They try to shoot in the target in turns, and SmallR shoots first. The probability of shooting the target each time is  for SmallR while  for Zanoes. The one who shoots in the target first should be the winner.Output the probability that SmallR will win the match.","['math', 'probabilities']","[0, 0, 0, 1, 0, 1, 0, 0]"790"Everyone was happy coding, until suddenly a power shortage happened and the best competitive programming site went down. Fortunately, a system administrator bought some new equipment recently, including some UPSs. Thus there are some servers that are still online, but we need all of them to be working in order to keep the round rated.Imagine the servers being a binary string $$$s$$$ of length $$$n$$$. If the $$$i$$$-th server is online, then $$$s_i = 1$$$, and $$$s_i = 0$$$ otherwise.A system administrator can do the following operation called electricity spread, that consists of the following phases:   Select two servers at positions $$$1 \le i < j \le n$$$ such that both are online (i.e. $$$s_i=s_j=1$$$). The spread starts only from online servers.  Check if we have enough power to make the spread. We consider having enough power if the number of turned on servers in range $$$[i, j]$$$ is at least the number of turned off servers in range $$$[i, j]$$$. More formally, check whether $$$2 \cdot (s_i + s_{i+1} + \ldots + s_j) \ge j - i + 1$$$. If the check is positive, turn on all the offline servers in range $$$[i, j]$$$. More formally, make $$$s_k := 1$$$ for all $$$k$$$ from $$$i$$$ to $$$j$$$. We call a binary string $$$s$$$ of length $$$n$$$ rated if we can turn on all servers (i.e. make $$$s_i = 1$$$ for $$$1 \le i \le n$$$) using the electricity spread operation any number of times (possibly, $$$0$$$). Your task is to find the number of rated strings of length $$$n$$$ modulo $$$m$$$.","['math', 'strings']","[0, 0, 0, 1, 0, 0, 1, 0]"791"Three years have passes and nothing changed. It is still raining in London, and Mr. Black has to close all the doors in his home in order to not be flooded. Once, however, Mr. Black became so nervous that he opened one door, then another, then one more and so on until he opened all the doors in his house.There are exactly two exits from Mr. Black's house, let's name them left and right exits. There are several doors in each of the exits, so each door in Mr. Black's house is located either in the left or in the right exit. You know where each door is located. Initially all the doors are closed. Mr. Black can exit the house if and only if all doors in at least one of the exits is open. You are given a sequence in which Mr. Black opened the doors, please find the smallest index $$$k$$$ such that Mr. Black can exit the house after opening the first $$$k$$$ doors.We have to note that Mr. Black opened each door at most once, and in the end all doors became open.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"792"Ujan has a lot of useless stuff in his drawers, a considerable part of which are his math notebooks: it is time to sort them out. This time he found an old dusty graph theory notebook with a description of a graph.It is an undirected weighted graph on $$$n$$$ vertices. It is a complete graph: each pair of vertices is connected by an edge. The weight of each edge is either $$$0$$$ or $$$1$$$; exactly $$$m$$$ edges have weight $$$1$$$, and all others have weight $$$0$$$.Since Ujan doesn't really want to organize his notes, he decided to find the weight of the minimum spanning tree of the graph. (The weight of a spanning tree is the sum of all its edges.) Can you find the answer for Ujan so he stops procrastinating?",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"793"It is the hard version of the problem. The only difference is that in this version $$$3 \le k \le n$$$.You are given a positive integer $$$n$$$. Find $$$k$$$ positive integers $$$a_1, a_2, \ldots, a_k$$$, such that:  $$$a_1 + a_2 + \ldots + a_k = n$$$  $$$LCM(a_1, a_2, \ldots, a_k) \le \frac{n}{2}$$$ Here $$$LCM$$$ is the least common multiple of numbers $$$a_1, a_2, \ldots, a_k$$$.We can show that for given constraints the answer always exists.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"794"The term of this problem is the same as the previous one, the only exception — increased restrictions.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"795"A bitstring is a string consisting only of the characters 0 and 1. A bitstring is called $$$k$$$-balanced if every substring of size $$$k$$$ of this bitstring has an equal amount of 0 and 1 characters ($$$\frac{k}{2}$$$ of each).You are given an integer $$$k$$$ and a string $$$s$$$ which is composed only of characters 0, 1, and ?. You need to determine whether you can make a $$$k$$$-balanced bitstring by replacing every ? characters in $$$s$$$ with either 0 or 1.A string $$$a$$$ is a substring of a string $$$b$$$ if $$$a$$$ can be obtained from $$$b$$$ by deletion of several (possibly, zero or all) characters from the beginning and several (possibly, zero or all) characters from the end.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"796"You are policeman and you are playing a game with Slavik. The game is turn-based and each turn consists of two phases. During the first phase you make your move and during the second phase Slavik makes his move.There are $$$n$$$ doors, the $$$i$$$-th door initially has durability equal to $$$a_i$$$.During your move you can try to break one of the doors. If you choose door $$$i$$$ and its current durability is $$$b_i$$$ then you reduce its durability to $$$max(0, b_i - x)$$$ (the value $$$x$$$ is given).During Slavik's move he tries to repair one of the doors. If he chooses door $$$i$$$ and its current durability is $$$b_i$$$ then he increases its durability to $$$b_i + y$$$ (the value $$$y$$$ is given). Slavik cannot repair doors with current durability equal to $$$0$$$.The game lasts $$$10^{100}$$$ turns. If some player cannot make his move then he has to skip it.Your goal is to maximize the number of doors with durability equal to $$$0$$$ at the end of the game. You can assume that Slavik wants to minimize the number of such doors. What is the number of such doors in the end if you both play optimally?",['games'],"[1, 0, 0, 0, 0, 0, 0, 0]"797"You are given three positive integers $$$n$$$, $$$a$$$ and $$$b$$$. You have to construct a string $$$s$$$ of length $$$n$$$ consisting of lowercase Latin letters such that each substring of length $$$a$$$ has exactly $$$b$$$ distinct letters. It is guaranteed that the answer exists.You have to answer $$$t$$$ independent test cases.Recall that the substring $$$s[l \dots r]$$$ is the string $$$s_l, s_{l+1}, \dots, s_{r}$$$ and its length is $$$r - l + 1$$$. In this problem you are only interested in substrings of length $$$a$$$.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"798"Lee bought some food for dinner time, but Lee's friends eat dinner in a deadly way. Lee is so scared, he doesn't want to die, at least not before seeing Online IOI 2020...There are $$$n$$$ different types of food and $$$m$$$ Lee's best friends. Lee has $$$w_i$$$ plates of the $$$i$$$-th type of food and each friend has two different favorite types of food: the $$$i$$$-th friend's favorite types of food are $$$x_i$$$ and $$$y_i$$$ ($$$x_i \ne y_i$$$).Lee will start calling his friends one by one. Whoever is called will go to the kitchen and will try to eat one plate of each of his favorite food types. Each of the friends will go to the kitchen exactly once.The only problem is the following: if a friend will eat at least one plate of food (in total) then he will be harmless. But if there is nothing left for him to eat (neither $$$x_i$$$ nor $$$y_i$$$), he will eat Lee instead $$$\times\_\times$$$.Lee can choose the order of friends to call, so he'd like to determine if he can survive dinner or not. Also, he'd like to know the order itself.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"799"People worry that computers will get too smart and take over the world, but the real problem is that they're too stupid and they've already taken over the world.— Pedro DomingosYou work for a well-known department store that uses leading technologies and employs mechanistic work — that is, robots!The department you work in sells $$$n \cdot k$$$ items. The first item costs $$$1$$$ dollar, the second item costs $$$2$$$ dollars, and so on: $$$i$$$-th item costs $$$i$$$ dollars. The items are situated on shelves. The items form a rectangular grid: there are $$$n$$$ shelves in total, and each shelf contains exactly $$$k$$$ items. We will denote by $$$a_{i,j}$$$ the price of $$$j$$$-th item (counting from the left) on the $$$i$$$-th shelf, $$$1 \le i \le n, 1 \le j \le k$$$.Occasionally robots get curious and ponder on the following question: what is the mean price (arithmetic average) of items $$$a_{i,l}, a_{i,l+1}, \ldots, a_{i,r}$$$ for some shelf $$$i$$$ and indices $$$l \le r$$$? Unfortunately, the old robots can only work with whole numbers. If the mean price turns out not to be an integer, they break down.You care about robots' welfare. You want to arrange the items in such a way that the robots cannot theoretically break. Formally, you want to choose such a two-dimensional array $$$a$$$ that:  Every number from $$$1$$$ to $$$n \cdot k$$$ (inclusively) occurs exactly once.  For each $$$i, l, r$$$, the mean price of items from $$$l$$$ to $$$r$$$ on $$$i$$$-th shelf is an integer. Find out if such an arrangement is possible, and if it is, give any example of such arrangement.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"800"There is a river of width $$$n$$$. The left bank of the river is cell $$$0$$$ and the right bank is cell $$$n + 1$$$ (more formally, the river can be represented as a sequence of $$$n + 2$$$ cells numbered from $$$0$$$ to $$$n + 1$$$). There are also $$$m$$$ wooden platforms on a river, the $$$i$$$-th platform has length $$$c_i$$$ (so the $$$i$$$-th platform takes $$$c_i$$$ consecutive cells of the river). It is guaranteed that the sum of lengths of platforms does not exceed $$$n$$$.You are standing at $$$0$$$ and want to reach $$$n+1$$$ somehow. If you are standing at the position $$$x$$$, you can jump to any position in the range $$$[x + 1; x + d]$$$. However you don't really like the water so you can jump only to such cells that belong to some wooden platform. For example, if $$$d=1$$$, you can jump only to the next position (if it belongs to the wooden platform). You can assume that cells $$$0$$$ and $$$n+1$$$ belong to wooden platforms.You want to know if it is possible to reach $$$n+1$$$ from $$$0$$$ if you can move any platform to the left or to the right arbitrary number of times (possibly, zero) as long as they do not intersect each other (but two platforms can touch each other). It also means that you cannot change the relative order of platforms.Note that you should move platforms until you start jumping (in other words, you first move the platforms and then start jumping).For example, if $$$n=7$$$, $$$m=3$$$, $$$d=2$$$ and $$$c = [1, 2, 1]$$$, then one of the ways to reach $$$8$$$ from $$$0$$$ is follow:    The first example: $$$n=7$$$. ",[],"[0, 0, 0, 0, 0, 0, 0, 0]"801"As you know, majority of students and teachers of Summer Informatics School live in Berland for the most part of the year. Since corruption there is quite widespread, the following story is not uncommon.Elections are coming. You know the number of voters and the number of parties — $$$n$$$ and $$$m$$$ respectively. For each voter you know the party he is going to vote for. However, he can easily change his vote given a certain amount of money. In particular, if you give $$$i$$$-th voter $$$c_i$$$ bytecoins you can ask him to vote for any other party you choose.The United Party of Berland has decided to perform a statistical study — you need to calculate the minimum number of bytecoins the Party needs to spend to ensure its victory. In order for a party to win the elections, it needs to receive strictly more votes than any other party.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"802"You are given a board of size $$$n \times n$$$, where $$$n$$$ is odd (not divisible by $$$2$$$). Initially, each cell of the board contains one figure.In one move, you can select exactly one figure presented in some cell and move it to one of the cells sharing a side or a corner with the current cell, i.e. from the cell $$$(i, j)$$$ you can move the figure to cells:   $$$(i - 1, j - 1)$$$;  $$$(i - 1, j)$$$;  $$$(i - 1, j + 1)$$$;  $$$(i, j - 1)$$$;  $$$(i, j + 1)$$$;  $$$(i + 1, j - 1)$$$;  $$$(i + 1, j)$$$;  $$$(i + 1, j + 1)$$$; Of course, you can not move figures to cells out of the board. It is allowed that after a move there will be several figures in one cell.Your task is to find the minimum number of moves needed to get all the figures into one cell (i.e. $$$n^2-1$$$ cells should contain $$$0$$$ figures and one cell should contain $$$n^2$$$ figures).You have to answer $$$t$$$ independent test cases.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"803"MisoilePunch♪ - 彩This is an interactive problem!On a normal day at the hidden office in A.R.C. Markland-N, Rin received an artifact, given to her by the exploration captain Sagar.After much analysis, she now realizes that this artifact contains data about a strange flower, which has existed way before the New Age. However, the information about its chemical structure has been encrypted heavily.The chemical structure of this flower can be represented as a string $$$p$$$. From the unencrypted papers included, Rin already knows the length $$$n$$$ of that string, and she can also conclude that the string contains at most three distinct letters: ""C"" (as in Carbon), ""H"" (as in Hydrogen), and ""O"" (as in Oxygen).At each moment, Rin can input a string $$$s$$$ of an arbitrary length into the artifact's terminal, and it will return every starting position of $$$s$$$ as a substring of $$$p$$$.However, the artifact has limited energy and cannot be recharged in any way, since the technology is way too ancient and is incompatible with any current A.R.C.'s devices. To be specific:  The artifact only contains $$$\frac{7}{5}$$$ units of energy.  For each time Rin inputs a string $$$s$$$ of length $$$t$$$, the artifact consumes $$$\frac{1}{t^2}$$$ units of energy.  If the amount of energy reaches below zero, the task will be considered failed immediately, as the artifact will go black forever. Since the artifact is so precious yet fragile, Rin is very nervous to attempt to crack the final data. Can you give her a helping hand?",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"804"You have a sequence $$$a$$$ with $$$n$$$ elements $$$1, 2, 3, \dots, k - 1, k, k - 1, k - 2, \dots, k - (n - k)$$$ ($$$k \le n < 2k$$$).Let's call as inversion in $$$a$$$ a pair of indices $$$i < j$$$ such that $$$a[i] > a[j]$$$.Suppose, you have some permutation $$$p$$$ of size $$$k$$$ and you build a sequence $$$b$$$ of size $$$n$$$ in the following manner: $$$b[i] = p[a[i]]$$$.Your goal is to find such permutation $$$p$$$ that the total number of inversions in $$$b$$$ doesn't exceed the total number of inversions in $$$a$$$, and $$$b$$$ is lexicographically maximum.Small reminder: the sequence of $$$k$$$ integers is called a permutation if it contains all integers from $$$1$$$ to $$$k$$$ exactly once.Another small reminder: a sequence $$$s$$$ is lexicographically smaller than another sequence $$$t$$$, if either $$$s$$$ is a prefix of $$$t$$$, or for the first $$$i$$$ such that $$$s_i \ne t_i$$$, $$$s_i < t_i$$$ holds (in the first position that these sequences are different, $$$s$$$ has smaller number than $$$t$$$).",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"805"Vanya and his friends are walking along the fence of height h and they do not want the guard to notice them. In order to achieve this the height of each of the friends should not exceed h. If the height of some person is greater than h he can bend down and then he surely won't be noticed by the guard. The height of the i-th person is equal to ai.Consider the width of the person walking as usual to be equal to 1, while the width of the bent person is equal to 2. Friends want to talk to each other while walking, so they would like to walk in a single row. What is the minimum width of the road, such that friends can walk in a row and remain unattended by the guard?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"806"After learning about polynomial hashing, Heidi decided to learn about shift-xor hashing. In particular, she came across this interesting problem.Given a bitstring $$$y \in \{0,1\}^n$$$ find out the number of different $$$k$$$ ($$$0 \leq k < n$$$) such that there exists $$$x \in \{0,1\}^n$$$ for which $$$y = x \oplus \mbox{shift}^k(x).$$$In the above, $$$\oplus$$$ is the xor operation and $$$\mbox{shift}^k$$$ is the operation of shifting a bitstring cyclically to the right $$$k$$$ times. For example, $$$001 \oplus 111 = 110$$$ and $$$\mbox{shift}^3(00010010111000) = 00000010010111$$$.",['number theory'],"[0, 0, 0, 0, 1, 0, 0, 0]"807"You are given an array consisting of n non-negative integers a1, a2, ..., an.You are going to destroy integers in the array one by one. Thus, you are given the permutation of integers from 1 to n defining the order elements of the array are destroyed.After each element is destroyed you have to find out the segment of the array, such that it contains no destroyed elements and the sum of its elements is maximum possible. The sum of elements in the empty segment is considered to be 0.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"808"You are given a rooted tree consisting of $$$n$$$ vertices numbered from $$$1$$$ to $$$n$$$. The root of the tree is a vertex number $$$1$$$.A tree is a connected undirected graph with $$$n-1$$$ edges.You are given $$$m$$$ queries. The $$$i$$$-th query consists of the set of $$$k_i$$$ distinct vertices $$$v_i[1], v_i[2], \dots, v_i[k_i]$$$. Your task is to say if there is a path from the root to some vertex $$$u$$$ such that each of the given $$$k$$$ vertices is either belongs to this path or has the distance $$$1$$$ to some vertex of this path.","['graphs', 'trees']","[0, 0, 1, 0, 0, 0, 0, 1]"809"You are given a connected undirected graph with $$$n$$$ vertices and $$$m$$$ edges. Vertices of the graph are numbered by integers from $$$1$$$ to $$$n$$$ and edges of the graph are numbered by integers from $$$1$$$ to $$$m$$$.Your task is to answer $$$q$$$ queries, each consisting of two integers $$$l$$$ and $$$r$$$. The answer to each query is the smallest non-negative integer $$$k$$$ such that the following condition holds:   For all pairs of integers $$$(a, b)$$$ such that $$$l\le a\le b\le r$$$, vertices $$$a$$$ and $$$b$$$ are reachable from one another using only the first $$$k$$$ edges (that is, edges $$$1, 2, \ldots, k$$$). ",['trees'],"[0, 0, 0, 0, 0, 0, 0, 1]"810"In the 2050 Conference, some people from the competitive programming community meet together and are going to take a photo. The $$$n$$$ people form a line. They are numbered from $$$1$$$ to $$$n$$$ from left to right. Each of them either holds a cardboard with the letter 'C' or a cardboard with the letter 'P'.Let $$$C=\{c_1,c_2,\dots,c_m\}$$$ $$$(c_1<c_2<\ldots <c_m)$$$ be the set of people who hold cardboards of 'C'. Let $$$P=\{p_1,p_2,\dots,p_k\}$$$ $$$(p_1<p_2<\ldots <p_k)$$$ be the set of people who hold cardboards of 'P'. The photo is good if and only if it satisfies the following constraints:   $$$C\cup P=\{1,2,\dots,n\}$$$  $$$C\cap P =\emptyset $$$.  $$$c_i-c_{i-1}\leq c_{i+1}-c_i(1< i <m)$$$.  $$$p_i-p_{i-1}\geq p_{i+1}-p_i(1< i <k)$$$. Given an array $$$a_1,\ldots, a_n$$$, please find the number of good photos satisfying the following condition: $$$$$$\sum\limits_{x\in C} a_x < \sum\limits_{y\in P} a_y.$$$$$$The answer can be large, so output it modulo $$$998\,244\,353$$$. Two photos are different if and only if there exists at least one person who holds a cardboard of 'C' in one photo but holds a cardboard of 'P' in the other.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"811"Consider every tree (connected undirected acyclic graph) with $$$n$$$ vertices ($$$n$$$ is odd, vertices numbered from $$$1$$$ to $$$n$$$), and for each $$$2 \le i \le n$$$ the $$$i$$$-th vertex is adjacent to exactly one vertex with a smaller index.For each $$$i$$$ ($$$1 \le i \le n$$$) calculate the number of trees for which the $$$i$$$-th vertex will be the centroid. The answer can be huge, output it modulo $$$998\,244\,353$$$.A vertex is called a centroid if its removal splits the tree into subtrees with at most $$$(n-1)/2$$$ vertices each.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"812"Dasha logged into the system and began to solve problems. One of them is as follows:Given two sequences a and b of length n each you need to write a sequence c of length n, the i-th element of which is calculated as follows: ci = bi - ai.About sequences a and b we know that their elements are in the range from l to r. More formally, elements satisfy the following conditions: l ≤ ai ≤ r and l ≤ bi ≤ r. About sequence c we know that all its elements are distinct.  Dasha wrote a solution to that problem quickly, but checking her work on the standard test was not so easy. Due to an error in the test system only the sequence a and the compressed sequence of the sequence c were known from that test.Let's give the definition to a compressed sequence. A compressed sequence of sequence c of length n is a sequence p of length n, so that pi equals to the number of integers which are less than or equal to ci in the sequence c. For example, for the sequence c = [250, 200, 300, 100, 50] the compressed sequence will be p = [4, 3, 5, 2, 1]. Pay attention that in c all integers are distinct. Consequently, the compressed sequence contains all integers from 1 to n inclusively.Help Dasha to find any sequence b for which the calculated compressed sequence of sequence c is correct.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"813"I, Fischl, Prinzessin der Verurteilung, descend upon this land by the call of fate an — Oh, you are also a traveler from another world? Very well, I grant you permission to travel with me.It is no surprise Fischl speaks with a strange choice of words. However, this time, not even Oz, her raven friend, can interpret her expressions! Maybe you can help us understand what this young princess is saying?You are given a string of $$$n$$$ lowercase Latin letters, the word that Fischl just spoke. You think that the MEX of this string may help you find the meaning behind this message. The MEX of the string is defined as the shortest string that doesn't appear as a contiguous substring in the input. If multiple strings exist, the lexicographically smallest one is considered the MEX. Note that the empty substring does NOT count as a valid MEX.A string $$$a$$$ is lexicographically smaller than a string $$$b$$$ if and only if one of the following holds:   $$$a$$$ is a prefix of $$$b$$$, but $$$a \ne b$$$;  in the first position where $$$a$$$ and $$$b$$$ differ, the string $$$a$$$ has a letter that appears earlier in the alphabet than the corresponding letter in $$$b$$$. A string $$$a$$$ is a substring of a string $$$b$$$ if $$$a$$$ can be obtained from $$$b$$$ by deletion of several (possibly, zero or all) characters from the beginning and several (possibly, zero or all) characters from the end.Find out what the MEX of the string is!",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"814"Inna and Dima decided to surprise Sereja. They brought a really huge candy matrix, it's big even for Sereja! Let's number the rows of the giant matrix from 1 to n from top to bottom and the columns — from 1 to m, from left to right. We'll represent the cell on the intersection of the i-th row and j-th column as (i, j). Just as is expected, some cells of the giant candy matrix contain candies. Overall the matrix has p candies: the k-th candy is at cell (xk, yk).The time moved closer to dinner and Inna was already going to eat p of her favourite sweets from the matrix, when suddenly Sereja (for the reason he didn't share with anyone) rotated the matrix x times clockwise by 90 degrees. Then he performed the horizontal rotate of the matrix y times. And then he rotated the matrix z times counterclockwise by 90 degrees. The figure below shows how the rotates of the matrix looks like.  Inna got really upset, but Duma suddenly understood two things: the candies didn't get damaged and he remembered which cells contained Inna's favourite sweets before Sereja's strange actions. Help guys to find the new coordinates in the candy matrix after the transformation Sereja made!",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"815"A sum of p rubles is charged from Arkady's mobile phone account every day in the morning. Among the following m days, there are n days when Arkady will top up the account: in the day di he will deposit ti rubles on his mobile phone account. Arkady will always top up the account before the daily payment will be done. There will be no other payments nor tops up in the following m days.Determine the number of days starting from the 1-st to the m-th such that the account will have a negative amount on it after the daily payment (i. e. in evening). Initially the account's balance is zero rubles.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"816"It's time polar bears Menshykov and Uslada from the zoo of St. Petersburg and elephant Horace from the zoo of Kiev got down to business. In total, there are n tasks for the day and each animal should do each of these tasks. For each task, they have evaluated its difficulty. Also animals decided to do the tasks in order of their difficulty. Unfortunately, some tasks can have the same difficulty, so the order in which one can perform the tasks may vary.Menshykov, Uslada and Horace ask you to deal with this nuisance and come up with individual plans for each of them. The plan is a sequence describing the order in which an animal should do all the n tasks. Besides, each of them wants to have its own unique plan. Therefore three plans must form three different sequences. You are to find the required plans, or otherwise deliver the sad news to them by stating that it is impossible to come up with three distinct plans for the given tasks.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"817"Acingel is a small town. There was only one doctor here — Miss Ada. She was very friendly and nobody has ever said something bad about her, so who could've expected that Ada will be found dead in her house? Mr Gawry, world-famous detective, is appointed to find the criminal. He asked $$$m$$$ neighbours of Ada about clients who have visited her in that unlucky day. Let's number the clients from $$$1$$$ to $$$n$$$. Each neighbour's testimony is a permutation of these numbers, which describes the order in which clients have been seen by the asked neighbour.However, some facts are very suspicious – how it is that, according to some of given permutations, some client has been seen in the morning, while in others he has been seen in the evening? ""In the morning some of neighbours must have been sleeping!"" — thinks Gawry — ""and in the evening there's been too dark to see somebody's face..."". Now he wants to delete some prefix and some suffix (both prefix and suffix can be empty) in each permutation, so that they'll be non-empty and equal to each other after that — some of the potential criminals may disappear, but the testimony won't stand in contradiction to each other.In how many ways he can do it? Two ways are called different if the remaining common part is different.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"818"Let $$$a$$$ and $$$b$$$ be some non-negative integers. Let's define strange addition of $$$a$$$ and $$$b$$$ as following:  write down the numbers one under another and align them by their least significant digit;  add them up digit by digit and concatenate the respective sums together. Assume that both numbers have an infinite number of leading zeros.For example, let's take a look at a strange addition of numbers $$$3248$$$ and $$$908$$$:  You are given a string $$$c$$$, consisting of $$$n$$$ digits from $$$0$$$ to $$$9$$$. You are also given $$$m$$$ updates of form:   $$$x~d$$$ — replace the digit at the $$$x$$$-th position of $$$c$$$ with a digit $$$d$$$. Note that string $$$c$$$ might have leading zeros at any point of time.After each update print the number of pairs $$$(a, b)$$$ such that both $$$a$$$ and $$$b$$$ are non-negative integers and the result of a strange addition of $$$a$$$ and $$$b$$$ is equal to $$$c$$$.Note that the numbers of pairs can be quite large, so print them modulo $$$998244353$$$.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"819"Luyi has n circles on the plane. The i-th circle is centered at (xi, yi). At the time zero circles start to grow simultaneously. In other words, the radius of each circle at time t (t > 0) is equal to t. The circles are drawn as black discs on an infinite white plane. So at each moment the plane consists of several black and white regions. Note that the circles may overlap while growing.  We define a hole as a closed, connected white region. For instance, the figure contains two holes shown by red border. During growing some holes may be created and it is easy to see that each created hole will disappear eventually. Luyi asks you to find moment of time such that the last hole disappears. In other words, you should find the first moment such that no hole can be seen after that.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"820"You were dreaming that you are traveling to a planet named Planetforces on your personal spaceship. Unfortunately, its piloting system was corrupted and now you need to fix it in order to reach Planetforces.  Space can be represented as the $$$XY$$$ plane. You are starting at point $$$(0, 0)$$$, and Planetforces is located in point $$$(p_x, p_y)$$$.The piloting system of your spaceship follows its list of orders which can be represented as a string $$$s$$$. The system reads $$$s$$$ from left to right. Suppose you are at point $$$(x, y)$$$ and current order is $$$s_i$$$:   if $$$s_i = \text{U}$$$, you move to $$$(x, y + 1)$$$;  if $$$s_i = \text{D}$$$, you move to $$$(x, y - 1)$$$;  if $$$s_i = \text{R}$$$, you move to $$$(x + 1, y)$$$;  if $$$s_i = \text{L}$$$, you move to $$$(x - 1, y)$$$. Since string $$$s$$$ could be corrupted, there is a possibility that you won't reach Planetforces in the end. Fortunately, you can delete some orders from $$$s$$$ but you can't change their positions.Can you delete several orders (possibly, zero) from $$$s$$$ in such a way, that you'll reach Planetforces after the system processes all orders?",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"821"Let's define a multiplication operation between a string $$$a$$$ and a positive integer $$$x$$$: $$$a \cdot x$$$ is the string that is a result of writing $$$x$$$ copies of $$$a$$$ one after another. For example, ""abc"" $$$\cdot~2~=$$$ ""abcabc"", ""a"" $$$\cdot~5~=$$$ ""aaaaa"".A string $$$a$$$ is divisible by another string $$$b$$$ if there exists an integer $$$x$$$ such that $$$b \cdot x = a$$$. For example, ""abababab"" is divisible by ""ab"", but is not divisible by ""ababab"" or ""aa"".LCM of two strings $$$s$$$ and $$$t$$$ (defined as $$$LCM(s, t)$$$) is the shortest non-empty string that is divisible by both $$$s$$$ and $$$t$$$.You are given two strings $$$s$$$ and $$$t$$$. Find $$$LCM(s, t)$$$ or report that it does not exist. It can be shown that if $$$LCM(s, t)$$$ exists, it is unique.","['math', 'number theory', 'strings']","[0, 0, 0, 1, 1, 0, 1, 0]"822"In a far away kingdom lived the King, the Prince, the Shoemaker, the Dressmaker and many other citizens. They lived happily until great trouble came into the Kingdom. The ACMers settled there.Most damage those strange creatures inflicted upon the kingdom was that they loved high precision numbers. As a result, the Kingdom healers had already had three appointments with the merchants who were asked to sell, say, exactly 0.273549107 beer barrels. To deal with the problem somehow, the King issued an order obliging rounding up all numbers to the closest integer to simplify calculations. Specifically, the order went like this: If a number's integer part does not end with digit 9 and its fractional part is strictly less than 0.5, then the rounded up number coincides with the number’s integer part.  If a number's integer part does not end with digit 9 and its fractional part is not less than 0.5, the rounded up number is obtained if we add 1 to the last digit of the number’s integer part. If the number’s integer part ends with digit 9, to round up the numbers one should go to Vasilisa the Wise. In the whole Kingdom she is the only one who can perform the tricky operation of carrying into the next position. Merchants found the algorithm very sophisticated and they asked you (the ACMers) to help them. Can you write a program that would perform the rounding according to the King’s order?",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"823"Maxim always goes to the supermarket on Sundays. Today the supermarket has a special offer of discount systems.There are m types of discounts. We assume that the discounts are indexed from 1 to m. To use the discount number i, the customer takes a special basket, where he puts exactly qi items he buys. Under the terms of the discount system, in addition to the items in the cart the customer can receive at most two items from the supermarket for free. The number of the ""free items"" (0, 1 or 2) to give is selected by the customer. The only condition imposed on the selected ""free items"" is as follows: each of them mustn't be more expensive than the cheapest item out of the qi items in the cart.Maxim now needs to buy n items in the shop. Count the minimum sum of money that Maxim needs to buy them, if he use the discount system optimally well.Please assume that the supermarket has enough carts for any actions. Maxim can use the same discount multiple times. Of course, Maxim can buy items without any discounts.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"824"There are $$$n$$$ shovels in the nearby shop. The $$$i$$$-th shovel costs $$$a_i$$$ bourles.Misha has to buy exactly $$$k$$$ shovels. Each shovel can be bought no more than once.Misha can buy shovels by several purchases. During one purchase he can choose any subset of remaining (non-bought) shovels and buy this subset.There are also $$$m$$$ special offers in the shop. The $$$j$$$-th of them is given as a pair $$$(x_j, y_j)$$$, and it means that if Misha buys exactly $$$x_j$$$ shovels during one purchase then $$$y_j$$$ most cheapest of them are for free (i.e. he will not pay for $$$y_j$$$ most cheapest shovels during the current purchase).Misha can use any offer any (possibly, zero) number of times, but he cannot use more than one offer during one purchase (but he can buy shovels without using any offers).Your task is to calculate the minimum cost of buying $$$k$$$ shovels, if Misha buys them optimally.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"825"Given an array a1, a2, ..., an of n integers, find the largest number in the array that is not a perfect square.A number x is said to be a perfect square if there exists an integer y such that x = y2.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"826"In the Catowice city next weekend the cat contest will be held. However, the jury members and the contestants haven't been selected yet. There are $$$n$$$ residents and $$$n$$$ cats in the Catowice, and each resident has exactly one cat living in his house. The residents and cats are numbered with integers from $$$1$$$ to $$$n$$$, where the $$$i$$$-th cat is living in the house of $$$i$$$-th resident.Each Catowice resident is in friendship with several cats, including the one living in his house. In order to conduct a contest, at least one jury member is needed and at least one cat contestant is needed. Of course, every jury member should know none of the contestants. For the contest to be successful, it's also needed that the number of jury members plus the number of contestants is equal to $$$n$$$.Please help Catowice residents to select the jury and the contestants for the upcoming competition, or determine that it's impossible to do.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"827"The problem uses a simplified TCP/IP address model, please make sure you've read the statement attentively.Polycarpus has found a job, he is a system administrator. One day he came across n IP addresses. Each IP address is a 32 bit number, represented as a group of four 8-bit numbers (without leading zeroes), separated by dots. For example, the record 0.255.1.123 shows a correct IP address and records 0.256.1.123 and 0.255.1.01 do not. In this problem an arbitrary group of four 8-bit numbers is a correct IP address.Having worked as an administrator for some time, Polycarpus learned that if you know the IP address, you can use the subnet mask to get the address of the network that has this IP addess.The subnet mask is an IP address that has the following property: if we write this IP address as a 32 bit string, that it is representable as ""11...11000..000"". In other words, the subnet mask first has one or more one bits, and then one or more zero bits (overall there are 32 bits). For example, the IP address 2.0.0.0 is not a correct subnet mask as its 32-bit record looks as 00000010000000000000000000000000.To get the network address of the IP address, you need to perform the operation of the bitwise ""and"" of the IP address and the subnet mask. For example, if the subnet mask is 255.192.0.0, and the IP address is 192.168.1.2, then the network address equals 192.128.0.0. In the bitwise ""and"" the result has a bit that equals 1 if and only if both operands have corresponding bits equal to one.Now Polycarpus wants to find all networks to which his IP addresses belong. Unfortunately, Polycarpus lost subnet mask. Fortunately, Polycarpus remembers that his IP addresses belonged to exactly k distinct networks. Help Polycarpus find the subnet mask, such that his IP addresses will belong to exactly k distinct networks. If there are several such subnet masks, find the one whose bit record contains the least number of ones. If such subnet mask do not exist, say so.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"828"$$$n$$$ boys and $$$m$$$ girls came to the party. Each boy presented each girl some integer number of sweets (possibly zero). All boys are numbered with integers from $$$1$$$ to $$$n$$$ and all girls are numbered with integers from $$$1$$$ to $$$m$$$. For all $$$1 \leq i \leq n$$$ the minimal number of sweets, which $$$i$$$-th boy presented to some girl is equal to $$$b_i$$$ and for all $$$1 \leq j \leq m$$$ the maximal number of sweets, which $$$j$$$-th girl received from some boy is equal to $$$g_j$$$.More formally, let $$$a_{i,j}$$$ be the number of sweets which the $$$i$$$-th boy give to the $$$j$$$-th girl. Then $$$b_i$$$ is equal exactly to the minimum among values $$$a_{i,1}, a_{i,2}, \ldots, a_{i,m}$$$ and $$$g_j$$$ is equal exactly to the maximum among values $$$b_{1,j}, b_{2,j}, \ldots, b_{n,j}$$$.You are interested in the minimum total number of sweets that boys could present, so you need to minimize the sum of $$$a_{i,j}$$$ for all $$$(i,j)$$$ such that $$$1 \leq i \leq n$$$ and $$$1 \leq j \leq m$$$. You are given the numbers $$$b_1, \ldots, b_n$$$ and $$$g_1, \ldots, g_m$$$, determine this number. ",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"829"We start with a permutation $$$a_1, a_2, \ldots, a_n$$$ and with an empty array $$$b$$$. We apply the following operation $$$k$$$ times.On the $$$i$$$-th iteration, we select an index $$$t_i$$$ ($$$1 \le t_i \le n-i+1$$$), remove $$$a_{t_i}$$$ from the array, and append one of the numbers $$$a_{t_i-1}$$$ or $$$a_{t_i+1}$$$ (if $$$t_i-1$$$ or $$$t_i+1$$$ are within the array bounds) to the right end of the array $$$b$$$. Then we move elements $$$a_{t_i+1}, \ldots, a_n$$$ to the left in order to fill in the empty space.You are given the initial permutation $$$a_1, a_2, \ldots, a_n$$$ and the resulting array $$$b_1, b_2, \ldots, b_k$$$. All elements of an array $$$b$$$ are distinct. Calculate the number of possible sequences of indices $$$t_1, t_2, \ldots, t_k$$$ modulo $$$998\,244\,353$$$.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"830"One day n friends gathered together to play ""Mafia"". During each round of the game some player must be the supervisor and other n - 1 people take part in the game. For each person we know in how many rounds he wants to be a player, not the supervisor: the i-th person wants to play ai rounds. What is the minimum number of rounds of the ""Mafia"" game they need to play to let each person play at least as many rounds as they want?",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"831"Mad scientist Mike does not use slow hard disks. His modification of a hard drive has not one, but n different heads that can read data in parallel.When viewed from the side, Mike's hard drive is an endless array of tracks. The tracks of the array are numbered from left to right with integers, starting with 1. In the initial state the i-th reading head is above the track number hi. For each of the reading heads, the hard drive's firmware can move the head exactly one track to the right or to the left, or leave it on the current track. During the operation each head's movement does not affect the movement of the other heads: the heads can change their relative order; there can be multiple reading heads above any of the tracks. A track is considered read if at least one head has visited this track. In particular, all of the tracks numbered h1, h2, ..., hn have been read at the beginning of the operation.  Mike needs to read the data on m distinct tracks with numbers p1, p2, ..., pm. Determine the minimum time the hard drive firmware needs to move the heads and read all the given tracks. Note that an arbitrary number of other tracks can also be read.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"832"Valera has n counters numbered from 1 to n. Some of them are connected by wires, and each of the counters has a special button.Initially, all the counters contain number 0. When you press a button on a certain counter, the value it has increases by one. Also, the values recorded in all the counters, directly connected to it by a wire, increase by one.Valera and Ignat started having a dispute, the dispute is as follows. Ignat thought of a sequence of n integers a1, a2, ..., an. Valera should choose some set of distinct counters and press buttons on each of them exactly once (on other counters the buttons won't be pressed). If after that there is a counter with the number i, which has value ai, then Valera loses the dispute, otherwise he wins the dispute.Help Valera to determine on which counters he needs to press a button to win the dispute.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"833"There are famous Russian nesting dolls named matryoshkas sold in one of the souvenir stores nearby, and you'd like to buy several of them. The store has $$$n$$$ different matryoshkas. Any matryoshka is a figure of volume $$$out_i$$$ with an empty space inside of volume $$$in_i$$$ (of course, $$$out_i > in_i$$$).You don't have much free space inside your bag, but, fortunately, you know that matryoshkas can be nested one inside another. Formally, let's call a set of matryoshkas nested if we can rearrange dolls in such a way, that the first doll can be nested inside the second one, the second doll — inside the third one and so on. Matryoshka $$$i$$$ can be nested inside matryoshka $$$j$$$ if $$$out_i \le in_j$$$. So only the last doll will take space inside your bag.Let's call extra space of a nested set of dolls as a total volume of empty space inside this structure. Obviously, it's equal to $$$in_{i_1} + (in_{i_2} - out_{i_1}) + (in_{i_3} - out_{i_2}) + \dots + (in_{i_k} - out_{i_{k-1}})$$$, where $$$i_1$$$, $$$i_2$$$, ..., $$$i_k$$$ are the indices of the chosen dolls in the order they are nested in each other.Finally, let's call a nested subset of the given sequence as big enough if there isn't any doll from the sequence that can be added to the nested subset without breaking its nested property.You want to buy many matryoshkas, so you should choose a big enough nested subset to buy it. But you will be disappointed if too much space in your bag will be wasted, so you want to choose a big enough subset so that its extra space is minimum possible among all big enough subsets. Now you wonder, how many different nested subsets meet these conditions (they are big enough, and there is no big enough subset such that its extra space is less than the extra space of the chosen subset). Two subsets are considered different if there exists at least one index $$$i$$$ such that one of the subsets contains the $$$i$$$-th doll, and another subset doesn't.Since the answer can be large, print it modulo $$$10^9 + 7$$$.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"834"You are given an array $$$a$$$ consisting of $$$n$$$ ($$$n \ge 3$$$) positive integers. It is known that in this array, all the numbers except one are the same (for example, in the array $$$[4, 11, 4, 4]$$$ all numbers except one are equal to $$$4$$$).Print the index of the element that does not equal others. The numbers in the array are numbered from one.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"835"The Berland State University is hosting a ballroom dance in celebration of its 100500-th anniversary! n boys and m girls are already busy rehearsing waltz, minuet, polonaise and quadrille moves.We know that several boy&girl pairs are going to be invited to the ball. However, the partners' dancing skill in each pair must differ by at most one.For each boy, we know his dancing skills. Similarly, for each girl we know her dancing skills. Write a code that can determine the largest possible number of pairs that can be formed from n boys and m girls.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"836"You are given an integer array $$$a_1, a_2, \dots, a_n$$$ and integer $$$k$$$.In one step you can   either choose some index $$$i$$$ and decrease $$$a_i$$$ by one (make $$$a_i = a_i - 1$$$);  or choose two indices $$$i$$$ and $$$j$$$ and set $$$a_i$$$ equal to $$$a_j$$$ (make $$$a_i = a_j$$$). What is the minimum number of steps you need to make the sum of array $$$\sum\limits_{i=1}^{n}{a_i} \le k$$$? (You are allowed to make values of array negative).",[],"[0, 0, 0, 0, 0, 0, 0, 0]"837"While resting on the ship after the ""Russian Code Cup"" a boy named Misha invented an interesting game. He promised to give his quadrocopter to whoever will be the first one to make a rectangular table of size n × m, consisting of positive integers such that the sum of the squares of numbers for each row and each column was also a square.Since checking the correctness of the table manually is difficult, Misha asks you to make each number in the table to not exceed 108.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"838"We have a string of letters 'a' and 'b'. We want to perform some operations on it. On each step we choose one of substrings ""ab"" in the string and replace it with the string ""bba"". If we have no ""ab"" as a substring, our job is done. Print the minimum number of steps we should perform to make our job done modulo 109 + 7.The string ""ab"" appears as a substring if there is a letter 'b' right after the letter 'a' somewhere in the string.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"839"Once upon a time in the galaxy of far, far away...Darth Wader found out the location of a rebels' base. Now he is going to destroy the base (and the whole planet that the base is located at), using the Death Star.When the rebels learnt that the Death Star was coming, they decided to use their new secret weapon — space mines. Let's describe a space mine's build.Each space mine is shaped like a ball (we'll call it the mine body) of a certain radius r with the center in the point O. Several spikes protrude from the center. Each spike can be represented as a segment, connecting the center of the mine with some point P, such that  (transporting long-spiked mines is problematic), where |OP| is the length of the segment connecting O and P. It is convenient to describe the point P by a vector p such that P = O + p.The Death Star is shaped like a ball with the radius of R (R exceeds any mine's radius). It moves at a constant speed along the v vector at the speed equal to |v|. At the moment the rebels noticed the Star of Death, it was located in the point A.The rebels located n space mines along the Death Star's way. You may regard the mines as being idle. The Death Star does not know about the mines' existence and cannot notice them, which is why it doesn't change the direction of its movement. As soon as the Star of Death touched the mine (its body or one of the spikes), the mine bursts and destroys the Star of Death. A touching is the situation when there is a point in space which belongs both to the mine and to the Death Star. It is considered that Death Star will not be destroyed if it can move infinitely long time without touching the mines.Help the rebels determine whether they will succeed in destroying the Death Star using space mines or not. If they will succeed, determine the moment of time when it will happen (starting from the moment the Death Star was noticed).",['geometry'],"[0, 1, 0, 0, 0, 0, 0, 0]"840"Three guys play a game: first, each person writes down $$$n$$$ distinct words of length $$$3$$$. Then, they total up the number of points as follows:   if a word was written by one person — that person gets 3 points,  if a word was written by two people — each of the two gets 1 point,  if a word was written by all — nobody gets any points.  In the end, how many points does each player have?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"841"You are an experienced Codeforces user. Today you found out that during your activity on Codeforces you have made y submissions, out of which x have been successful. Thus, your current success rate on Codeforces is equal to x / y.Your favorite rational number in the [0;1] range is p / q. Now you wonder: what is the smallest number of submissions you have to make if you want your success rate to be p / q?",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"842"This is an easier version of the problem. In this version $$$n \le 1000$$$The outskirts of the capital are being actively built up in Berland. The company ""Kernel Panic"" manages the construction of a residential complex of skyscrapers in New Berlskva. All skyscrapers are built along the highway. It is known that the company has already bought $$$n$$$ plots along the highway and is preparing to build $$$n$$$ skyscrapers, one skyscraper per plot.Architects must consider several requirements when planning a skyscraper. Firstly, since the land on each plot has different properties, each skyscraper has a limit on the largest number of floors it can have. Secondly, according to the design code of the city, it is unacceptable for a skyscraper to simultaneously have higher skyscrapers both to the left and to the right of it.Formally, let's number the plots from $$$1$$$ to $$$n$$$. Then if the skyscraper on the $$$i$$$-th plot has $$$a_i$$$ floors, it must hold that $$$a_i$$$ is at most $$$m_i$$$ ($$$1 \le a_i \le m_i$$$). Also there mustn't be integers $$$j$$$ and $$$k$$$ such that $$$j < i < k$$$ and $$$a_j > a_i < a_k$$$. Plots $$$j$$$ and $$$k$$$ are not required to be adjacent to $$$i$$$.The company wants the total number of floors in the built skyscrapers to be as large as possible. Help it to choose the number of floors for each skyscraper in an optimal way, i.e. in such a way that all requirements are fulfilled, and among all such construction plans choose any plan with the maximum possible total number of floors.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"843"You are given an array of integers $$$a_1, a_2, \ldots, a_n$$$ and an integer $$$x$$$.You need to select the maximum number of elements in the array, such that for every subsegment $$$a_l, a_{l + 1}, \ldots, a_r$$$ containing strictly more than one element $$$(l < r)$$$, either:   At least one element on this subsegment is not selected, or  $$$a_l + a_{l+1} + \ldots + a_r \geq x \cdot (r - l + 1)$$$. ",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"844"In one very large and very respectable company there is a cloakroom with a coat hanger. It is represented by n hooks, positioned in a row. The hooks are numbered with positive integers from 1 to n from the left to the right.The company workers have a very complicated work schedule. At the beginning of a work day all the employees are not there and the coat hanger in the cloakroom is empty. At some moments of time the employees arrive and some of them leave.When some employee arrives, he hangs his cloak on one of the available hooks. To be of as little discomfort to his colleagues as possible, the hook where the coat will hang, is chosen like this. First the employee chooses the longest segment among available hooks following in a row. If there are several of such segments, then he chooses the one closest to the right. After that the coat is hung on the hook located in the middle of this segment. If the segment has an even number of hooks, then among two central hooks we choose the one closest to the right.When an employee leaves, he takes his coat. As all the company workers deeply respect each other, no one takes somebody else's coat.From time to time the director of this respectable company gets bored and he sends his secretary to see how many coats hang on the coat hanger from the i-th to the j-th hook inclusive. And this whim is always to be fulfilled, otherwise the director gets angry and has a mental breakdown.Not to spend too much time traversing from the director's office to the cloakroom and back again, the secretary asked you to write a program, emulating the company cloakroom's work.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"845"The zombies are gathering in their secret lair! Heidi will strike hard to destroy them once and for all. But there is a little problem... Before she can strike, she needs to know where the lair is. And the intel she has is not very good.Heidi knows that the lair can be represented as a rectangle on a lattice, with sides parallel to the axes. Each vertex of the polygon occupies an integer point on the lattice. For each cell of the lattice, Heidi can check the level of Zombie Contamination. This level is an integer between 0 and 4, equal to the number of corners of the cell that are inside or on the border of the rectangle.As a test, Heidi wants to check that her Zombie Contamination level checker works. Given the output of the checker, Heidi wants to know whether it could have been produced by a single non-zero area rectangular-shaped lair (with axis-parallel sides). ",[],"[0, 0, 0, 0, 0, 0, 0, 0]"846"Anton loves transforming one permutation into another one by swapping elements for money, and Ira doesn't like paying for stupid games. Help them obtain the required permutation by paying as little money as possible.More formally, we have two permutations, p and s of numbers from 1 to n. We can swap pi and pj, by paying |i - j| coins for it. Find and print the smallest number of coins required to obtain permutation s from permutation p. Also print the sequence of swap operations at which we obtain a solution. ",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"847"There are $$$n$$$ people in a horizontal line, each looking either to the left or the right. Each person counts the number of people in the direction they are looking. The value of the line is the sum of each person's count.For example, in the arrangement LRRLL, where L stands for a person looking left and R stands for a person looking right, the counts for each person are $$$[0, 3, 2, 3, 4]$$$, and the value is $$$0+3+2+3+4=12$$$.You are given the initial arrangement of people in the line. For each $$$k$$$ from $$$1$$$ to $$$n$$$, determine the maximum value of the line if you can change the direction of at most $$$k$$$ people.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"848"Sam is a kindergartener, and there are $$$n$$$ children in his group. He decided to create a team with some of his children to play ""brawl:go 2"".Sam has $$$n$$$ power-ups, the $$$i$$$-th has type $$$a_i$$$. A child's strength is equal to the number of different types among power-ups he has.For a team of size $$$k$$$, Sam will distribute all $$$n$$$ power-ups to $$$k$$$ children in such a way that each of the $$$k$$$ children receives at least one power-up, and each power-up is given to someone.For each integer $$$k$$$ from $$$1$$$ to $$$n$$$, find the minimum sum of strengths of a team of $$$k$$$ children Sam can get.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"849"Bogdan has a birthday today and mom gave him a tree consisting of n vertecies. For every edge of the tree i, some number xi was written on it. In case you forget, a tree is a connected non-directed graph without cycles. After the present was granted, m guests consecutively come to Bogdan's party. When the i-th guest comes, he performs exactly one of the two possible operations:   Chooses some number yi, and two vertecies ai and bi. After that, he moves along the edges of the tree from vertex ai to vertex bi using the shortest path (of course, such a path is unique in the tree). Every time he moves along some edge j, he replaces his current number yi by , that is, by the result of integer division yi div xj.  Chooses some edge pi and replaces the value written in it xpi by some positive integer ci < xpi. As Bogdan cares about his guests, he decided to ease the process. Write a program that performs all the operations requested by guests and outputs the resulting value yi for each i of the first type.","['graphs', 'math', 'trees']","[0, 0, 1, 1, 0, 0, 0, 1]"850"Sherlock has a new girlfriend (so unlike him!). Valentine's day is coming and he wants to gift her some jewelry.He bought n pieces of jewelry. The i-th piece has price equal to i + 1, that is, the prices of the jewelry are 2, 3, 4, ... n + 1.Watson gave Sherlock a challenge to color these jewelry pieces such that two pieces don't have the same color if the price of one piece is a prime divisor of the price of the other piece. Also, Watson asked him to minimize the number of different colors used.Help Sherlock complete this trivial task.",['number theory'],"[0, 0, 0, 0, 1, 0, 0, 0]"851Ann and Borya have n piles with candies and n is even number. There are ai candies in pile with number i.Ann likes numbers which are square of some integer and Borya doesn't like numbers which are square of any integer. During one move guys can select some pile with candies and add one candy to it (this candy is new and doesn't belong to any other pile) or remove one candy (if there is at least one candy in this pile). Find out minimal number of moves that is required to make exactly n / 2 piles contain number of candies that is a square of some integer and exactly n / 2 piles contain number of candies that is not a square of any integer.,[],"[0, 0, 0, 0, 0, 0, 0, 0]"852"Some time ago Mister B detected a strange signal from the space, which he started to study.After some transformation the signal turned out to be a permutation p of length n or its cyclic shift. For the further investigation Mister B need some basis, that's why he decided to choose cyclic shift of this permutation which has the minimum possible deviation.Let's define the deviation of a permutation p as .Find a cyclic shift of permutation p with minimum possible deviation. If there are multiple solutions, print any of them.Let's denote id k (0 ≤ k < n) of a cyclic shift of permutation p as the number of right shifts needed to reach this shift, for example:  k = 0: shift p1, p2, ... pn,  k = 1: shift pn, p1, ... pn - 1,  ...,  k = n - 1: shift p2, p3, ... pn, p1. ",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"853"You are given a text consisting of n lines. Each line contains some space-separated words, consisting of lowercase English letters.We define a syllable as a string that contains exactly one vowel and any arbitrary number (possibly none) of consonants. In English alphabet following letters are considered to be vowels: 'a', 'e', 'i', 'o', 'u' and 'y'.Each word of the text that contains at least one vowel can be divided into syllables. Each character should be a part of exactly one syllable. For example, the word ""mamma"" can be divided into syllables as ""ma"" and ""mma"", ""mam"" and ""ma"", and ""mamm"" and ""a"". Words that consist of only consonants should be ignored.The verse patterns for the given text is a sequence of n integers p1, p2, ..., pn. Text matches the given verse pattern if for each i from 1 to n one can divide words of the i-th line in syllables in such a way that the total number of syllables is equal to pi.You are given the text and the verse pattern. Check, if the given text matches the given verse pattern.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"854"The title is a reference to the very first Educational Round from our writers team, Educational Round 18.There is a bag, containing colored balls. There are $$$n$$$ different colors of balls, numbered from $$$1$$$ to $$$n$$$. There are $$$\mathit{cnt}_i$$$ balls of color $$$i$$$ in the bag. The total amount of balls in the bag is odd (e. g. $$$\mathit{cnt}_1 + \mathit{cnt}_2 + \dots + \mathit{cnt}_n$$$ is odd).In one move, you can choose two balls with different colors and take them out of the bag.At some point, all the remaining balls in the bag will have the same color. That's when you can't make moves anymore.Find any possible color of the remaining balls.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"855"A family consisting of father bear, mother bear and son bear owns three cars. Father bear can climb into the largest car and he likes it. Also, mother bear can climb into the middle car and she likes it. Moreover, son bear can climb into the smallest car and he likes it. It's known that the largest car is strictly larger than the middle car, and the middle car is strictly larger than the smallest car. Masha came to test these cars. She could climb into all cars, but she liked only the smallest car. It's known that a character with size a can climb into some car with size b if and only if a ≤ b, he or she likes it if and only if he can climb into this car and 2a ≥ b.You are given sizes of bears and Masha. Find out some possible integer non-negative sizes of cars.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"856"You are given an array $$$a$$$ consisting of $$$n$$$ integers. Let's denote monotonic renumeration of array $$$a$$$ as an array $$$b$$$ consisting of $$$n$$$ integers such that all of the following conditions are met:  $$$b_1 = 0$$$;  for every pair of indices $$$i$$$ and $$$j$$$ such that $$$1 \le i, j \le n$$$, if $$$a_i = a_j$$$, then $$$b_i = b_j$$$ (note that if $$$a_i \ne a_j$$$, it is still possible that $$$b_i = b_j$$$);  for every index $$$i \in [1, n - 1]$$$ either $$$b_i = b_{i + 1}$$$ or $$$b_i + 1 = b_{i + 1}$$$. For example, if $$$a = [1, 2, 1, 2, 3]$$$, then two possible monotonic renumerations of $$$a$$$ are $$$b = [0, 0, 0, 0, 0]$$$ and $$$b = [0, 0, 0, 0, 1]$$$.Your task is to calculate the number of different monotonic renumerations of $$$a$$$. The answer may be large, so print it modulo $$$998244353$$$.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"857"You wanted to write a text $$$t$$$ consisting of $$$m$$$ lowercase Latin letters. But instead, you have written a text $$$s$$$ consisting of $$$n$$$ lowercase Latin letters, and now you want to fix it by obtaining the text $$$t$$$ from the text $$$s$$$.Initially, the cursor of your text editor is at the end of the text $$$s$$$ (after its last character). In one move, you can do one of the following actions:  press the ""left"" button, so the cursor is moved to the left by one position (or does nothing if it is pointing at the beginning of the text, i. e. before its first character);  press the ""right"" button, so the cursor is moved to the right by one position (or does nothing if it is pointing at the end of the text, i. e. after its last character);  press the ""home"" button, so the cursor is moved to the beginning of the text (before the first character of the text);  press the ""end"" button, so the cursor is moved to the end of the text (after the last character of the text);  press the ""backspace"" button, so the character before the cursor is removed from the text (if there is no such character, nothing happens). Your task is to calculate the minimum number of moves required to obtain the text $$$t$$$ from the text $$$s$$$ using the given set of actions, or determine it is impossible to obtain the text $$$t$$$ from the text $$$s$$$.You have to answer $$$T$$$ independent test cases.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"858"In Summer Informatics School, if a student doesn't behave well, teachers make a hole in his badge. And today one of the teachers caught a group of $$$n$$$ students doing yet another trick. Let's assume that all these students are numbered from $$$1$$$ to $$$n$$$. The teacher came to student $$$a$$$ and put a hole in his badge. The student, however, claimed that the main culprit is some other student $$$p_a$$$.After that, the teacher came to student $$$p_a$$$ and made a hole in his badge as well. The student in reply said that the main culprit was student $$$p_{p_a}$$$.This process went on for a while, but, since the number of students was finite, eventually the teacher came to the student, who already had a hole in his badge.After that, the teacher put a second hole in the student's badge and decided that he is done with this process, and went to the sauna.You don't know the first student who was caught by the teacher. However, you know all the numbers $$$p_i$$$. Your task is to find out for every student $$$a$$$, who would be the student with two holes in the badge if the first caught student was $$$a$$$.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"859"You are given two sets of positive integers $$$A$$$ and $$$B$$$. You have to find two non-empty subsets $$$S_A \subseteq A$$$, $$$S_B \subseteq B$$$ so that the least common multiple (LCM) of the elements of $$$S_A$$$ is equal to the least common multiple (LCM) of the elements of $$$S_B$$$.","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"860"RedDreamer has an array $$$a$$$ consisting of $$$n$$$ non-negative integers, and an unlucky integer $$$T$$$.Let's denote the misfortune of array $$$b$$$ having length $$$m$$$ as $$$f(b)$$$ — the number of pairs of integers $$$(i, j)$$$ such that $$$1 \le i < j \le m$$$ and $$$b_i + b_j = T$$$. RedDreamer has to paint each element of $$$a$$$ into one of two colors, white and black (for each element, the color is chosen independently), and then create two arrays $$$c$$$ and $$$d$$$ so that all white elements belong to $$$c$$$, and all black elements belong to $$$d$$$ (it is possible that one of these two arrays becomes empty). RedDreamer wants to paint the elements in such a way that $$$f(c) + f(d)$$$ is minimum possible.For example:  if $$$n = 6$$$, $$$T = 7$$$ and $$$a = [1, 2, 3, 4, 5, 6]$$$, it is possible to paint the $$$1$$$-st, the $$$4$$$-th and the $$$5$$$-th elements white, and all other elements black. So $$$c = [1, 4, 5]$$$, $$$d = [2, 3, 6]$$$, and $$$f(c) + f(d) = 0 + 0 = 0$$$;  if $$$n = 3$$$, $$$T = 6$$$ and $$$a = [3, 3, 3]$$$, it is possible to paint the $$$1$$$-st element white, and all other elements black. So $$$c = [3]$$$, $$$d = [3, 3]$$$, and $$$f(c) + f(d) = 0 + 1 = 1$$$. Help RedDreamer to paint the array optimally!",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"861"Easy and hard versions are actually different problems, so we advise you to read both statements carefully.You are given a weighted rooted tree, vertex $$$1$$$ is the root of this tree. Also, each edge has its own cost.A tree is a connected graph without cycles. A rooted tree has a special vertex called the root. A parent of a vertex $$$v$$$ is the last different from $$$v$$$ vertex on the path from the root to the vertex $$$v$$$. Children of vertex $$$v$$$ are all vertices for which $$$v$$$ is the parent. A vertex is a leaf if it has no children. The weighted tree is such a tree that each edge of this tree has some weight.The weight of the path is the sum of edges weights on this path. The weight of the path from the vertex to itself is $$$0$$$.You can make a sequence of zero or more moves. On each move, you select an edge and divide its weight by $$$2$$$ rounding down. More formally, during one move, you choose some edge $$$i$$$ and divide its weight by $$$2$$$ rounding down ($$$w_i := \left\lfloor\frac{w_i}{2}\right\rfloor$$$).Each edge $$$i$$$ has an associated cost $$$c_i$$$ which is either $$$1$$$ or $$$2$$$ coins. Each move with edge $$$i$$$ costs $$$c_i$$$ coins.Your task is to find the minimum total cost to make the sum of weights of paths from the root to each leaf at most $$$S$$$. In other words, if $$$w(i, j)$$$ is the weight of the path from the vertex $$$i$$$ to the vertex $$$j$$$, then you have to make $$$\sum\limits_{v \in leaves} w(root, v) \le S$$$, where $$$leaves$$$ is the list of all leaves.You have to answer $$$t$$$ independent test cases.",['trees'],"[0, 0, 0, 0, 0, 0, 0, 1]"862"Kristina has two arrays $$$a$$$ and $$$b$$$, each containing $$$n$$$ non-negative integers. She can perform the following operation on array $$$a$$$ any number of times:   apply a decrement to each non-zero element of the array, that is, replace the value of each element $$$a_i$$$ such that $$$a_i > 0$$$ with the value $$$a_i - 1$$$ ($$$1 \le i \le n$$$). If $$$a_i$$$ was $$$0$$$, its value does not change. Determine whether Kristina can get an array $$$b$$$ from an array $$$a$$$ in some number of operations (probably zero). In other words, can she make $$$a_i = b_i$$$ after some number of operations for each $$$1 \le i \le n$$$?For example, let $$$n = 4$$$, $$$a = [3, 5, 4, 1]$$$ and $$$b = [1, 3, 2, 0]$$$. In this case, she can apply the operation twice:   after the first application of the operation she gets $$$a = [2, 4, 3, 0]$$$;  after the second use of the operation she gets $$$a = [1, 3, 2, 0]$$$. Thus, in two operations, she can get an array $$$b$$$ from an array $$$a$$$.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"863"You are given an array $$$a$$$ with $$$n$$$ integers. You can perform the following operation at most $$$k$$$ times:  Choose two indices $$$i$$$ and $$$j$$$, in which $$$i \,\bmod\, k = j \,\bmod\, k$$$ ($$$1 \le i < j \le n$$$).  Swap $$$a_i$$$ and $$$a_j$$$. After performing all operations, you have to select $$$k$$$ consecutive elements, and the sum of the $$$k$$$ elements becomes your score. Find the maximum score you can get.Here $$$x \bmod y$$$ denotes the remainder from dividing $$$x$$$ by $$$y$$$.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"864"Mike and Joe are playing a game with some stones. Specifically, they have $$$n$$$ piles of stones of sizes $$$a_1, a_2, \ldots, a_n$$$. These piles are arranged in a circle.The game goes as follows. Players take turns removing some positive number of stones from a pile in clockwise order starting from pile $$$1$$$. Formally, if a player removed stones from pile $$$i$$$ on a turn, the other player removes stones from pile $$$((i\bmod n) + 1)$$$ on the next turn.If a player cannot remove any stones on their turn (because the pile is empty), they lose. Mike goes first.If Mike and Joe play optimally, who will win?",['games'],"[1, 0, 0, 0, 0, 0, 0, 0]"865"The new academic year has started, and Berland's university has $$$n$$$ first-year students. They are divided into $$$k$$$ academic groups, however, some of the groups might be empty. Among the students, there are $$$m$$$ pairs of acquaintances, and each acquaintance pair might be both in a common group or be in two different groups.Alice is the curator of the first years, she wants to host an entertaining game to make everyone know each other. To do that, she will select two different academic groups and then divide the students of those groups into two teams. The game requires that there are no acquaintance pairs inside each of the teams.Alice wonders how many pairs of groups she can select, such that it'll be possible to play a game after that. All students of the two selected groups must take part in the game.Please note, that the teams Alice will form for the game don't need to coincide with groups the students learn in. Moreover, teams may have different sizes (or even be empty).",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"866"A non-empty digit string is diverse if the number of occurrences of each character in it doesn't exceed the number of distinct characters in it.For example:   string ""7"" is diverse because 7 appears in it $$$1$$$ time and the number of distinct characters in it is $$$1$$$;  string ""77"" is not diverse because 7 appears in it $$$2$$$ times and the number of distinct characters in it is $$$1$$$;  string ""1010"" is diverse because both 0 and 1 appear in it $$$2$$$ times and the number of distinct characters in it is $$$2$$$;  string ""6668"" is not diverse because 6 appears in it $$$3$$$ times and the number of distinct characters in it is $$$2$$$. You are given a string $$$s$$$ of length $$$n$$$, consisting of only digits $$$0$$$ to $$$9$$$. Find how many of its $$$\frac{n(n+1)}{2}$$$ substrings are diverse.A string $$$a$$$ is a substring of a string $$$b$$$ if $$$a$$$ can be obtained from $$$b$$$ by deletion of several (possibly, zero or all) characters from the beginning and several (possibly, zero or all) characters from the end.Note that if the same diverse string appears in $$$s$$$ multiple times, each occurrence should be counted independently. For example, there are two diverse substrings in ""77"" both equal to ""7"", so the answer for the string ""77"" is $$$2$$$.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"867"One day Sasha visited the farmer 2D and his famous magnetic farm. On this farm, the crop grows due to the influence of a special magnetic field. Maintaining of the magnetic field is provided by $$$n$$$ machines, and the power of the $$$i$$$-th machine is $$$a_i$$$. This year 2D decided to cultivate a new culture, but what exactly he didn't say. For the successful growth of the new culture, it is necessary to slightly change the powers of the machines. 2D can at most once choose an arbitrary integer $$$x$$$, then choose one machine and reduce the power of its machine by $$$x$$$ times, and at the same time increase the power of one another machine by $$$x$$$ times (powers of all the machines must stay positive integers). Note that he may not do that if he wants. More formally, 2D can choose two such indices $$$i$$$ and $$$j$$$, and one integer $$$x$$$ such that $$$x$$$ is a divisor of $$$a_i$$$, and change powers as following: $$$a_i = \frac{a_i}{x}$$$, $$$a_j = a_j \cdot x$$$Sasha is very curious, that's why he wants to calculate the minimum total power the farmer can reach. There are too many machines, and Sasha can't cope with computations, help him!",['number theory'],"[0, 0, 0, 0, 1, 0, 0, 0]"868"You are a usual chat user on the most famous streaming platform. Of course, there are some moments when you just want to chill and spam something.More precisely, you want to spam the emote triangle of size $$$k$$$. It consists of $$$2k-1$$$ messages. The first message consists of one emote, the second one — of two emotes, ..., the $$$k$$$-th one — of $$$k$$$ emotes, the $$$k+1$$$-th one — of $$$k-1$$$ emotes, ..., and the last one — of one emote.For example, the emote triangle for $$$k=3$$$ consists of $$$5$$$ messages:  Of course, most of the channels have auto moderation. Auto moderator of the current chat will ban you right after you spam at least $$$x$$$ emotes in succession (you can assume you are the only user in the chat). Now you are interested — how many messages will you write before getting banned? Or maybe you will not get banned at all (i.e. will write all $$$2k-1$$$ messages and complete your emote triangle successfully)? Note that if you get banned as a result of writing a message, this message is also counted.You have to answer $$$t$$$ independent test cases.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"869"Every person likes prime numbers. Alice is a person, thus she also shares the love for them. Bob wanted to give her an affectionate gift but couldn't think of anything inventive. Hence, he will be giving her a graph. How original, Bob! Alice will surely be thrilled!When building the graph, he needs four conditions to be satisfied:   It must be a simple undirected graph, i.e. without multiple (parallel) edges and self-loops.  The number of vertices must be exactly $$$n$$$ — a number he selected. This number is not necessarily prime.  The total number of edges must be prime.  The degree (i.e. the number of edges connected to the vertex) of each vertex must be prime. Below is an example for $$$n = 4$$$. The first graph (left one) is invalid as the degree of vertex $$$2$$$ (and $$$4$$$) equals to $$$1$$$, which is not prime. The second graph (middle one) is invalid as the total number of edges is $$$4$$$, which is not a prime number. The third graph (right one) is a valid answer for $$$n = 4$$$.   Note that the graph can be disconnected.Please help Bob to find any such graph!","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"870"You are given three integers $$$a$$$, $$$b$$$, and $$$d$$$. Your task is to find any integer $$$x$$$ which satisfies all of the following conditions, or determine that no such integers exist:  $$$0 \le x \lt 2^{60}$$$;  $$$a|x$$$ is divisible by $$$d$$$;  $$$b|x$$$ is divisible by $$$d$$$. Here, $$$|$$$ denotes the bitwise OR operation.","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"871"Let $$$LCM(x, y)$$$ be the minimum positive integer that is divisible by both $$$x$$$ and $$$y$$$. For example, $$$LCM(13, 37) = 481$$$, $$$LCM(9, 6) = 18$$$.You are given two integers $$$l$$$ and $$$r$$$. Find two integers $$$x$$$ and $$$y$$$ such that $$$l \le x < y \le r$$$ and $$$l \le LCM(x, y) \le r$$$.","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"872"The string $$$s$$$ is given, the string length is odd number. The string consists of lowercase letters of the Latin alphabet.As long as the string length is greater than $$$1$$$, the following operation can be performed on it: select any two adjacent letters in the string $$$s$$$ and delete them from the string. For example, from the string ""lemma"" in one operation, you can get any of the four strings: ""mma"", ""lma"", ""lea"" or ""lem"" In particular, in one operation, the length of the string reduces by $$$2$$$.Formally, let the string $$$s$$$ have the form $$$s=s_1s_2 \dots s_n$$$ ($$$n>1$$$). During one operation, you choose an arbitrary index $$$i$$$ ($$$1 \le i < n$$$) and replace $$$s=s_1s_2 \dots s_{i-1}s_{i+2} \dots s_n$$$.For the given string $$$s$$$ and the letter $$$c$$$, determine whether it is possible to make such a sequence of operations that in the end the equality $$$s=c$$$ will be true? In other words, is there such a sequence of operations that the process will end with a string of length $$$1$$$, which consists of the letter $$$c$$$?",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"873"The Third Doctor Who once correctly said that travel between parallel universes is ""like travelling sideways"". However, he incorrectly thought that there were infinite parallel universes, whereas in fact, as we now all know, there will never be more than 250.Heidi recently got her hands on a multiverse observation tool. She was able to see all $$$n$$$ universes lined up in a row, with non-existent links between them. She also noticed that the Doctor was in the $$$k$$$-th universe.The tool also points out that due to restrictions originating from the space-time discontinuum, the number of universes will never exceed $$$m$$$.Obviously, the multiverse is unstable because of free will. Each time a decision is made, one of two events will randomly happen: a new parallel universe is created, or a non-existent link is broken.More specifically,    When a universe is created, it will manifest itself between any two adjacent universes or at one of the ends.   When a link is broken, it could be cut between any two adjacent universes. After separating the multiverse into two segments, the segment NOT containing the Doctor will cease to exist. Heidi wants to perform a simulation of $$$t$$$ decisions. Each time a decision is made, Heidi wants to know the length of the multiverse (i.e. the number of universes), and the position of the Doctor.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"874"Polycarp lives on the coordinate axis $$$Ox$$$ and travels from the point $$$x=a$$$ to $$$x=b$$$. It moves uniformly rectilinearly at a speed of one unit of distance per minute.On the axis $$$Ox$$$ at the point $$$x=c$$$ the base station of the mobile operator is placed. It is known that the radius of its coverage is $$$r$$$. Thus, if Polycarp is at a distance less than or equal to $$$r$$$ from the point $$$x=c$$$, then he is in the network coverage area, otherwise — no. The base station can be located both on the route of Polycarp and outside it.Print the time in minutes during which Polycarp will not be in the coverage area of the network, with a rectilinear uniform movement from $$$x=a$$$ to $$$x=b$$$. His speed — one unit of distance per minute.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"875"Rahul and Tina are looking forward to starting their new year at college. As they enter their new classroom, they observe the seats of students are arranged in a $$$n \times m$$$ grid. The seat in row $$$r$$$ and column $$$c$$$ is denoted by $$$(r, c)$$$, and the distance between two seats $$$(a,b)$$$ and $$$(c,d)$$$ is $$$|a-c| + |b-d|$$$. As the class president, Tina has access to exactly $$$k$$$ buckets of pink paint. The following process occurs.   First, Tina chooses exactly $$$k$$$ seats in the classroom to paint with pink paint. One bucket of paint can paint exactly one seat.  After Tina has painted $$$k$$$ seats in the previous step, Rahul chooses where he sits. He will not choose a seat that has been painted pink due to his hatred of the colour pink.  After Rahul has chosen his seat, Tina chooses a seat for herself. She can choose any of the seats, painted or not, other than the one chosen by Rahul. Rahul wants to choose a seat such that he sits as close to Tina as possible. However, Tina wants to sit as far away from Rahul as possible due to some complicated relationship history that we couldn't fit into the statement!Now, Rahul wonders for $$$k = 0, 1, \dots, n \cdot m - 1$$$, if Tina has $$$k$$$ buckets of paint, how close can Rahul sit to Tina, if both Rahul and Tina are aware of each other's intentions and they both act as strategically as possible? Please help satisfy Rahul's curiosity! ",['games'],"[1, 0, 0, 0, 0, 0, 0, 0]"876"Berland year consists of $$$m$$$ months with $$$d$$$ days each. Months are numbered from $$$1$$$ to $$$m$$$. Berland week consists of $$$w$$$ days. The first day of the year is also the first day of the week. Note that the last week of the year might be shorter than $$$w$$$ days.A pair $$$(x, y)$$$ such that $$$x < y$$$ is ambiguous if day $$$x$$$ of month $$$y$$$ is the same day of the week as day $$$y$$$ of month $$$x$$$.Count the number of ambiguous pairs.","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"877"Theofanis decided to visit his uncle's farm. There are $$$s$$$ animals and $$$n$$$ animal pens on the farm. For utility purpose, animal pens are constructed in one row.Uncle told Theofanis that a farm is lucky if you can distribute all animals in all pens in such a way that there are no empty pens and there is at least one continuous segment of pens that has exactly $$$k$$$ animals in total.Moreover, a farm is ideal if it's lucky for any distribution without empty pens.Neither Theofanis nor his uncle knows if their farm is ideal or not. Can you help them to figure it out?",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"878"Sereja and Dima play a game. The rules of the game are very simple. The players have n cards in a row. Each card contains a number, all numbers on the cards are distinct. The players take turns, Sereja moves first. During his turn a player can take one card: either the leftmost card in a row, or the rightmost one. The game ends when there is no more cards. The player who has the maximum sum of numbers on his cards by the end of the game, wins.Sereja and Dima are being greedy. Each of them chooses the card with the larger number during his move.Inna is a friend of Sereja and Dima. She knows which strategy the guys are using, so she wants to determine the final score, given the initial state of the game. Help her.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"879"The Zoo in the Grid Kingdom is represented by an infinite grid. The Zoo has n observation binoculars located at the OX axis. For each i between 1 and n, inclusive, there exists a single binocular located at the point with coordinates (i, 0). There are m flamingos in the Zoo, located at points with positive coordinates. The flamingos are currently sleeping and you can assume that they don't move.In order to get a good view over the flamingos, each of the binoculars can be independently rotated to face any angle (not necessarily integer). Then, the binocular can be used to observe all flamingos that is located at the straight line passing through the binocular at the angle it is set. In other words, you can assign each binocular a direction corresponding to any straight line passing through the binocular, and the binocular will be able to see all flamingos located on that line.Today, some kids from the prestigious Codeforces kindergarten went on a Field Study to the Zoo. Their teacher would like to set each binocular an angle to maximize the number of flamingos that can be seen by the binocular. The teacher is very interested in the sum of these values over all binoculars. Please help him find this sum.",['geometry'],"[0, 1, 0, 0, 0, 0, 0, 0]"880"Appleman has a very big sheet of paper. This sheet has a form of rectangle with dimensions 1 × n. Your task is help Appleman with folding of such a sheet. Actually, you need to perform q queries. Each query will have one of the following types:  Fold the sheet of paper at position pi. After this query the leftmost part of the paper with dimensions 1 × pi must be above the rightmost part of the paper with dimensions 1 × ([current width of sheet] - pi).  Count what is the total width of the paper pieces, if we will make two described later cuts and consider only the pieces between the cuts. We will make one cut at distance li from the left border of the current sheet of paper and the other at distance ri from the left border of the current sheet of paper. Please look at the explanation of the first test example for better understanding of the problem.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"881"Sasha and Kolya decided to get drunk with Coke, again. This time they have k types of Coke. i-th type is characterised by its carbon dioxide concentration . Today, on the party in honour of Sergiy of Vancouver they decided to prepare a glass of Coke with carbon dioxide concentration . The drink should also be tasty, so the glass can contain only integer number of liters of each Coke type (some types can be not presented in the glass). Also, they want to minimize the total volume of Coke in the glass.Carbon dioxide concentration is defined as the volume of carbone dioxide in the Coke divided by the total volume of Coke. When you mix two Cokes, the volume of carbon dioxide sums up, and the total volume of Coke sums up as well.Help them, find the minimal natural number of liters needed to create a glass with carbon dioxide concentration . Assume that the friends have unlimited amount of each Coke type.","['graphs', 'math']","[0, 0, 1, 1, 0, 0, 0, 0]"882"Given three numbers $$$n, a, b$$$. You need to find an adjacency matrix of such an undirected graph that the number of components in it is equal to $$$a$$$, and the number of components in its complement is $$$b$$$. The matrix must be symmetric, and all digits on the main diagonal must be zeroes.In an undirected graph loops (edges from a vertex to itself) are not allowed. It can be at most one edge between a pair of vertices.The adjacency matrix of an undirected graph is a square matrix of size $$$n$$$ consisting only of ""0"" and ""1"", where $$$n$$$ is the number of vertices of the graph and the $$$i$$$-th row and the $$$i$$$-th column correspond to the $$$i$$$-th vertex of the graph. The cell $$$(i,j)$$$ of the adjacency matrix contains $$$1$$$ if and only if the $$$i$$$-th and $$$j$$$-th vertices in the graph are connected by an edge.A connected component is a set of vertices $$$X$$$ such that for every two vertices from this set there exists at least one path in the graph connecting this pair of vertices, but adding any other vertex to $$$X$$$ violates this rule.The complement or inverse of a graph $$$G$$$ is a graph $$$H$$$ on the same vertices such that two distinct vertices of $$$H$$$ are adjacent if and only if they are not adjacent in $$$G$$$.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"883"There are $$$n$$$ chests. The $$$i$$$-th chest contains $$$a_i$$$ coins. You need to open all $$$n$$$ chests in order from chest $$$1$$$ to chest $$$n$$$.There are two types of keys you can use to open a chest:   a good key, which costs $$$k$$$ coins to use;  a bad key, which does not cost any coins, but will halve all the coins in each unopened chest, including the chest it is about to open. The halving operation will round down to the nearest integer for each chest halved. In other words using a bad key to open chest $$$i$$$ will do $$$a_i = \lfloor{\frac{a_i}{2}\rfloor}$$$, $$$a_{i+1} = \lfloor\frac{a_{i+1}}{2}\rfloor, \dots, a_n = \lfloor \frac{a_n}{2}\rfloor$$$;  any key (both good and bad) breaks after a usage, that is, it is a one-time use. You need to use in total $$$n$$$ keys, one for each chest. Initially, you have no coins and no keys. If you want to use a good key, then you need to buy it.During the process, you are allowed to go into debt; for example, if you have $$$1$$$ coin, you are allowed to buy a good key worth $$$k=3$$$ coins, and your balance will become $$$-2$$$ coins.Find the maximum number of coins you can have after opening all $$$n$$$ chests in order from chest $$$1$$$ to chest $$$n$$$.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"884"You have a permutation: an array $$$a = [a_1, a_2, \ldots, a_n]$$$ of distinct integers from $$$1$$$ to $$$n$$$. The length of the permutation $$$n$$$ is odd.Consider the following algorithm of sorting the permutation in increasing order.A helper procedure of the algorithm, $$$f(i)$$$, takes a single argument $$$i$$$ ($$$1 \le i \le n-1$$$) and does the following. If $$$a_i > a_{i+1}$$$, the values of $$$a_i$$$ and $$$a_{i+1}$$$ are exchanged. Otherwise, the permutation doesn't change.The algorithm consists of iterations, numbered with consecutive integers starting with $$$1$$$. On the $$$i$$$-th iteration, the algorithm does the following:   if $$$i$$$ is odd, call $$$f(1), f(3), \ldots, f(n - 2)$$$;  if $$$i$$$ is even, call $$$f(2), f(4), \ldots, f(n - 1)$$$. It can be proven that after a finite number of iterations the permutation will be sorted in increasing order.After how many iterations will this happen for the first time?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"885"All cities of Lineland are located on the Ox coordinate axis. Thus, each city is associated with its position xi — a coordinate on the Ox axis. No two cities are located at a single point.Lineland residents love to send letters to each other. A person may send a letter only if the recipient lives in another city (because if they live in the same city, then it is easier to drop in).Strange but true, the cost of sending the letter is exactly equal to the distance between the sender's city and the recipient's city.For each city calculate two values ​​mini and maxi, where mini is the minimum cost of sending a letter from the i-th city to some other city, and maxi is the the maximum cost of sending a letter from the i-th city to some other city",[],"[0, 0, 0, 0, 0, 0, 0, 0]"886"You're given an array $$$a$$$ of length $$$n$$$. You can perform the following operation on it as many times as you want:  Pick two integers $$$i$$$ and $$$j$$$ $$$(1 \le i,j \le n)$$$ such that $$$a_i+a_j$$$ is odd, then swap $$$a_i$$$ and $$$a_j$$$. What is lexicographically the smallest array you can obtain?An array $$$x$$$ is lexicographically smaller than an array $$$y$$$ if there exists an index $$$i$$$ such that $$$x_i<y_i$$$, and $$$x_j=y_j$$$ for all $$$1 \le j < i$$$. Less formally, at the first index $$$i$$$ in which they differ, $$$x_i<y_i$$$",[],"[0, 0, 0, 0, 0, 0, 0, 0]"887"In the snake exhibition, there are $$$n$$$ rooms (numbered $$$0$$$ to $$$n - 1$$$) arranged in a circle, with a snake in each room. The rooms are connected by $$$n$$$ conveyor belts, and the $$$i$$$-th conveyor belt connects the rooms $$$i$$$ and $$$(i+1) \bmod n$$$. In the other words, rooms $$$0$$$ and $$$1$$$, $$$1$$$ and $$$2$$$, $$$\ldots$$$, $$$n-2$$$ and $$$n-1$$$, $$$n-1$$$ and $$$0$$$ are connected with conveyor belts.The $$$i$$$-th conveyor belt is in one of three states:  If it is clockwise, snakes can only go from room $$$i$$$ to $$$(i+1) \bmod n$$$.  If it is anticlockwise, snakes can only go from room $$$(i+1) \bmod n$$$ to $$$i$$$.  If it is off, snakes can travel in either direction.   Above is an example with $$$4$$$ rooms, where belts $$$0$$$ and $$$3$$$ are off, $$$1$$$ is clockwise, and $$$2$$$ is anticlockwise.Each snake wants to leave its room and come back to it later. A room is returnable if the snake there can leave the room, and later come back to it using the conveyor belts. How many such returnable rooms are there?",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"888"You might have remembered Theatre square from the problem 1A. Now it's finally getting repaved.The square still has a rectangular shape of $$$n \times m$$$ meters. However, the picture is about to get more complicated now. Let $$$a_{i,j}$$$ be the $$$j$$$-th square in the $$$i$$$-th row of the pavement.You are given the picture of the squares:  if $$$a_{i,j} = $$$ ""*"", then the $$$j$$$-th square in the $$$i$$$-th row should be black;  if $$$a_{i,j} = $$$ ""."", then the $$$j$$$-th square in the $$$i$$$-th row should be white. The black squares are paved already. You have to pave the white squares. There are two options for pavement tiles:  $$$1 \times 1$$$ tiles — each tile costs $$$x$$$ burles and covers exactly $$$1$$$ square;  $$$1 \times 2$$$ tiles — each tile costs $$$y$$$ burles and covers exactly $$$2$$$ adjacent squares of the same row. Note that you are not allowed to rotate these tiles or cut them into $$$1 \times 1$$$ tiles. You should cover all the white squares, no two tiles should overlap and no black squares should be covered by tiles.What is the smallest total price of the tiles needed to cover all the white squares?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"889"The programmers from the R2 company love playing 2048. One day, they decided to invent their own simplified version of this game — 2k on a stripe.Imagine an infinite in one direction stripe, consisting of unit squares (the side of each square is equal to the height of the stripe). Each square can either be empty or contain some number.Initially, all squares are empty. Then at infinity one of the unit squares number 2 or 4 appears. Then the player presses a button once, and the appeared number begins to move towards the beginning of the stripe. Let's assume that some number x moves to the beginning of the stripe, then it will stop if:  it either gets in the first square of the stripe;  or it is in the square that is preceded by a square with number y (y ≠ x). But if number x at some point of time gets to the square with the same number then both numbers add to each other and result in 2x. The new number 2x continues moving to the beginning of the stripe by the same rules. After the final stop of the number moving process, the infinity gets a new number 2 or 4 and the process repeats. Read the notes to the test samples to better understand the moving strategy.I guess you've understood that the game progress fully depends on the order in which numbers 2 and 4 appear. Let's look at some sequence of numbers 2 and 4 in the game. We assume that the sequence is winning if it results in at least one square getting the number greater or equal than 2k. The goal of the game is to make up a winning sequence of n numbers. But not everything is so simple, some numbers in the sequence are identified beforehand. You are given a sequence consisting of numbers 0, 2, 4. Count how many ways there are to replace each 0 of the sequence with 2 or 4 to get a winning sequence.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"890"Wabbit is playing a game with $$$n$$$ bosses numbered from $$$1$$$ to $$$n$$$. The bosses can be fought in any order. Each boss needs to be defeated exactly once. There is a parameter called boss bonus which is initially $$$0$$$.When the $$$i$$$-th boss is defeated, the current boss bonus is added to Wabbit's score, and then the value of the boss bonus increases by the point increment $$$c_i$$$. Note that $$$c_i$$$ can be negative, which means that other bosses now give fewer points.However, Wabbit has found a glitch in the game. At any point in time, he can reset the playthrough and start a New Game Plus playthrough. This will set the current boss bonus to $$$0$$$, while all defeated bosses remain defeated. The current score is also saved and does not reset to zero after this operation. This glitch can be used at most $$$k$$$ times. He can reset after defeating any number of bosses (including before or after defeating all of them), and he also can reset the game several times in a row without defeating any boss.Help Wabbit determine the maximum score he can obtain if he has to defeat all $$$n$$$ bosses.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"891"$$$n$$$ distinct integers $$$x_1,x_2,\ldots,x_n$$$ are written on the board. Nezzar can perform the following operation multiple times.  Select two integers $$$x,y$$$ (not necessarily distinct) on the board, and write down $$$2x-y$$$. Note that you don't remove selected numbers. Now, Nezzar wonders if it is possible to have his favorite number $$$k$$$ on the board after applying above operation multiple times.","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"892"There are $$$n$$$ detachments on the surface, numbered from $$$1$$$ to $$$n$$$, the $$$i$$$-th detachment is placed in a point with coordinates $$$(x_i, y_i)$$$. All detachments are placed in different points.Brimstone should visit each detachment at least once. You can choose the detachment where Brimstone starts.To move from one detachment to another he should first choose one of four directions of movement (up, right, left or down) and then start moving with the constant speed of one unit interval in a second until he comes to a detachment. After he reaches an arbitrary detachment, he can repeat the same process.Each $$$t$$$ seconds an orbital strike covers the whole surface, so at that moment Brimstone should be in a point where some detachment is located. He can stay with any detachment as long as needed.Brimstone is a good commander, that's why he can create at most one detachment and place it in any empty point with integer coordinates he wants before his trip. Keep in mind that Brimstone will need to visit this detachment, too.Help Brimstone and find such minimal $$$t$$$ that it is possible to check each detachment. If there is no such $$$t$$$ report about it.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"893"Polycarp has decided to decorate his room because the New Year is soon. One of the main decorations that Polycarp will install is the garland he is going to solder himself.Simple garlands consisting of several lamps connected by one wire are too boring for Polycarp. He is going to solder a garland consisting of $$$n$$$ lamps and $$$n - 1$$$ wires. Exactly one lamp will be connected to power grid, and power will be transmitted from it to other lamps by the wires. Each wire connectes exactly two lamps; one lamp is called the main lamp for this wire (the one that gets power from some other wire and transmits it to this wire), the other one is called the auxiliary lamp (the one that gets power from this wire). Obviously, each lamp has at most one wire that brings power to it (and this lamp is the auxiliary lamp for this wire, and the main lamp for all other wires connected directly to it).Each lamp has a brightness value associated with it, the $$$i$$$-th lamp has brightness $$$2^i$$$. We define the importance of the wire as the sum of brightness values over all lamps that become disconnected from the grid if the wire is cut (and all other wires are still working).Polycarp has drawn the scheme of the garland he wants to make (the scheme depicts all $$$n$$$ lamp and $$$n - 1$$$ wires, and the lamp that will be connected directly to the grid is marked; the wires are placed in such a way that the power can be transmitted to each lamp). After that, Polycarp calculated the importance of each wire, enumerated them from $$$1$$$ to $$$n - 1$$$ in descending order of their importance, and then wrote the index of the main lamp for each wire (in the order from the first wire to the last one).The following day Polycarp bought all required components of the garland and decided to solder it — but he could not find the scheme. Fortunately, Polycarp found the list of indices of main lamps for all wires. Can you help him restore the original scheme?",['trees'],"[0, 0, 0, 0, 0, 0, 0, 1]"894"After getting kicked out of her reporting job for not knowing the alphabet, Bessie has decided to attend school at the Fillet and Eggs Eater Academy. She has been making good progress with her studies and now knows the first k English letters.Each morning, Bessie travels to school along a sidewalk consisting of m + n tiles. In order to help Bessie review, Mr. Moozing has labeled each of the first m sidewalk tiles with one of the first k lowercase English letters, spelling out a string t. Mr. Moozing, impressed by Bessie's extensive knowledge of farm animals, plans to let her finish labeling the last n tiles of the sidewalk by herself.Consider the resulting string s (|s| = m + n) consisting of letters labeled on tiles in order from home to school. For any sequence of indices p1 < p2 < ... < pq we can define subsequence of the string s as string sp1sp2... spq. Two subsequences are considered to be distinct if they differ as strings. Bessie wants to label the remaining part of the sidewalk such that the number of distinct subsequences of tiles is maximum possible. However, since Bessie hasn't even finished learning the alphabet, she needs your help!Note that empty subsequence also counts.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"895"One day Vasya went out for a walk in the yard but there weren't any of his friends outside and he had no one to play touch and run. But the boy didn't lose the high spirits and decided to play touch and run with himself. You may ask: ""How did he do that?"" The answer is simple.Vasya noticed that the yard is a rectangular n × m field. The squares have coordinates (x, y) (1 ≤ x ≤ n, 1 ≤ y ≤ m), where x is the index of the row and y is the index of the column.Initially Vasya stands in the square with coordinates (xc, yc). To play, he has got a list of k vectors (dxi, dyi) of non-zero length. The game goes like this. The boy considers all vectors in the order from 1 to k, and consecutively chooses each vector as the current one. After the boy has chosen a current vector, he makes the maximally possible number of valid steps in the vector's direction (it is possible that he makes zero steps).A step is defined as one movement from the square where the boy is standing now, in the direction of the current vector. That is, if Vasya is positioned in square (x, y), and the current vector is (dx, dy), one step moves Vasya to square (x + dx, y + dy). A step is considered valid, if the boy does not go out of the yard if he performs the step.Vasya stepped on and on, on and on until he ran out of vectors in his list. Ha had been stepping for so long that he completely forgot how many steps he had made. Help the boy and count how many steps he had made.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"896"Fox Ciel is playing a card game with her friend Fox Jiro. There are n piles of cards on the table. And there is a positive integer on each card.The players take turns and Ciel takes the first turn. In Ciel's turn she takes a card from the top of any non-empty pile, and in Jiro's turn he takes a card from the bottom of any non-empty pile. Each player wants to maximize the total sum of the cards he took. The game ends when all piles become empty.Suppose Ciel and Jiro play optimally, what is the score of the game?",['games'],"[1, 0, 0, 0, 0, 0, 0, 0]"897"Bakry got bored of solving problems related to xor, so he asked you to solve this problem for him.You are given an array $$$a$$$ of $$$n$$$ integers $$$[a_1, a_2, \ldots, a_n]$$$.Let's call a subarray $$$a_{l}, a_{l+1}, a_{l+2}, \ldots, a_r$$$ good if $$$a_l \, \& \, a_{l+1} \, \& \, a_{l+2} \, \ldots \, \& \, a_r > a_l \oplus a_{l+1} \oplus a_{l+2} \ldots \oplus a_r$$$, where $$$\oplus$$$ denotes the bitwise XOR operation and $$$\&$$$ denotes the bitwise AND operation.Find the length of the longest good subarray of $$$a$$$, or determine that no such subarray exists.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"898"Having watched the last Harry Potter film, little Gerald also decided to practice magic. He found in his father's magical book a spell that turns any number in the sum of its digits. At the moment Gerald learned that, he came across a number n. How many times can Gerald put a spell on it until the number becomes one-digit?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"899"A tuple of positive integers {x1, x2, ..., xk} is called simple if for all pairs of positive integers (i,  j) (1  ≤ i  <  j ≤ k), xi  +  xj is a prime.You are given an array a with n positive integers a1,  a2,  ...,  an (not necessary distinct). You want to find a simple subset of the array a with the maximum size.A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself.Let's define a subset of the array a as a tuple that can be obtained from a by removing some (possibly all) elements of it.",['number theory'],"[0, 0, 0, 0, 1, 0, 0, 0]"900"Andrew and Jerry are playing a game with Harry as the scorekeeper. The game consists of three rounds. In each round, Andrew and Jerry draw randomly without replacement from a jar containing n balls, each labeled with a distinct positive integer. Without looking, they hand their balls to Harry, who awards the point to the player with the larger number and returns the balls to the jar. The winner of the game is the one who wins at least two of the three rounds.Andrew wins rounds 1 and 2 while Jerry wins round 3, so Andrew wins the game. However, Jerry is unhappy with this system, claiming that he will often lose the match despite having the higher overall total. What is the probability that the sum of the three balls Jerry drew is strictly higher than the sum of the three balls Andrew drew?",['probabilities'],"[0, 0, 0, 0, 0, 1, 0, 0]"901"After battling Shikamaru, Tayuya decided that her flute is too predictable, and replaced it with a guitar. The guitar has $$$6$$$ strings and an infinite number of frets numbered from $$$1$$$. Fretting the fret number $$$j$$$ on the $$$i$$$-th string produces the note $$$a_{i} + j$$$.Tayuya wants to play a melody of $$$n$$$ notes. Each note can be played on different string-fret combination. The easiness of performance depends on the difference between the maximal and the minimal indices of used frets. The less this difference is, the easier it is to perform the technique. Please determine the minimal possible difference.For example, if $$$a = [1, 1, 2, 2, 3, 3]$$$, and the sequence of notes is $$$4, 11, 11, 12, 12, 13, 13$$$ (corresponding to the second example), we can play the first note on the first string, and all the other notes on the sixth string. Then the maximal fret will be $$$10$$$, the minimal one will be $$$3$$$, and the answer is $$$10 - 3 = 7$$$, as shown on the picture.  ",[],"[0, 0, 0, 0, 0, 0, 0, 0]"902"Alice and Bob are playing a game. They have an array of positive integers $$$a$$$ of size $$$n$$$.Before starting the game, Alice chooses an integer $$$k \ge 0$$$. The game lasts for $$$k$$$ stages, the stages are numbered from $$$1$$$ to $$$k$$$. During the $$$i$$$-th stage, Alice must remove an element from the array that is less than or equal to $$$k - i + 1$$$. After that, if the array is not empty, Bob must add $$$k - i + 1$$$ to an arbitrary element of the array. Note that both Alice's move and Bob's move are two parts of the same stage of the game. If Alice can't delete an element during some stage, she loses. If the $$$k$$$-th stage ends and Alice hasn't lost yet, she wins.Your task is to determine the maximum value of $$$k$$$ such that Alice can win if both players play optimally. Bob plays against Alice, so he tries to make her lose the game, if it's possible.",['games'],"[1, 0, 0, 0, 0, 0, 0, 0]"903"This problem is an extension of the problem ""Wonderful Coloring - 1"". It has quite many differences, so you should read this statement completely.Recently, Paul and Mary have found a new favorite sequence of integers $$$a_1, a_2, \dots, a_n$$$. They want to paint it using pieces of chalk of $$$k$$$ colors. The coloring of a sequence is called wonderful if the following conditions are met:  each element of the sequence is either painted in one of $$$k$$$ colors or isn't painted;  each two elements which are painted in the same color are different (i. e. there's no two equal values painted in the same color);  let's calculate for each of $$$k$$$ colors the number of elements painted in the color — all calculated numbers must be equal;  the total number of painted elements of the sequence is the maximum among all colorings of the sequence which meet the first three conditions. E. g. consider a sequence $$$a=[3, 1, 1, 1, 1, 10, 3, 10, 10, 2]$$$ and $$$k=3$$$. One of the wonderful colorings of the sequence is shown in the figure.    The example of a wonderful coloring of the sequence $$$a=[3, 1, 1, 1, 1, 10, 3, 10, 10, 2]$$$ and $$$k=3$$$. Note that one of the elements isn't painted. Help Paul and Mary to find a wonderful coloring of a given sequence $$$a$$$.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"904Alice and Bob are playing a game. They start with a positive integer $$$n$$$ and take alternating turns doing operations on it. Each turn a player can subtract from $$$n$$$ one of its divisors that isn't $$$1$$$ or $$$n$$$. The player who cannot make a move on his/her turn loses. Alice always moves first.Note that they subtract a divisor of the current number in each turn.You are asked to find out who will win the game if both players play optimally.,"['games', 'math', 'number theory']","[1, 0, 0, 1, 1, 0, 0, 0]"905"There is an automatic door at the entrance of a factory. The door works in the following way:  when one or several people come to the door and it is closed, the door immediately opens automatically and all people immediately come inside,  when one or several people come to the door and it is open, all people immediately come inside,  opened door immediately closes in d seconds after its opening,  if the door is closing and one or several people are coming to the door at the same moment, then all of them will have enough time to enter and only after that the door will close. For example, if d = 3 and four people are coming at four different moments of time t1 = 4, t2 = 7, t3 = 9 and t4 = 13 then the door will open three times: at moments 4, 9 and 13. It will close at moments 7 and 12.It is known that n employees will enter at moments a, 2·a, 3·a, ..., n·a (the value a is positive integer). Also m clients will enter at moments t1, t2, ..., tm.Write program to find the number of times the automatic door will open. Assume that the door is initially closed.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"906"There are n sharks who grow flowers for Wet Shark. They are all sitting around the table, such that sharks i and i + 1 are neighbours for all i from 1 to n - 1. Sharks n and 1 are neighbours too.Each shark will grow some number of flowers si. For i-th shark value si is random integer equiprobably chosen in range from li to ri. Wet Shark has it's favourite prime number p, and he really likes it! If for any pair of neighbouring sharks i and j the product si·sj is divisible by p, then Wet Shark becomes happy and gives 1000 dollars to each of these sharks.At the end of the day sharks sum all the money Wet Shark granted to them. Find the expectation of this value.","['math', 'number theory', 'probabilities']","[0, 0, 0, 1, 1, 1, 0, 0]"907"Little X has n distinct integers: p1, p2, ..., pn. He wants to divide all of them into two sets A and B. The following two conditions must be satisfied:  If number x belongs to set A, then number a - x must also belong to set A.  If number x belongs to set B, then number b - x must also belong to set B. Help Little X divide the numbers into two sets or determine that it's impossible.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"908"Hr0d1y has $$$q$$$ queries on a binary string $$$s$$$ of length $$$n$$$. A binary string is a string containing only characters '0' and '1'.A query is described by a pair of integers $$$l_i$$$, $$$r_i$$$ $$$(1 \leq l_i \lt r_i \leq n)$$$. For each query, he has to determine whether there exists a good subsequence in $$$s$$$ that is equal to the substring $$$s[l_i\ldots r_i]$$$.   A substring $$$s[i\ldots j]$$$ of a string $$$s$$$ is the string formed by characters $$$s_i s_{i+1} \ldots s_j$$$. String $$$a$$$ is said to be a subsequence of string $$$b$$$ if $$$a$$$ can be obtained from $$$b$$$ by deleting some characters without changing the order of the remaining characters. A subsequence is said to be good if it is not contiguous and has length $$$\ge 2$$$. For example, if $$$s$$$ is ""1100110"", then the subsequences $$$s_1s_2s_4$$$ (""1100110"") and $$$s_1s_5s_7$$$ (""1100110"") are good, while $$$s_1s_2s_3$$$ (""1100110"") is not good. Can you help Hr0d1y answer each query?",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"909"Andrew prefers taxi to other means of transport, but recently most taxi drivers have been acting inappropriately. In order to earn more money, taxi drivers started to drive in circles. Roads in Andrew's city are one-way, and people are not necessary able to travel from one part to another, but it pales in comparison to insidious taxi drivers.The mayor of the city decided to change the direction of certain roads so that the taxi drivers wouldn't be able to increase the cost of the trip endlessly. More formally, if the taxi driver is on a certain crossroads, they wouldn't be able to reach it again if he performs a nonzero trip. Traffic controllers are needed in order to change the direction the road goes. For every road it is known how many traffic controllers are needed to change the direction of the road to the opposite one. It is allowed to change the directions of roads one by one, meaning that each traffic controller can participate in reversing two or more roads.You need to calculate the minimum number of traffic controllers that you need to hire to perform the task and the list of the roads that need to be reversed.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"910"Finally! Vasya have come of age and that means he can finally get a passport! To do it, he needs to visit the passport office, but it's not that simple. There's only one receptionist at the passport office and people can queue up long before it actually opens. Vasya wants to visit the passport office tomorrow.He knows that the receptionist starts working after ts minutes have passed after midnight and closes after tf minutes have passed after midnight (so that (tf - 1) is the last minute when the receptionist is still working). The receptionist spends exactly t minutes on each person in the queue. If the receptionist would stop working within t minutes, he stops serving visitors (other than the one he already serves). Vasya also knows that exactly n visitors would come tomorrow. For each visitor Vasya knows the point of time when he would come to the passport office. Each visitor queues up and doesn't leave until he was served. If the receptionist is free when a visitor comes (in particular, if the previous visitor was just served and the queue is empty), the receptionist begins to serve the newcomer immediately.    ""Reception 1"" For each visitor, the point of time when he would come to the passport office is positive. Vasya can come to the office at the time zero (that is, at midnight) if he needs so, but he can come to the office only at integer points of time. If Vasya arrives at the passport office at the same time with several other visitors, he yields to them and stand in the queue after the last of them.Vasya wants to come at such point of time that he will be served by the receptionist, and he would spend the minimum possible time in the queue. Help him!",[],"[0, 0, 0, 0, 0, 0, 0, 0]"911"In Berland there are n cities and n - 1 bidirectional roads. Each road connects some pair of cities, from any city you can get to any other one using only the given roads.In each city there is exactly one repair brigade. To repair some road, you need two teams based in the cities connected by the road to work simultaneously for one day. Both brigades repair one road for the whole day and cannot take part in repairing other roads on that day. But the repair brigade can do nothing on that day.Determine the minimum number of days needed to repair all the roads. The brigades cannot change the cities where they initially are.","['graphs', 'trees']","[0, 0, 1, 0, 0, 0, 0, 1]"912"Om Nom is the main character of a game ""Cut the Rope"". He is a bright little monster who likes visiting friends living at the other side of the park. However the dark old parks can scare even somebody as fearless as Om Nom, so he asks you to help him.  The park consists of 2n + 1 - 1 squares connected by roads so that the scheme of the park is a full binary tree of depth n. More formally, the entrance to the park is located at the square 1. The exits out of the park are located at squares 2n, 2n + 1, ..., 2n + 1 - 1 and these exits lead straight to the Om Nom friends' houses. From each square i (2 ≤ i < 2n + 1) there is a road to the square . Thus, it is possible to go from the park entrance to each of the exits by walking along exactly n roads.    To light the path roads in the evening, the park keeper installed street lights along each road. The road that leads from square i to square  has ai lights.Om Nom loves counting lights on the way to his friend. Om Nom is afraid of spiders who live in the park, so he doesn't like to walk along roads that are not enough lit. What he wants is that the way to any of his friends should have in total the same number of lights. That will make him feel safe. He asked you to help him install additional lights. Determine what minimum number of lights it is needed to additionally place on the park roads so that a path from the entrance to any exit of the park contains the same number of street lights. You may add an arbitrary number of street lights to each of the roads.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"913"You are given $$$q$$$ queries in the following form:Given three integers $$$l_i$$$, $$$r_i$$$ and $$$d_i$$$, find minimum positive integer $$$x_i$$$ such that it is divisible by $$$d_i$$$ and it does not belong to the segment $$$[l_i, r_i]$$$.Can you answer all the queries?Recall that a number $$$x$$$ belongs to segment $$$[l, r]$$$ if $$$l \le x \le r$$$.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"914"You are given $$$n$$$ strings $$$s_1, s_2, \ldots, s_n$$$, each consisting of lowercase and uppercase English letters. In addition, it's guaranteed that each character occurs in each string at most twice. Find the longest common subsequence of these strings.A string $$$t$$$ is a subsequence of a string $$$s$$$ if $$$t$$$ can be obtained from $$$s$$$ by deletion of several (possibly, zero or all) symbols.","['graphs', 'strings']","[0, 0, 1, 0, 0, 0, 1, 0]"915"You are given an array $$$a_1, a_2, \dots a_n$$$. Count the number of pairs of indices $$$1 \leq i, j \leq n$$$ such that $$$a_i < i < a_j < j$$$.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"916"Today there is going to be an unusual performance at the circus — hamsters and tigers will perform together! All of them stand in circle along the arena edge and now the trainer faces a difficult task: he wants to swap the animals' positions so that all the hamsters stood together and all the tigers also stood together. The trainer swaps the animals in pairs not to create a mess. He orders two animals to step out of the circle and swap places. As hamsters feel highly uncomfortable when tigers are nearby as well as tigers get nervous when there's so much potential prey around (consisting not only of hamsters but also of yummier spectators), the trainer wants to spend as little time as possible moving the animals, i.e. he wants to achieve it with the minimal number of swaps. Your task is to help him.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"917"You are playing the following game. There are n points on a plane. They are the vertices of a regular n-polygon. Points are labeled with integer numbers from 1 to n. Each pair of distinct points is connected by a diagonal, which is colored in one of 26 colors. Points are denoted by lowercase English letters. There are three stones positioned on three distinct vertices. All stones are the same. With one move you can move the stone to another free vertex along some diagonal. The color of this diagonal must be the same as the color of the diagonal, connecting another two stones. Your goal is to move stones in such way that the only vertices occupied by stones are 1, 2 and 3. You must achieve such position using minimal number of moves. Write a program which plays this game in an optimal way.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"918"One way to create task is to learn from game. You should pick a game and focus on part of the mechanic of that game, then it might be a good task.Let's have a try. Puzzle and Dragon was a popular game in Japan, we focus on the puzzle part of that game, it is a tile-matching puzzle.(Picture from Wikipedia page: http://en.wikipedia.org/wiki/Puzzle_&_Dragons)There is an n × m board which consists of orbs. During the game you can do the following move. In the beginning of move you touch a cell of the board, then you can move your finger to one of the adjacent cells (a cell not on the boundary has 8 adjacent cells), then you can move your finger from the current cell to one of the adjacent cells one more time, and so on. Each time you move your finger from a cell to another cell, the orbs in these cells swap with each other. In other words whatever move you make, the orb in the cell you are touching never changes.The goal is to achieve such kind of pattern that the orbs will be cancelled and your monster will attack the enemy, but we don't care about these details. Instead, we will give you the initial board as an input and the target board as an output. Your goal is to determine whether there is a way to reach the target in a single move. ",[],"[0, 0, 0, 0, 0, 0, 0, 0]"919"There are n cities and n - 1 roads in the Seven Kingdoms, each road connects two cities and we can reach any city from any other by the roads.Theon and Yara Greyjoy are on a horse in the first city, they are starting traveling through the roads. But the weather is foggy, so they can’t see where the horse brings them. When the horse reaches a city (including the first one), it goes to one of the cities connected to the current city. But it is a strange horse, it only goes to cities in which they weren't before. In each such city, the horse goes with equal probabilities and it stops when there are no such cities. Let the length of each road be 1. The journey starts in the city 1. What is the expected length (expected value of length) of their journey? You can read about expected (average) value by the link https://en.wikipedia.org/wiki/Expected_value.","['graphs', 'probabilities', 'trees']","[0, 0, 1, 0, 0, 1, 0, 1]"920"Polycarp plans to conduct a load testing of its new project Fakebook. He already agreed with his friends that at certain points in time they will send requests to Fakebook. The load testing will last n minutes and in the i-th minute friends will send ai requests.Polycarp plans to test Fakebook under a special kind of load. In case the information about Fakebook gets into the mass media, Polycarp hopes for a monotone increase of the load, followed by a monotone decrease of the interest to the service. Polycarp wants to test this form of load.Your task is to determine how many requests Polycarp must add so that before some moment the load on the server strictly increases and after that moment strictly decreases. Both the increasing part and the decreasing part can be empty (i. e. absent). The decrease should immediately follow the increase. In particular, the load with two equal neigbouring values is unacceptable.For example, if the load is described with one of the arrays [1, 2, 8, 4, 3], [1, 3, 5] or [10], then such load satisfies Polycarp (in each of the cases there is an increasing part, immediately followed with a decreasing part). If the load is described with one of the arrays [1, 2, 2, 1], [2, 1, 2] or [10, 10], then such load does not satisfy Polycarp.Help Polycarp to make the minimum number of additional requests, so that the resulting load satisfies Polycarp. He can make any number of additional requests at any minute from 1 to n.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"921"You are given two rectangles on a plane. The centers of both rectangles are located in the origin of coordinates (meaning the center of the rectangle's symmetry). The first rectangle's sides are parallel to the coordinate axes: the length of the side that is parallel to the Ox axis, equals w, the length of the side that is parallel to the Oy axis, equals h. The second rectangle can be obtained by rotating the first rectangle relative to the origin of coordinates by angle α.  Your task is to find the area of the region which belongs to both given rectangles. This region is shaded in the picture.",['geometry'],"[0, 1, 0, 0, 0, 0, 0, 0]"922"Ramesses came to university to algorithms practice, and his professor, who is a fairly known programmer, gave him the following task.You are given two matrices $$$A$$$ and $$$B$$$ of size $$$n \times m$$$, each of which consists of $$$0$$$ and $$$1$$$ only. You can apply the following operation to the matrix $$$A$$$ arbitrary number of times: take any submatrix of the matrix $$$A$$$ that has at least two rows and two columns, and invert the values in its corners (i.e. all corners of the submatrix that contain $$$0$$$, will be replaced by $$$1$$$, and all corners of the submatrix that contain $$$1$$$, will be replaced by $$$0$$$). You have to answer whether you can obtain the matrix $$$B$$$ from the matrix $$$A$$$.  An example of the operation. The chosen submatrix is shown in blue and yellow, its corners are shown in yellow. Ramesses don't want to perform these operations by himself, so he asks you to answer this question.A submatrix of matrix $$$M$$$ is a matrix which consist of all elements which come from one of the rows with indices $$$x_1, x_1+1, \ldots, x_2$$$ of matrix $$$M$$$ and one of the columns with indices $$$y_1, y_1+1, \ldots, y_2$$$ of matrix $$$M$$$, where $$$x_1, x_2, y_1, y_2$$$ are the edge rows and columns of the submatrix. In other words, a submatrix is a set of elements of source matrix which form a solid rectangle (i.e. without holes) with sides parallel to the sides of the original matrix. The corners of the submatrix are cells $$$(x_1, y_1)$$$, $$$(x_1, y_2)$$$, $$$(x_2, y_1)$$$, $$$(x_2, y_2)$$$, where the cell $$$(i,j)$$$ denotes the cell on the intersection of the $$$i$$$-th row and the $$$j$$$-th column.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"923"There is a street with $$$n$$$ houses in a line, numbered from $$$1$$$ to $$$n$$$. The house $$$i$$$ is initially painted in color $$$c_i$$$. The street is considered beautiful if all houses are painted in the same color. Tom, the painter, is in charge of making the street beautiful. Tom's painting capacity is defined by an integer, let's call it $$$k$$$.On one day, Tom can do the following repainting process that consists of two steps:   He chooses two integers $$$l$$$ and $$$r$$$ such that $$$ 1 \le l \le r \le n $$$ and $$$ r - l + 1 = k $$$.  For each house $$$i$$$ such that $$$l \le i \le r$$$, he can either repaint it with any color he wants, or ignore it and let it keep its current color. Note that in the same day Tom can use different colors to repaint different houses.Tom wants to know the minimum number of days needed to repaint the street so that it becomes beautiful.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"924"You are given an array $$$a$$$ of $$$n$$$ non-negative integers, numbered from $$$1$$$ to $$$n$$$.Let's define the cost of the array $$$a$$$ as $$$\displaystyle \min_{i \neq j} a_i | a_j$$$, where $$$|$$$ denotes the bitwise OR operation.There are $$$q$$$ queries. For each query you are given two integers $$$l$$$ and $$$r$$$ ($$$l < r$$$). For each query you should find the cost of the subarray $$$a_{l}, a_{l + 1}, \ldots, a_{r}$$$.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"925"Let us call two integers $$$x$$$ and $$$y$$$ adjacent if $$$\frac{lcm(x, y)}{gcd(x, y)}$$$ is a perfect square. For example, $$$3$$$ and $$$12$$$ are adjacent, but $$$6$$$ and $$$9$$$ are not.Here $$$gcd(x, y)$$$ denotes the greatest common divisor (GCD) of integers $$$x$$$ and $$$y$$$, and $$$lcm(x, y)$$$ denotes the least common multiple (LCM) of integers $$$x$$$ and $$$y$$$.You are given an array $$$a$$$ of length $$$n$$$. Each second the following happens: each element $$$a_i$$$ of the array is replaced by the product of all elements of the array (including itself), that are adjacent to the current value. Let $$$d_i$$$ be the number of adjacent elements to $$$a_i$$$ (including $$$a_i$$$ itself). The beauty of the array is defined as $$$\max_{1 \le i \le n} d_i$$$. You are given $$$q$$$ queries: each query is described by an integer $$$w$$$, and you have to output the beauty of the array after $$$w$$$ seconds.","['graphs', 'math', 'number theory']","[0, 0, 1, 1, 1, 0, 0, 0]"926"You play a strategic video game (yeah, we ran out of good problem legends). In this game you control a large army, and your goal is to conquer $$$n$$$ castles of your opponent.Let's describe the game process in detail. Initially you control an army of $$$k$$$ warriors. Your enemy controls $$$n$$$ castles; to conquer the $$$i$$$-th castle, you need at least $$$a_i$$$ warriors (you are so good at this game that you don't lose any warriors while taking over a castle, so your army stays the same after the fight). After you take control over a castle, you recruit new warriors into your army — formally, after you capture the $$$i$$$-th castle, $$$b_i$$$ warriors join your army. Furthermore, after capturing a castle (or later) you can defend it: if you leave at least one warrior in a castle, this castle is considered defended. Each castle has an importance parameter $$$c_i$$$, and your total score is the sum of importance values over all defended castles. There are two ways to defend a castle:  if you are currently in the castle $$$i$$$, you may leave one warrior to defend castle $$$i$$$;  there are $$$m$$$ one-way portals connecting the castles. Each portal is characterised by two numbers of castles $$$u$$$ and $$$v$$$ (for each portal holds $$$u > v$$$). A portal can be used as follows: if you are currently in the castle $$$u$$$, you may send one warrior to defend castle $$$v$$$. Obviously, when you order your warrior to defend some castle, he leaves your army.You capture the castles in fixed order: you have to capture the first one, then the second one, and so on. After you capture the castle $$$i$$$ (but only before capturing castle $$$i + 1$$$) you may recruit new warriors from castle $$$i$$$, leave a warrior to defend castle $$$i$$$, and use any number of portals leading from castle $$$i$$$ to other castles having smaller numbers. As soon as you capture the next castle, these actions for castle $$$i$$$ won't be available to you.If, during some moment in the game, you don't have enough warriors to capture the next castle, you lose. Your goal is to maximize the sum of importance values over all defended castles (note that you may hire new warriors in the last castle, defend it and use portals leading from it even after you capture it — your score will be calculated afterwards).Can you determine an optimal strategy of capturing and defending the castles?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"927"Levko loves tables that consist of n rows and n columns very much. He especially loves beautiful tables. A table is beautiful to Levko if the sum of elements in each row and column of the table equals k.Unfortunately, he doesn't know any such table. Your task is to help him to find at least one of them. ",[],"[0, 0, 0, 0, 0, 0, 0, 0]"928"You are wandering in the explorer space of the 2050 Conference.The explorer space can be viewed as an undirected weighted grid graph with size $$$n\times m$$$. The set of vertices is $$$\{(i, j)|1\le i\le n, 1\le j\le m\}$$$. Two vertices $$$(i_1,j_1)$$$ and $$$(i_2, j_2)$$$ are connected by an edge if and only if $$$|i_1-i_2|+|j_1-j_2|=1$$$.At each step, you can walk to any vertex connected by an edge with your current vertex. On each edge, there are some number of exhibits. Since you already know all the exhibits, whenever you go through an edge containing $$$x$$$ exhibits, your boredness increases by $$$x$$$.For each starting vertex $$$(i, j)$$$, please answer the following question: What is the minimum possible boredness if you walk from $$$(i, j)$$$ and go back to it after exactly $$$k$$$ steps?You can use any edge for multiple times but the boredness on those edges are also counted for multiple times. At each step, you cannot stay on your current vertex. You also cannot change direction while going through an edge. Before going back to your starting vertex $$$(i, j)$$$ after $$$k$$$ steps, you can visit $$$(i, j)$$$ (or not) freely.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"929"This is the hard version of this problem. The only difference is the constraint on $$$k$$$ — the number of gifts in the offer. In this version: $$$2 \le k \le n$$$.Vasya came to the store to buy goods for his friends for the New Year. It turned out that he was very lucky — today the offer ""$$$k$$$ of goods for the price of one"" is held in store.Using this offer, Vasya can buy exactly $$$k$$$ of any goods, paying only for the most expensive of them. Vasya decided to take this opportunity and buy as many goods as possible for his friends with the money he has.More formally, for each good, its price is determined by $$$a_i$$$ — the number of coins it costs. Initially, Vasya has $$$p$$$ coins. He wants to buy the maximum number of goods. Vasya can perform one of the following operations as many times as necessary:  Vasya can buy one good with the index $$$i$$$ if he currently has enough coins (i.e $$$p \ge a_i$$$). After buying this good, the number of Vasya's coins will decrease by $$$a_i$$$, (i.e it becomes $$$p := p - a_i$$$).  Vasya can buy a good with the index $$$i$$$, and also choose exactly $$$k-1$$$ goods, the price of which does not exceed $$$a_i$$$, if he currently has enough coins (i.e $$$p \ge a_i$$$). Thus, he buys all these $$$k$$$ goods, and his number of coins decreases by $$$a_i$$$ (i.e it becomes $$$p := p - a_i$$$). Please note that each good can be bought no more than once.For example, if the store now has $$$n=5$$$ goods worth $$$a_1=2, a_2=4, a_3=3, a_4=5, a_5=7$$$, respectively, $$$k=2$$$, and Vasya has $$$6$$$ coins, then he can buy $$$3$$$ goods. A good with the index $$$1$$$ will be bought by Vasya without using the offer and he will pay $$$2$$$ coins. Goods with the indices $$$2$$$ and $$$3$$$ Vasya will buy using the offer and he will pay $$$4$$$ coins. It can be proved that Vasya can not buy more goods with six coins.Help Vasya to find out the maximum number of goods he can buy.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"930"The main characters have been omitted to be short.You are given a directed unweighted graph without loops with $$$n$$$ vertexes and a path in it (that path is not necessary simple) given by a sequence $$$p_1, p_2, \ldots, p_m$$$ of $$$m$$$ vertexes; for each $$$1 \leq i < m$$$ there is an arc from $$$p_i$$$ to $$$p_{i+1}$$$.Define the sequence $$$v_1, v_2, \ldots, v_k$$$ of $$$k$$$ vertexes as good, if $$$v$$$ is a subsequence of $$$p$$$, $$$v_1 = p_1$$$, $$$v_k = p_m$$$, and $$$p$$$ is one of the shortest paths passing through the vertexes $$$v_1$$$, $$$\ldots$$$, $$$v_k$$$ in that order.A sequence $$$a$$$ is a subsequence of a sequence $$$b$$$ if $$$a$$$ can be obtained from $$$b$$$ by deletion of several (possibly, zero or all) elements. It is obvious that the sequence $$$p$$$ is good but your task is to find the shortest good subsequence.If there are multiple shortest good subsequences, output any of them. ",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"931"В самолёте есть n рядов мест. Если смотреть на ряды сверху, то в каждом ряду есть 3 места слева, затем проход между рядами, затем 4 центральных места, затем ещё один проход между рядами, а затем ещё 3 места справа.Известно, что некоторые места уже заняты пассажирами. Всего есть два вида пассажиров — статусные (те, которые часто летают) и обычные. Перед вами стоит задача рассадить ещё k обычных пассажиров так, чтобы суммарное число соседей у статусных пассажиров было минимально возможным. Два пассажира считаются соседями, если они сидят в одном ряду и между ними нет других мест и прохода между рядами. Если пассажир является соседним пассажиром для двух статусных пассажиров, то его следует учитывать в сумме соседей дважды.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"932"This is the easy version of the problem. The difference between the versions is the constraint on $$$n$$$ and the required number of operations. You can make hacks only if all versions of the problem are solved.There are two binary strings $$$a$$$ and $$$b$$$ of length $$$n$$$ (a binary string is a string consisting of symbols $$$0$$$ and $$$1$$$). In an operation, you select a prefix of $$$a$$$, and simultaneously invert the bits in the prefix ($$$0$$$ changes to $$$1$$$ and $$$1$$$ changes to $$$0$$$) and reverse the order of the bits in the prefix.For example, if $$$a=001011$$$ and you select the prefix of length $$$3$$$, it becomes $$$011011$$$. Then if you select the entire string, it becomes $$$001001$$$.Your task is to transform the string $$$a$$$ into $$$b$$$ in at most $$$3n$$$ operations. It can be proved that it is always possible.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"933"You're given an array $$$a_1, \ldots, a_n$$$ of $$$n$$$ non-negative integers.Let's call it sharpened if and only if there exists an integer $$$1 \le k \le n$$$ such that $$$a_1 < a_2 < \ldots < a_k$$$ and $$$a_k > a_{k+1} > \ldots > a_n$$$. In particular, any strictly increasing or strictly decreasing array is sharpened. For example:  The arrays $$$[4]$$$, $$$[0, 1]$$$, $$$[12, 10, 8]$$$ and $$$[3, 11, 15, 9, 7, 4]$$$ are sharpened;  The arrays $$$[2, 8, 2, 8, 6, 5]$$$, $$$[0, 1, 1, 0]$$$ and $$$[2, 5, 6, 9, 8, 8]$$$ are not sharpened. You can do the following operation as many times as you want: choose any strictly positive element of the array, and decrease it by one. Formally, you can choose any $$$i$$$ ($$$1 \le i \le n$$$) such that $$$a_i>0$$$ and assign $$$a_i := a_i - 1$$$.Tell if it's possible to make the given array sharpened using some number (possibly zero) of these operations.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"934"As meticulous Gerald sets the table and caring Alexander sends the postcards, Sergey makes snowmen. Each showman should consist of three snowballs: a big one, a medium one and a small one. Sergey's twins help him: they've already made n snowballs with radii equal to r1, r2, ..., rn. To make a snowman, one needs any three snowballs whose radii are pairwise different. For example, the balls with radii 1, 2 and 3 can be used to make a snowman but 2, 2, 3 or 2, 2, 2 cannot. Help Sergey and his twins to determine what maximum number of snowmen they can make from those snowballs.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"935"After a hard-working week Polycarp prefers to have fun. Polycarp's favorite entertainment is drawing snakes. He takes a rectangular checkered sheet of paper of size $$$n \times m$$$ (where $$$n$$$ is the number of rows, $$$m$$$ is the number of columns) and starts to draw snakes in cells.Polycarp draws snakes with lowercase Latin letters. He always draws the first snake with the symbol 'a', the second snake with the symbol 'b', the third snake with the symbol 'c' and so on. All snakes have their own unique symbol. There are only $$$26$$$ letters in the Latin alphabet, Polycarp is very tired and he doesn't want to invent new symbols, so the total number of drawn snakes doesn't exceed $$$26$$$.Since by the end of the week Polycarp is very tired, he draws snakes as straight lines without bends. So each snake is positioned either vertically or horizontally. Width of any snake equals $$$1$$$, i.e. each snake has size either $$$1 \times l$$$ or $$$l \times 1$$$, where $$$l$$$ is snake's length. Note that snakes can't bend.When Polycarp draws a new snake, he can use already occupied cells for drawing the snake. In this situation, he draws the snake ""over the top"" and overwrites the previous value in the cell.Recently when Polycarp was at work he found a checkered sheet of paper with Latin letters. He wants to know if it is possible to get this sheet of paper from an empty sheet by drawing some snakes according to the rules described above. If it is possible, he is interested in a way to draw snakes.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"936"A system of n vessels with water is given. Several pairs of vessels are connected by tubes with transfusion mechanisms. One may transfer an integer amount of liters of water between two vessels connected by such tube (tube works in both directions). There might be multiple tubes between two vessels. Total number of tubes equals e. Volume of each vessel equals v liters. Of course, the amount of the water in any vessel cannot exceed v liters in the process of transfusions.Given the initial amounts ai of water in the vessels and the desired amounts bi find a sequence of transfusions that deals with the task. Total number of transfusions must not exceed 2·n2.","['graphs', 'trees']","[0, 0, 1, 0, 0, 0, 0, 1]"937"No Great Victory anniversary in Berland has ever passed without the war parade. This year is not an exception. That’s why the preparations are on in full strength. Tanks are building a line, artillery mounts are ready to fire, soldiers are marching on the main square... And the air forces general Mr. Generalov is in trouble again. This year a lot of sky-scrapers have been built which makes it difficult for the airplanes to fly above the city. It was decided that the planes should fly strictly from south to north. Moreover, there must be no sky scraper on a plane’s route, otherwise the anniversary will become a tragedy. The Ministry of Building gave the data on n sky scrapers (the rest of the buildings are rather small and will not be a problem to the planes). When looking at the city from south to north as a geometrical plane, the i-th building is a rectangle of height hi. Its westernmost point has the x-coordinate of li and the easternmost — of ri. The terrain of the area is plain so that all the buildings stand on one level. Your task as the Ministry of Defence’s head programmer is to find an enveloping polyline using the data on the sky-scrapers. The polyline’s properties are as follows:  If you look at the city from south to north as a plane, then any part of any building will be inside or on the boarder of the area that the polyline encloses together with the land surface.  The polyline starts and ends on the land level, i.e. at the height equal to 0.  The segments of the polyline are parallel to the coordinate axes, i.e. they can only be vertical or horizontal.  The polyline’s vertices should have integer coordinates.  If you look at the city from south to north the polyline (together with the land surface) must enclose the minimum possible area.  The polyline must have the smallest length among all the polylines, enclosing the minimum possible area with the land.  The consecutive segments of the polyline must be perpendicular.    Picture to the second sample test (the enveloping polyline is marked on the right). ",[],"[0, 0, 0, 0, 0, 0, 0, 0]"938"In the school computer room there are n servers which are responsible for processing several computing tasks. You know the number of scheduled tasks for each server: there are mi tasks assigned to the i-th server.In order to balance the load for each server, you want to reassign some tasks to make the difference between the most loaded server and the least loaded server as small as possible. In other words you want to minimize expression ma - mb, where a is the most loaded server and b is the least loaded one.In one second you can reassign a single task. Thus in one second you can choose any pair of servers and move a single task from one server to another.Write a program to find the minimum number of seconds needed to balance the load of servers.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"939"Let A = {a1, a2, ..., an} be any permutation of the first n natural numbers {1, 2, ..., n}. You are given a positive integer k and another sequence B = {b1, b2, ..., bn}, where bi is the number of elements aj in A to the left of the element at = i such that aj ≥ (i + k).For example, if n = 5, a possible A is {5, 1, 4, 2, 3}. For k = 2, B is given by {1, 2, 1, 0, 0}. But if k = 3, then B = {1, 1, 0, 0, 0}.For two sequences X = {x1, x2, ..., xn} and Y = {y1, y2, ..., yn}, let i-th elements be the first elements such that xi ≠ yi. If xi < yi, then X is lexicographically smaller than Y, while if xi > yi, then X is lexicographically greater than Y.Given n, k and B, you need to determine the lexicographically smallest A.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"940"There are n incoming messages for Vasya. The i-th message is going to be received after ti minutes. Each message has a cost, which equals to A initially. After being received, the cost of a message decreases by B each minute (it can become negative). Vasya can read any message after receiving it at any moment of time. After reading the message, Vasya's bank account receives the current cost of this message. Initially, Vasya's bank account is at 0.Also, each minute Vasya's bank account receives C·k, where k is the amount of received but unread messages.Vasya's messages are very important to him, and because of that he wants to have all messages read after T minutes.Determine the maximum amount of money Vasya's bank account can hold after T minutes.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"941"While looking at the kitchen fridge, the little boy Tyler noticed magnets with symbols, that can be aligned into a string $$$s$$$.Tyler likes strings, and especially those that are lexicographically smaller than another string, $$$t$$$. After playing with magnets on the fridge, he is wondering, how many distinct strings can be composed out of letters of string $$$s$$$ by rearranging them, so that the resulting string is lexicographically smaller than the string $$$t$$$? Tyler is too young, so he can't answer this question. The alphabet Tyler uses is very large, so for your convenience he has already replaced the same letters in $$$s$$$ and $$$t$$$ to the same integers, keeping that different letters have been replaced to different integers.We call a string $$$x$$$ lexicographically smaller than a string $$$y$$$ if one of the followings conditions is fulfilled:   There exists such position of symbol $$$m$$$ that is presented in both strings, so that before $$$m$$$-th symbol the strings are equal, and the $$$m$$$-th symbol of string $$$x$$$ is smaller than $$$m$$$-th symbol of string $$$y$$$.  String $$$x$$$ is the prefix of string $$$y$$$ and $$$x \neq y$$$. Because the answer can be too large, print it modulo $$$998\,244\,353$$$.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"942"This is an interactive problem.In good old times dwarves tried to develop extrasensory abilities:  Exactly n dwarves entered completely dark cave.  Each dwarf received a hat — white or black. While in cave, none of the dwarves was able to see either his own hat or hats of other Dwarves.  Dwarves went out of the cave to the meadow and sat at an arbitrary place one after the other. When a dwarf leaves the cave, he sees the colors of all hats of all dwarves that are seating on the meadow (i.e. left the cave before him). However, he is not able to see the color of his own hat and none of the dwarves can give him this information.  The task for dwarves was to got diverged into two parts — one with dwarves with white hats and one with black hats. After many centuries, dwarves finally managed to select the right place on the meadow without error. Will you be able to repeat their success?You are asked to successively name n different integer points on the plane. After naming each new point you will be given its color — black or white. Your task is to ensure that the named points can be split by a line in such a way that all points of one color lie on the same side from the line and points of different colors lie on different sides. Moreover, no points can belong to the line. Also, you need to report any such line at the end of the process.In this problem, the interactor is adaptive — the colors of the points in the tests are not fixed beforehand and the jury program can select them arbitrarily, in particular, depending on your program output.",['geometry'],"[0, 1, 0, 0, 0, 0, 0, 0]"943"You are given two huge binary integer numbers $$$a$$$ and $$$b$$$ of lengths $$$n$$$ and $$$m$$$ respectively. You will repeat the following process: if $$$b > 0$$$, then add to the answer the value $$$a~ \&~ b$$$ and divide $$$b$$$ by $$$2$$$ rounding down (i.e. remove the last digit of $$$b$$$), and repeat the process again, otherwise stop the process.The value $$$a~ \&~ b$$$ means bitwise AND of $$$a$$$ and $$$b$$$. Your task is to calculate the answer modulo $$$998244353$$$.Note that you should add the value $$$a~ \&~ b$$$ to the answer in decimal notation, not in binary. So your task is to calculate the answer in decimal notation. For example, if $$$a = 1010_2~ (10_{10})$$$ and $$$b = 1000_2~ (8_{10})$$$, then the value $$$a~ \&~ b$$$ will be equal to $$$8$$$, not to $$$1000$$$.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"944"Theofanis really likes sequences of positive integers, thus his teacher (Yeltsa Kcir) gave him a problem about a sequence that consists of only special numbers.Let's call a positive number special if it can be written as a sum of different non-negative powers of $$$n$$$. For example, for $$$n = 4$$$ number $$$17$$$ is special, because it can be written as $$$4^0 + 4^2 = 1 + 16 = 17$$$, but $$$9$$$ is not.Theofanis asks you to help him find the $$$k$$$-th special number if they are sorted in increasing order. Since this number may be too large, output it modulo $$$10^9+7$$$.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"945"You are given a binary string $$$s$$$ (recall that a string is binary if each character is either $$$0$$$ or $$$1$$$).Let $$$f(t)$$$ be the decimal representation of integer $$$t$$$ written in binary form (possibly with leading zeroes). For example $$$f(011) = 3, f(00101) = 5, f(00001) = 1, f(10) = 2, f(000) = 0$$$ and $$$f(000100) = 4$$$.The substring $$$s_{l}, s_{l+1}, \dots , s_{r}$$$ is good if $$$r - l + 1 = f(s_l \dots s_r)$$$.For example string $$$s = 1011$$$ has $$$5$$$ good substrings: $$$s_1 \dots s_1 = 1$$$, $$$s_3 \dots s_3 = 1$$$, $$$s_4 \dots s_4 = 1$$$, $$$s_1 \dots s_2 = 10$$$ and $$$s_2 \dots s_4 = 011$$$. Your task is to calculate the number of good substrings of string $$$s$$$.You have to answer $$$t$$$ independent queries.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"946"Ziota found a video game called ""Monster Invaders"".Similar to every other shooting RPG game, ""Monster Invaders"" involves killing monsters and bosses with guns.For the sake of simplicity, we only consider two different types of monsters and three different types of guns.Namely, the two types of monsters are:   a normal monster with $$$1$$$ hp.  a boss with $$$2$$$ hp. And the three types of guns are:   Pistol, deals $$$1$$$ hp in damage to one monster, $$$r_1$$$ reloading time  Laser gun, deals $$$1$$$ hp in damage to all the monsters in the current level (including the boss), $$$r_2$$$ reloading time  AWP, instantly kills any monster, $$$r_3$$$ reloading time The guns are initially not loaded, and the Ziota can only reload 1 gun at a time.The levels of the game can be considered as an array $$$a_1, a_2, \ldots, a_n$$$, in which the $$$i$$$-th stage has $$$a_i$$$ normal monsters and 1 boss. Due to the nature of the game, Ziota cannot use the Pistol (the first type of gun) or AWP (the third type of gun) to shoot the boss before killing all of the $$$a_i$$$ normal monsters.If Ziota damages the boss but does not kill it immediately, he is forced to move out of the current level to an arbitrary adjacent level (adjacent levels of level $$$i$$$ $$$(1 < i < n)$$$ are levels $$$i - 1$$$ and $$$i + 1$$$, the only adjacent level of level $$$1$$$ is level $$$2$$$, the only adjacent level of level $$$n$$$ is level $$$n - 1$$$). Ziota can also choose to move to an adjacent level at any time. Each move between adjacent levels are managed by portals with $$$d$$$ teleportation time.In order not to disrupt the space-time continuum within the game, it is strictly forbidden to reload or shoot monsters during teleportation. Ziota starts the game at level 1. The objective of the game is rather simple, to kill all the bosses in all the levels. He is curious about the minimum time to finish the game (assuming it takes no time to shoot the monsters with a loaded gun and Ziota has infinite ammo on all the three guns). Please help him find this value.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"947"Recently Duff has been a soldier in the army. Malek is her commander.Their country, Andarz Gu has n cities (numbered from 1 to n) and n - 1 bidirectional roads. Each road connects two different cities. There exist a unique path between any two cities.There are also m people living in Andarz Gu (numbered from 1 to m). Each person has and ID number. ID number of i - th person is i and he/she lives in city number ci. Note that there may be more than one person in a city, also there may be no people living in the city.  Malek loves to order. That's why he asks Duff to answer to q queries. In each query, he gives her numbers v, u and a.To answer a query:Assume there are x people living in the cities lying on the path from city v to city u. Assume these people's IDs are p1, p2, ..., px in increasing order. If k = min(x, a), then Duff should tell Malek numbers k, p1, p2, ..., pk in this order. In the other words, Malek wants to know a minimums on that path (or less, if there are less than a people).Duff is very busy at the moment, so she asked you to help her and answer the queries.",['trees'],"[0, 0, 0, 0, 0, 0, 0, 1]"948"You have a card deck of $$$n$$$ cards, numbered from top to bottom, i. e. the top card has index $$$1$$$ and bottom card — index $$$n$$$. Each card has its color: the $$$i$$$-th card has color $$$a_i$$$.You should process $$$q$$$ queries. The $$$j$$$-th query is described by integer $$$t_j$$$. For each query you should:   find the highest card in the deck with color $$$t_j$$$, i. e. the card with minimum index;  print the position of the card you found;  take the card and place it on top of the deck. ",['trees'],"[0, 0, 0, 0, 0, 0, 0, 1]"949"You are given an undirected graph without self-loops or multiple edges which consists of $$$n$$$ vertices and $$$m$$$ edges. Also you are given three integers $$$n_1$$$, $$$n_2$$$ and $$$n_3$$$.Can you label each vertex with one of three numbers 1, 2 or 3 in such way, that:   Each vertex should be labeled by exactly one number 1, 2 or 3;  The total number of vertices with label 1 should be equal to $$$n_1$$$;  The total number of vertices with label 2 should be equal to $$$n_2$$$;  The total number of vertices with label 3 should be equal to $$$n_3$$$;  $$$|col_u - col_v| = 1$$$ for each edge $$$(u, v)$$$, where $$$col_x$$$ is the label of vertex $$$x$$$. If there are multiple valid labelings, print any of them.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"950"The Super Duper Secret Meeting of the Super Duper Secret Military Squad takes place in a Super Duper Secret Place. The place is an infinite plane with introduced Cartesian coordinate system. The meeting table is represented as a rectangle whose sides are parallel to the coordinate axes and whose vertexes are located at the integer points of the plane. At each integer point which belongs to the table perimeter there is a chair in which a general sits.Some points on the plane contain radiators for the generals not to freeze in winter. Each radiator is characterized by the number ri — the radius of the area this radiator can heat. That is, if the distance between some general and the given radiator is less than or equal to ri, than the general feels comfortable and warm. Here distance is defined as Euclidean distance, so the distance between points (x1, y1) and (x2, y2) is Each general who is located outside the radiators' heating area can get sick. Thus, you should bring him a warm blanket. Your task is to count the number of warm blankets you should bring to the Super Duper Secret Place.The generals who are already comfortable do not need a blanket. Also the generals never overheat, ever if they are located in the heating area of several radiators. The radiators can be located at any integer points on the plane, even inside the rectangle (under the table) or on the perimeter (directly under some general). Even in this case their radius does not change.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"951"The Bytelandian Institute for Biological Research (BIBR) is investigating the properties of two species of bacteria, named simply 0 and 1. Even under a microscope, bacteria of those two species are very difficult to distinguish. In fact, the only thing the scientists possess that is able to differentiate between them is a plant called Formurosa.If the scientists place a sample of colonies of bacteria on each on Formurosa's leaves, it will activate a complicated nutrition process. During that process color of Formurosa changes to reflect the result of a — possibly very complicated — logical formula on the species of bacteria, involving constants and the operators | (OR), & (AND) and ^ (XOR). If it is 0, the plant will turn red, otherwise — it will turn blue.For example, if the nutrition process of Formurosa is described by the formula: (((?^?)|?)&(1^?)); then Formurosa has four leaves (the ""?"" signs denote the leaves). If we place 0, 1, 0, 0 on the respective leaves, the result of the nutrition process will be (((0^1)|0)&(1^0)) = 1, therefore the plant will turn blue.The scientists have n colonies of bacteria. They do not know their types; the only thing they know for sure is that not all colonies are of the same type. They want to attempt to determine the bacteria's species by repeated evaluations with Formurosa. During each evaluation they must place exactly one sample on every leaf of the plant. However, they may use multiple samples of one colony during a single evaluation; they can even cover the whole plant with bacteria from one colony!Is it possible for them to always determine the species of each colony, no matter what they are (assuming they are not all the same)?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"952"This problem is given in two versions that differ only by constraints. If you can solve this problem in large constraints, then you can just write a single solution to the both versions. If you find the problem too difficult in large constraints, you can write solution to the simplified version only.Waking up in the morning, Apollinaria decided to bake cookies. To bake one cookie, she needs n ingredients, and for each ingredient she knows the value ai — how many grams of this ingredient one needs to bake a cookie. To prepare one cookie Apollinaria needs to use all n ingredients.Apollinaria has bi gram of the i-th ingredient. Also she has k grams of a magic powder. Each gram of magic powder can be turned to exactly 1 gram of any of the n ingredients and can be used for baking cookies.Your task is to determine the maximum number of cookies, which Apollinaria is able to bake using the ingredients that she has and the magic powder.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"953"Mikhail walks on a Cartesian plane. He starts at the point $$$(0, 0)$$$, and in one move he can go to any of eight adjacent points. For example, if Mikhail is currently at the point $$$(0, 0)$$$, he can go to any of the following points in one move:   $$$(1, 0)$$$;  $$$(1, 1)$$$;  $$$(0, 1)$$$;  $$$(-1, 1)$$$;  $$$(-1, 0)$$$;  $$$(-1, -1)$$$;  $$$(0, -1)$$$;  $$$(1, -1)$$$. If Mikhail goes from the point $$$(x1, y1)$$$ to the point $$$(x2, y2)$$$ in one move, and $$$x1 \ne x2$$$ and $$$y1 \ne y2$$$, then such a move is called a diagonal move.Mikhail has $$$q$$$ queries. For the $$$i$$$-th query Mikhail's target is to go to the point $$$(n_i, m_i)$$$ from the point $$$(0, 0)$$$ in exactly $$$k_i$$$ moves. Among all possible movements he want to choose one with the maximum number of diagonal moves. Your task is to find the maximum number of diagonal moves or find that it is impossible to go from the point $$$(0, 0)$$$ to the point $$$(n_i, m_i)$$$ in $$$k_i$$$ moves.Note that Mikhail can visit any point any number of times (even the destination point!).",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"954"Polycarp came up with a new programming language. There are only two types of statements in it:   ""x := s"": assign the variable named x the value s (where s is a string). For example, the statement var := hello assigns the variable named var the value hello. Note that s is the value of a string, not the name of a variable. Between the variable name, the := operator and the string contains exactly one space each. ""x = a + b"": assign the variable named x the concatenation of values of two variables a and b. For example, if the program consists of three statements a := hello, b := world, c = a + b, then the variable c will contain the string helloworld. It is guaranteed that the program is correct and the variables a and b were previously defined. There is exactly one space between the variable names and the = and + operators. All variable names and strings only consist of lowercase letters of the English alphabet and do not exceed $$$5$$$ characters.The result of the program is the number of occurrences of string haha in the string that was written to the variable in the last statement.Polycarp was very tired while inventing that language. He asks you to implement it. Your task is — for given program statements calculate the number of occurrences of string haha in the last assigned variable.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"955"An IPv6-address is a 128-bit number. For convenience, this number is recorded in blocks of 16 bits in hexadecimal record, the blocks are separated by colons — 8 blocks in total, each block has four hexadecimal digits. Here is an example of the correct record of a IPv6 address: ""0124:5678:90ab:cdef:0124:5678:90ab:cdef"". We'll call such format of recording an IPv6-address full.Besides the full record of an IPv6 address there is a short record format. The record of an IPv6 address can be shortened by removing one or more leading zeroes at the beginning of each block. However, each block should contain at least one digit in the short format. For example, the leading zeroes can be removed like that: ""a56f:00d3:0000:0124:0001:f19a:1000:0000""  →  ""a56f:d3:0:0124:01:f19a:1000:00"". There are more ways to shorten zeroes in this IPv6 address.Some IPv6 addresses contain long sequences of zeroes. Continuous sequences of 16-bit zero blocks can be shortened to ""::"". A sequence can consist of one or several consecutive blocks, with all 16 bits equal to 0. You can see examples of zero block shortenings below:  ""a56f:00d3:0000:0124:0001:0000:0000:0000""  →  ""a56f:00d3:0000:0124:0001::"";  ""a56f:0000:0000:0124:0001:0000:1234:0ff0""  →  ""a56f::0124:0001:0000:1234:0ff0"";  ""a56f:0000:0000:0000:0001:0000:1234:0ff0""  →  ""a56f:0000::0000:0001:0000:1234:0ff0"";  ""a56f:00d3:0000:0124:0001:0000:0000:0000""  →  ""a56f:00d3:0000:0124:0001::0000"";  ""0000:0000:0000:0000:0000:0000:0000:0000""  →  ""::"". It is not allowed to shorten zero blocks in the address more than once. This means that the short record can't contain the sequence of characters ""::"" more than once. Otherwise, it will sometimes be impossible to determine the number of zero blocks, each represented by a double colon.The format of the record of the IPv6 address after removing the leading zeroes and shortening the zero blocks is called short.You've got several short records of IPv6 addresses. Restore their full record.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"956"There are two main kinds of events in the life of top-model: fashion shows and photo shoots. Participating in any of these events affects the rating of appropriate top-model. After each photo shoot model's rating increases by a and after each fashion show decreases by b (designers do too many experiments nowadays). Moreover, sometimes top-models participates in talk shows. After participating in talk show model becomes more popular and increasing of her rating after photo shoots become c and decreasing of her rating after fashion show becomes d.Izabella wants to participate in a talk show, but she wants to do it in such a way that her rating will never become negative. Help her to find a suitable moment for participating in the talk show. Let's assume that model's career begins in moment 0. At that moment Izabella's rating was equal to start. If talk show happens in moment t if will affect all events in model's life in interval of time [t..t + len) (including t and not including t + len), where len is duration of influence.Izabella wants to participate in a talk show, but she wants to do it in such a way that her rating will not become become negative before talk show or during period of influence of talk show. Help her to find a suitable moment for participating in the talk show. ",[],"[0, 0, 0, 0, 0, 0, 0, 0]"957"Vasya had a strictly increasing sequence of positive integers a1, ..., an. Vasya used it to build a new sequence b1, ..., bn, where bi is the sum of digits of ai's decimal representation. Then sequence ai got lost and all that remained is sequence bi.Vasya wonders what the numbers ai could be like. Of all the possible options he likes the one sequence with the minimum possible last number an. Help Vasya restore the initial sequence.It is guaranteed that such a sequence always exists.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"958"You are given n points on Cartesian plane. Every point is a lattice point (i. e. both of its coordinates are integers), and all points are distinct.You may draw two straight lines (not necessarily distinct). Is it possible to do this in such a way that every point lies on at least one of these lines?",['geometry'],"[0, 1, 0, 0, 0, 0, 0, 0]"959"This is the hard version of this problem. The difference between easy and hard versions is only the constraints on $$$a_i$$$ and on $$$n$$$. You can make hacks only if both versions of the problem are solved.Burenka is the crown princess of Buryatia, and soon she will become the $$$n$$$-th queen of the country. There is an ancient tradition in Buryatia — before the coronation, the ruler must show their strength to the inhabitants. To determine the strength of the $$$n$$$-th ruler, the inhabitants of the country give them an array of $$$a$$$ of exactly $$$n$$$ numbers, after which the ruler must turn all the elements of the array into zeros in the shortest time. The ruler can do the following two-step operation any number of times:   select two indices $$$l$$$ and $$$r$$$, so that $$$1 \le l \le r \le n$$$ and a non-negative integer $$$x$$$, then  for all $$$l \leq i \leq r$$$ assign $$$a_i := a_i \oplus x$$$, where $$$\oplus$$$ denotes the bitwise XOR operation. It takes $$$\left\lceil \frac{r-l+1}{2} \right\rceil$$$ seconds to do this operation, where $$$\lceil y \rceil$$$ denotes $$$y$$$ rounded up to the nearest integer. Help Burenka calculate how much time she will need.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"960"The Smart Beaver from ABBYY has once again surprised us! He has developed a new calculating device, which he called the ""Beaver's Calculator 1.0"". It is very peculiar and it is planned to be used in a variety of scientific problems.To test it, the Smart Beaver invited n scientists, numbered from 1 to n. The i-th scientist brought ki calculating problems for the device developed by the Smart Beaver from ABBYY. The problems of the i-th scientist are numbered from 1 to ki, and they must be calculated sequentially in the described order, since calculating each problem heavily depends on the results of calculating of the previous ones.Each problem of each of the n scientists is described by one integer ai, j, where i (1 ≤ i ≤ n) is the number of the scientist, j (1 ≤ j ≤ ki) is the number of the problem, and ai, j is the number of resource units the calculating device needs to solve this problem.The calculating device that is developed by the Smart Beaver is pretty unusual. It solves problems sequentially, one after another. After some problem is solved and before the next one is considered, the calculating device allocates or frees resources.The most expensive operation for the calculating device is freeing resources, which works much slower than allocating them. It is therefore desirable that each next problem for the calculating device requires no less resources than the previous one.You are given the information about the problems the scientists offered for the testing. You need to arrange these problems in such an order that the number of adjacent ""bad"" pairs of problems in this list is minimum possible. We will call two consecutive problems in this list a ""bad pair"" if the problem that is performed first requires more resources than the one that goes after it. Do not forget that the problems of the same scientist must be solved in a fixed order.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"961"Cowboy Vlad has a birthday today! There are $$$n$$$ children who came to the celebration. In order to greet Vlad, the children decided to form a circle around him. Among the children who came, there are both tall and low, so if they stand in a circle arbitrarily, it may turn out, that there is a tall and low child standing next to each other, and it will be difficult for them to hold hands. Therefore, children want to stand in a circle so that the maximum difference between the growth of two neighboring children would be minimal possible.Formally, let's number children from $$$1$$$ to $$$n$$$ in a circle order, that is, for every $$$i$$$ child with number $$$i$$$ will stand next to the child with number $$$i+1$$$, also the child with number $$$1$$$ stands next to the child with number $$$n$$$. Then we will call the discomfort of the circle the maximum absolute difference of heights of the children, who stand next to each other.Please help children to find out how they should reorder themselves, so that the resulting discomfort is smallest possible.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"962"Anton likes to play chess. Also he likes to do programming. No wonder that he decided to attend chess classes and programming classes.Anton has n variants when he will attend chess classes, i-th variant is given by a period of time (l1, i, r1, i). Also he has m variants when he will attend programming classes, i-th variant is given by a period of time (l2, i, r2, i).Anton needs to choose exactly one of n possible periods of time when he will attend chess classes and exactly one of m possible periods of time when he will attend programming classes. He wants to have a rest between classes, so from all the possible pairs of the periods he wants to choose the one where the distance between the periods is maximal.The distance between periods (l1, r1) and (l2, r2) is the minimal possible distance between a point in the first period and a point in the second period, that is the minimal possible |i - j|, where l1 ≤ i ≤ r1 and l2 ≤ j ≤ r2. In particular, when the periods intersect, the distance between them is 0.Anton wants to know how much time his rest between the classes will last in the best case. Help Anton and find this number!",[],"[0, 0, 0, 0, 0, 0, 0, 0]"963"Vasya is pressing the keys on the keyboard reluctantly, squeezing out his ideas on the classical epos depicted in Homer's Odysseus... How can he explain to his literature teacher that he isn't going to become a writer? In fact, he is going to become a programmer. So, he would take great pleasure in writing a program, but none — in writing a composition.As Vasya was fishing for a sentence in the dark pond of his imagination, he suddenly wondered: what is the least number of times he should push a key to shift the cursor from one position to another one?Let's describe his question more formally: to type a text, Vasya is using the text editor. He has already written n lines, the i-th line contains ai characters (including spaces). If some line contains k characters, then this line overall contains (k + 1) positions where the cursor can stand: before some character or after all characters (at the end of the line). Thus, the cursor's position is determined by a pair of integers (r, c), where r is the number of the line and c is the cursor's position in the line (the positions are indexed starting from one from the beginning of the line).Vasya doesn't use the mouse to move the cursor. He uses keys ""Up"", ""Down"", ""Right"" and ""Left"". When he pushes each of these keys, the cursor shifts in the needed direction. Let's assume that before the corresponding key is pressed, the cursor was located in the position (r, c), then Vasya pushed key: ""Up"": if the cursor was located in the first line (r = 1), then it does not move. Otherwise, it moves to the previous line (with number r - 1), to the same position. At that, if the previous line was short, that is, the cursor couldn't occupy position c there, the cursor moves to the last position of the line with number r - 1; ""Down"": if the cursor was located in the last line (r = n), then it does not move. Otherwise, it moves to the next line (with number r + 1), to the same position. At that, if the next line was short, that is, the cursor couldn't occupy position c there, the cursor moves to the last position of the line with number r + 1; ""Right"": if the cursor can move to the right in this line (c < ar + 1), then it moves to the right (to position c + 1). Otherwise, it is located at the end of the line and doesn't move anywhere when Vasya presses the ""Right"" key; ""Left"": if the cursor can move to the left in this line (c > 1), then it moves to the left (to position c - 1). Otherwise, it is located at the beginning of the line and doesn't move anywhere when Vasya presses the ""Left"" key.You've got the number of lines in the text file and the number of characters, written in each line of this file. Find the least number of times Vasya should push the keys, described above, to shift the cursor from position (r1, c1) to position (r2, c2).",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"964"You are given an array $$$a$$$ consisting of $$$n$$$ positive integers, and an array $$$b$$$, with length $$$n$$$. Initially $$$b_i=0$$$ for each $$$1 \leq i \leq n$$$.In one move you can choose an integer $$$i$$$ ($$$1 \leq i \leq n$$$), and add $$$a_i$$$ to $$$b_i$$$ or subtract $$$a_i$$$ from $$$b_i$$$. What is the minimum number of moves needed to make $$$b$$$ increasing (that is, every element is strictly greater than every element before it)?",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"965"You can perfectly predict the price of a certain stock for the next N days. You would like to profit on this knowledge, but only want to transact one share of stock per day. That is, each day you will either buy one share, sell one share, or do nothing. Initially you own zero shares, and you cannot sell shares when you don't own any. At the end of the N days you would like to again own zero shares, but want to have as much money as possible.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"966"Kawasiro Nitori is excellent in engineering. Thus she has been appointed to help maintain trains.There are $$$n$$$ models of trains, and Nitori's department will only have at most one train of each model at any moment. In the beginning, there are no trains, at each of the following $$$m$$$ days, one train will be added, or one train will be removed. When a train of model $$$i$$$ is added at day $$$t$$$, it works for $$$x_i$$$ days (day $$$t$$$ inclusive), then it is in maintenance for $$$y_i$$$ days, then in work for $$$x_i$$$ days again, and so on until it is removed.In order to make management easier, Nitori wants you to help her calculate how many trains are in maintenance in each day.On a day a train is removed, it is not counted as in maintenance.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"967"The stardate is 1983, and Princess Heidi is getting better at detecting the Death Stars. This time, two Rebel spies have yet again given Heidi two maps with the possible locations of the Death Star. Since she got rid of all double agents last time, she knows that both maps are correct, and indeed show the map of the solar system that contains the Death Star. However, this time the Empire has hidden the Death Star very well, and Heidi needs to find a place that appears on both maps in order to detect the Death Star.The first map is an N × M grid, each cell of which shows some type of cosmic object that is present in the corresponding quadrant of space. The second map is an M × N grid. Heidi needs to align those two maps in such a way that they overlap over some M × M section in which all cosmic objects are identical. Help Heidi by identifying where such an M × M section lies within both maps.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"968"The map of the capital of Berland can be viewed on the infinite coordinate plane. Each point with integer coordinates contains a building, and there are streets connecting every building to four neighbouring buildings. All streets are parallel to the coordinate axes.The main school of the capital is located in $$$(s_x, s_y)$$$. There are $$$n$$$ students attending this school, the $$$i$$$-th of them lives in the house located in $$$(x_i, y_i)$$$. It is possible that some students live in the same house, but no student lives in $$$(s_x, s_y)$$$.After classes end, each student walks from the school to his house along one of the shortest paths. So the distance the $$$i$$$-th student goes from the school to his house is $$$|s_x - x_i| + |s_y - y_i|$$$.The Provision Department of Berland has decided to open a shawarma tent somewhere in the capital (at some point with integer coordinates). It is considered that the $$$i$$$-th student will buy a shawarma if at least one of the shortest paths from the school to the $$$i$$$-th student's house goes through the point where the shawarma tent is located. It is forbidden to place the shawarma tent at the point where the school is located, but the coordinates of the shawarma tent may coincide with the coordinates of the house of some student (or even multiple students).You want to find the maximum possible number of students buying shawarma and the optimal location for the tent itself.",['geometry'],"[0, 1, 0, 0, 0, 0, 0, 0]"969"You are given two strings $$$s$$$ and $$$t$$$. In a single move, you can choose any of two strings and delete the first (that is, the leftmost) character. After a move, the length of the string decreases by $$$1$$$. You can't choose a string if it is empty.For example:  by applying a move to the string ""where"", the result is the string ""here"",  by applying a move to the string ""a"", the result is an empty string """". You are required to make two given strings equal using the fewest number of moves. It is possible that, in the end, both strings will be equal to the empty string, and so, are equal to each other. In this case, the answer is obviously the sum of the lengths of the initial strings.Write a program that finds the minimum number of moves to make two given strings $$$s$$$ and $$$t$$$ equal.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"970"Multiset —is a set of numbers in which there can be equal elements, and the order of the numbers does not matter. Two multisets are equal when each value occurs the same number of times. For example, the multisets $$$\{2,2,4\}$$$ and $$$\{2,4,2\}$$$ are equal, but the multisets $$$\{1,2,2\}$$$ and $$$\{1,1,2\}$$$ — are not.You are given two multisets $$$a$$$ and $$$b$$$, each consisting of $$$n$$$ integers.In a single operation, any element of the $$$b$$$ multiset can be doubled or halved (rounded down). In other words, you have one of the following operations available for an element $$$x$$$ of the $$$b$$$ multiset:   replace $$$x$$$ with $$$x \cdot 2$$$,  or replace $$$x$$$ with $$$\lfloor \frac{x}{2} \rfloor$$$ (round down).  Note that you cannot change the elements of the $$$a$$$ multiset.See if you can make the multiset $$$b$$$ become equal to the multiset $$$a$$$ in an arbitrary number of operations (maybe $$$0$$$).For example, if $$$n = 4$$$, $$$a = \{4, 24, 5, 2\}$$$, $$$b = \{4, 1, 6, 11\}$$$, then the answer is yes. We can proceed as follows:   Replace $$$1$$$ with $$$1 \cdot 2 = 2$$$. We get $$$b = \{4, 2, 6, 11\}$$$.  Replace $$$11$$$ with $$$\lfloor \frac{11}{2} \rfloor = 5$$$. We get $$$b = \{4, 2, 6, 5\}$$$.  Replace $$$6$$$ with $$$6 \cdot 2 = 12$$$. We get $$$b = \{4, 2, 12, 5\}$$$.  Replace $$$12$$$ with $$$12 \cdot 2 = 24$$$. We get $$$b = \{4, 2, 24, 5\}$$$.  Got equal multisets $$$a = \{4, 24, 5, 2\}$$$ and $$$b = \{4, 2, 24, 5\}$$$. ","['math', 'number theory']","[0, 0, 0, 1, 1, 0, 0, 0]"971"ZS the Coder is coding on a crazy computer. If you don't type in a word for a c consecutive seconds, everything you typed disappear! More formally, if you typed a word at second a and then the next word at second b, then if b - a ≤ c, just the new word is appended to other words on the screen. If b - a > c, then everything on the screen disappears and after that the word you have typed appears on the screen.For example, if c = 5 and you typed words at seconds 1, 3, 8, 14, 19, 20 then at the second 8 there will be 3 words on the screen. After that, everything disappears at the second 13 because nothing was typed. At the seconds 14 and 19 another two words are typed, and finally, at the second 20, one more word is typed, and a total of 3 words remain on the screen.You're given the times when ZS the Coder typed the words. Determine how many words remain on the screen after he finished typing everything.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"972"Recently an official statement of the world Olympic Committee said that the Olympic Winter Games 2030 will be held in Tomsk. The city officials decided to prepare for the Olympics thoroughly and to build all the necessary Olympic facilities as early as possible. First, a biathlon track will be built.To construct a biathlon track a plot of land was allocated, which is a rectangle divided into n × m identical squares. Each of the squares has two coordinates: the number of the row (from 1 to n), where it is located, the number of the column (from 1 to m), where it is located. Also each of the squares is characterized by its height. During the sports the biathletes will have to move from one square to another. If a biathlete moves from a higher square to a lower one, he makes a descent. If a biathlete moves from a lower square to a higher one, he makes an ascent. If a biathlete moves between two squares with the same height, then he moves on flat ground.The biathlon track should be a border of some rectangular area of the allocated land on which biathletes will move in the clockwise direction. It is known that on one move on flat ground an average biathlete spends tp seconds, an ascent takes tu seconds, a descent takes td seconds. The Tomsk Administration wants to choose the route so that the average biathlete passes it in as close to t seconds as possible. In other words, the difference between time ts of passing the selected track and t should be minimum.For a better understanding you can look at the first sample of the input data. In this sample n = 6, m = 7, and the administration wants the track covering time to be as close to t = 48 seconds as possible, also, tp = 3, tu = 6 and td = 2. If we consider the rectangle shown on the image by arrows, the average biathlete can move along the boundary in a clockwise direction in exactly 48 seconds. The upper left corner of this track is located in the square with the row number 4, column number 3 and the lower right corner is at square with row number 6, column number 7.  Among other things the administration wants all sides of the rectangle which boundaries will be the biathlon track to consist of no less than three squares and to be completely contained within the selected land.You are given the description of the given plot of land and all necessary time values. You are to write the program to find the most suitable rectangle for a biathlon track. If there are several such rectangles, you are allowed to print any of them.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"973"There are $$$n$$$ players sitting at the card table. Each player has a favorite number. The favorite number of the $$$j$$$-th player is $$$f_j$$$.There are $$$k \cdot n$$$ cards on the table. Each card contains a single integer: the $$$i$$$-th card contains number $$$c_i$$$. Also, you are given a sequence $$$h_1, h_2, \dots, h_k$$$. Its meaning will be explained below.The players have to distribute all the cards in such a way that each of them will hold exactly $$$k$$$ cards. After all the cards are distributed, each player counts the number of cards he has that contains his favorite number. The joy level of a player equals $$$h_t$$$ if the player holds $$$t$$$ cards containing his favorite number. If a player gets no cards with his favorite number (i.e., $$$t=0$$$), his joy level is $$$0$$$.Print the maximum possible total joy levels of the players after the cards are distributed. Note that the sequence $$$h_1, \dots, h_k$$$ is the same for all the players.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"974"Moamen has an array of $$$n$$$ distinct integers. He wants to sort that array in non-decreasing order by doing the following operations in order exactly once:  Split the array into exactly $$$k$$$ non-empty subarrays such that each element belongs to exactly one subarray.  Reorder these subarrays arbitrary.  Merge the subarrays in their new order. A sequence $$$a$$$ is a subarray of a sequence $$$b$$$ if $$$a$$$ can be obtained from $$$b$$$ by deletion of several (possibly, zero or all) elements from the beginning and several (possibly, zero or all) elements from the end.Can you tell Moamen if there is a way to sort the array in non-decreasing order using the operations written above?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"975"Andrew was very excited to participate in Olympiad of Metropolises. Days flew by quickly, and Andrew is already at the airport, ready to go home. He has $$$n$$$ rubles left, and would like to exchange them to euro and dollar bills. Andrew can mix dollar bills and euro bills in whatever way he wants. The price of one dollar is $$$d$$$ rubles, and one euro costs $$$e$$$ rubles.Recall that there exist the following dollar bills: $$$1$$$, $$$2$$$, $$$5$$$, $$$10$$$, $$$20$$$, $$$50$$$, $$$100$$$, and the following euro bills — $$$5$$$, $$$10$$$, $$$20$$$, $$$50$$$, $$$100$$$, $$$200$$$ (note that, in this problem we do not consider the $$$500$$$ euro bill, it is hard to find such bills in the currency exchange points). Andrew can buy any combination of bills, and his goal is to minimize the total number of rubles he will have after the exchange.Help him — write a program that given integers $$$n$$$, $$$e$$$ and $$$d$$$, finds the minimum number of rubles Andrew can get after buying dollar and euro bills.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"976"Sonya was unable to think of a story for this problem, so here comes the formal description.You are given the array containing n positive integers. At one turn you can pick any element and increase or decrease it by 1. The goal is the make the array strictly increasing by making the minimum possible number of operations. You are allowed to change elements in any way, they can become negative or equal to 0.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"977"Monocarp is playing a computer game. Now he wants to complete the first level of this game.A level is a rectangular grid of $$$2$$$ rows and $$$n$$$ columns. Monocarp controls a character, which starts in cell $$$(1, 1)$$$ — at the intersection of the $$$1$$$-st row and the $$$1$$$-st column.Monocarp's character can move from one cell to another in one step if the cells are adjacent by side and/or corner. Formally, it is possible to move from cell $$$(x_1, y_1)$$$ to cell $$$(x_2, y_2)$$$ in one step if $$$|x_1 - x_2| \le 1$$$ and $$$|y_1 - y_2| \le 1$$$. Obviously, it is prohibited to go outside the grid.There are traps in some cells. If Monocarp's character finds himself in such a cell, he dies, and the game ends.To complete a level, Monocarp's character should reach cell $$$(2, n)$$$ — at the intersection of row $$$2$$$ and column $$$n$$$.Help Monocarp determine if it is possible to complete the level.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"978"Let's call a string ""s-palindrome"" if it is symmetric about the middle of the string. For example, the string ""oHo"" is ""s-palindrome"", but the string ""aa"" is not. The string ""aa"" is not ""s-palindrome"", because the second half of it is not a mirror reflection of the first half.  English alphabet You are given a string s. Check if the string is ""s-palindrome"".",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"979"There are $$$n$$$ logs, the $$$i$$$-th log has a length of $$$a_i$$$ meters. Since chopping logs is tiring work, errorgorn and maomao90 have decided to play a game.errorgorn and maomao90 will take turns chopping the logs with errorgorn chopping first. On his turn, the player will pick a log and chop it into $$$2$$$ pieces. If the length of the chosen log is $$$x$$$, and the lengths of the resulting pieces are $$$y$$$ and $$$z$$$, then $$$y$$$ and $$$z$$$ have to be positive integers, and $$$x=y+z$$$ must hold. For example, you can chop a log of length $$$3$$$ into logs of lengths $$$2$$$ and $$$1$$$, but not into logs of lengths $$$3$$$ and $$$0$$$, $$$2$$$ and $$$2$$$, or $$$1.5$$$ and $$$1.5$$$.The player who is unable to make a chop will be the loser. Assuming that both errorgorn and maomao90 play optimally, who will be the winner?","['games', 'math']","[1, 0, 0, 1, 0, 0, 0, 0]"980"You are given a string $$$s$$$ of length $$$n$$$ consisting only of the characters 0 and 1.You perform the following operation until the string becomes empty: choose some consecutive substring of equal characters, erase it from the string and glue the remaining two parts together (any of them can be empty) in the same order. For example, if you erase the substring 111 from the string 111110, you will get the string 110. When you delete a substring of length $$$l$$$, you get $$$a \cdot l + b$$$ points.Your task is to calculate the maximum number of points that you can score in total, if you have to make the given string empty.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"981"There are $$$n$$$ blocks arranged in a row and numbered from left to right, starting from one. Each block is either black or white. You may perform the following operation zero or more times: choose two adjacent blocks and invert their colors (white block becomes black, and vice versa). You want to find a sequence of operations, such that they make all the blocks having the same color. You don't have to minimize the number of operations, but it should not exceed $$$3 \cdot n$$$. If it is impossible to find such a sequence of operations, you need to report it.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"982"The only difference with E2 is the question of the problem..Vlad built a maze out of $$$n$$$ rooms and $$$n-1$$$ bidirectional corridors. From any room $$$u$$$ any other room $$$v$$$ can be reached through a sequence of corridors. Thus, the room system forms an undirected tree.Vlad invited $$$k$$$ friends to play a game with them.Vlad starts the game in the room $$$1$$$ and wins if he reaches a room other than $$$1$$$, into which exactly one corridor leads.Friends are placed in the maze: the friend with number $$$i$$$ is in the room $$$x_i$$$, and no two friends are in the same room (that is, $$$x_i \neq x_j$$$ for all $$$i \neq j$$$). Friends win if one of them meets Vlad in any room or corridor before he wins.For one unit of time, each participant of the game can go through one corridor. All participants move at the same time. Participants may not move. Each room can fit all participants at the same time. Friends know the plan of a maze and intend to win. Vlad is a bit afraid of their ardor. Determine if he can guarantee victory (i.e. can he win in any way friends play).In other words, determine if there is such a sequence of Vlad's moves that lets Vlad win in any way friends play.",['trees'],"[0, 0, 0, 0, 0, 0, 0, 1]"983"You are given a keyboard that consists of $$$26$$$ keys. The keys are arranged sequentially in one row in a certain order. Each key corresponds to a unique lowercase Latin letter.You have to type the word $$$s$$$ on this keyboard. It also consists only of lowercase Latin letters.To type a word, you need to type all its letters consecutively one by one. To type each letter you must position your hand exactly over the corresponding key and press it.Moving the hand between the keys takes time which is equal to the absolute value of the difference between positions of these keys (the keys are numbered from left to right). No time is spent on pressing the keys and on placing your hand over the first letter of the word.For example, consider a keyboard where the letters from 'a' to 'z' are arranged in consecutive alphabetical order. The letters 'h', 'e', 'l' and 'o' then are on the positions $$$8$$$, $$$5$$$, $$$12$$$ and $$$15$$$, respectively. Therefore, it will take $$$|5 - 8| + |12 - 5| + |12 - 12| + |15 - 12| = 13$$$ units of time to type the word ""hello"". Determine how long it will take to print the word $$$s$$$.",['strings'],"[0, 0, 0, 0, 0, 0, 1, 0]"984"A motorcade of n trucks, driving from city «Z» to city «З», has approached a tunnel, known as Tunnel of Horror. Among truck drivers there were rumours about monster DravDe, who hunts for drivers in that tunnel. Some drivers fear to go first, others - to be the last, but let's consider the general case. Each truck is described with four numbers:   v — value of the truck, of its passangers and cargo  c — amount of passanger on the truck, the driver included  l — total amount of people that should go into the tunnel before this truck, so that the driver can overcome his fear («if the monster appears in front of the motorcade, he'll eat them first»)  r — total amount of people that should follow this truck, so that the driver can overcome his fear («if the monster appears behind the motorcade, he'll eat them first»). Since the road is narrow, it's impossible to escape DravDe, if he appears from one side. Moreover, the motorcade can't be rearranged. The order of the trucks can't be changed, but it's possible to take any truck out of the motorcade, and leave it near the tunnel for an indefinite period. You, as the head of the motorcade, should remove some of the trucks so, that the rest of the motorcade can move into the tunnel and the total amount of the left trucks' values is maximal. ",[],"[0, 0, 0, 0, 0, 0, 0, 0]"985You are given n × m table. Each cell of the table is colored white or black. Find the number of non-empty sets of cells such that:  All cells in a set have the same color.  Every two cells in a set share row or column. ,['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"986"You have $$$r$$$ red and $$$b$$$ blue beans. You'd like to distribute them among several (maybe, one) packets in such a way that each packet:   has at least one red bean (or the number of red beans $$$r_i \ge 1$$$);  has at least one blue bean (or the number of blue beans $$$b_i \ge 1$$$);  the number of red and blue beans should differ in no more than $$$d$$$ (or $$$|r_i - b_i| \le d$$$) Can you distribute all beans?",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"987"The Romans have attacked again. This time they are much more than the Persians but Shapur is ready to defeat them. He says: ""A lion is never afraid of a hundred sheep"". Nevertheless Shapur has to find weaknesses in the Roman army to defeat them. So he gives the army a weakness number.In Shapur's opinion the weakness of an army is equal to the number of triplets i, j, k such that i < j < k and ai > aj > ak where ax is the power of man standing at position x. The Roman army has one special trait — powers of all the people in it are distinct.Help Shapur find out how weak the Romans are.",['trees'],"[0, 0, 0, 0, 0, 0, 0, 1]"988"Bill is a famous mathematician in BubbleLand. Thanks to his revolutionary math discoveries he was able to make enough money to build a beautiful house. Unfortunately, for not paying property tax on time, court decided to punish Bill by making him lose a part of his property.Bill’s property can be observed as a convex regular 2n-sided polygon A0 A1... A2n - 1 A2n,  A2n =  A0, with sides of the exactly 1 meter in length. Court rules for removing part of his property are as follows:   Split every edge Ak Ak + 1,  k = 0... 2n - 1 in n equal parts of size 1 / n with points P0, P1, ..., Pn - 1   On every edge A2k A2k + 1,  k = 0... n - 1 court will choose one point B2k =  Pi for some i = 0, ...,  n - 1 such that    On every edge A2k + 1A2k + 2,  k = 0...n - 1 Bill will choose one point B2k + 1 =  Pi for some i = 0, ...,  n - 1 such that    Bill gets to keep property inside of 2n-sided polygon B0 B1... B2n - 1 Luckily, Bill found out which B2k points the court chose. Even though he is a great mathematician, his house is very big and he has a hard time calculating. Therefore, he is asking you to help him choose points so he maximizes area of property he can keep.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"989"You found a useless array $$$a$$$ of $$$2n$$$ positive integers. You have realized that you actually don't need this array, so you decided to throw out all elements of $$$a$$$.It could have been an easy task, but it turned out that you should follow some rules:   In the beginning, you select any positive integer $$$x$$$. Then you do the following operation $$$n$$$ times:   select two elements of array with sum equals $$$x$$$;  remove them from $$$a$$$ and replace $$$x$$$ with maximum of that two numbers.  For example, if initially $$$a = [3, 5, 1, 2]$$$, you can select $$$x = 6$$$. Then you can select the second and the third elements of $$$a$$$ with sum $$$5 + 1 = 6$$$ and throw them out. After this operation, $$$x$$$ equals $$$5$$$ and there are two elements in array: $$$3$$$ and $$$2$$$. You can throw them out on the next operation.Note, that you choose $$$x$$$ before the start and can't change it as you want between the operations.Determine how should you behave to throw out all elements of $$$a$$$.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"990"The King of Berland Polycarp LXXXIV has $$$n$$$ daughters. To establish his power to the neighbouring kingdoms he wants to marry his daughters to the princes of these kingdoms. As a lucky coincidence there are $$$n$$$ other kingdoms as well.So Polycarp LXXXIV has enumerated his daughters from $$$1$$$ to $$$n$$$ and the kingdoms from $$$1$$$ to $$$n$$$. For each daughter he has compiled a list of kingdoms princes of which she wanted to marry.Polycarp LXXXIV is very busy, so he finds a couple for his daughters greedily one after another.For the first daughter he takes the kingdom with the lowest number from her list and marries the daughter to their prince. For the second daughter he takes the kingdom with the lowest number from her list, prince of which hasn't been taken already. If there are no free princes in the list then the daughter marries nobody and Polycarp LXXXIV proceeds to the next daughter. The process ends after the $$$n$$$-th daughter.For example, let there be $$$4$$$ daughters and kingdoms, the lists daughters have are $$$[2, 3]$$$, $$$[1, 2]$$$, $$$[3, 4]$$$, $$$[3]$$$, respectively.  In that case daughter $$$1$$$ marries the prince of kingdom $$$2$$$, daughter $$$2$$$ marries the prince of kingdom $$$1$$$, daughter $$$3$$$ marries the prince of kingdom $$$3$$$, leaving daughter $$$4$$$ nobody to marry to.Actually, before starting the marriage process Polycarp LXXXIV has the time to convince one of his daughters that some prince is also worth marrying to. Effectively, that means that he can add exactly one kingdom to exactly one of his daughter's list. Note that this kingdom should not be present in the daughter's list.Polycarp LXXXIV wants to increase the number of married couples.Unfortunately, what he doesn't have the time for is determining what entry to add. If there is no way to increase the total number of married couples then output that the marriages are already optimal. Otherwise, find such an entry that the total number of married couples increases if Polycarp LXXXIV adds it.If there are multiple ways to add an entry so that the total number of married couples increases then print any of them.For your and our convenience you are asked to answer $$$t$$$ independent test cases.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"991"Vasya's birthday is approaching and Lena decided to sew a patterned handkerchief to him as a present. Lena chose digits from 0 to n as the pattern. The digits will form a rhombus. The largest digit n should be located in the centre. The digits should decrease as they approach the edges. For example, for n = 5 the handkerchief pattern should look like that:           0        0 1 0      0 1 2 1 0    0 1 2 3 2 1 0  0 1 2 3 4 3 2 1 00 1 2 3 4 5 4 3 2 1 0  0 1 2 3 4 3 2 1 0    0 1 2 3 2 1 0      0 1 2 1 0        0 1 0          0Your task is to determine the way the handkerchief will look like by the given n.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"992"Alice has a cake, and she is going to cut it. She will perform the following operation $$$n-1$$$ times: choose a piece of the cake (initially, the cake is all one piece) with weight $$$w\ge 2$$$ and cut it into two smaller pieces of weight $$$\lfloor\frac{w}{2}\rfloor$$$ and $$$\lceil\frac{w}{2}\rceil$$$ ($$$\lfloor x \rfloor$$$ and $$$\lceil x \rceil$$$ denote floor and ceiling functions, respectively).After cutting the cake in $$$n$$$ pieces, she will line up these $$$n$$$ pieces on a table in an arbitrary order. Let $$$a_i$$$ be the weight of the $$$i$$$-th piece in the line.You are given the array $$$a$$$. Determine whether there exists an initial weight and sequence of operations which results in $$$a$$$.",[],"[0, 0, 0, 0, 0, 0, 0, 0]"993"There are $$$n$$$ Christmas trees on an infinite number line. The $$$i$$$-th tree grows at the position $$$x_i$$$. All $$$x_i$$$ are guaranteed to be distinct.Each integer point can be either occupied by the Christmas tree, by the human or not occupied at all. Non-integer points cannot be occupied by anything.There are $$$m$$$ people who want to celebrate Christmas. Let $$$y_1, y_2, \dots, y_m$$$ be the positions of people (note that all values $$$x_1, x_2, \dots, x_n, y_1, y_2, \dots, y_m$$$ should be distinct and all $$$y_j$$$ should be integer). You want to find such an arrangement of people that the value $$$\sum\limits_{j=1}^{m}\min\limits_{i=1}^{n}|x_i - y_j|$$$ is the minimum possible (in other words, the sum of distances to the nearest Christmas tree for all people should be minimized).In other words, let $$$d_j$$$ be the distance from the $$$j$$$-th human to the nearest Christmas tree ($$$d_j = \min\limits_{i=1}^{n} |y_j - x_i|$$$). Then you need to choose such positions $$$y_1, y_2, \dots, y_m$$$ that $$$\sum\limits_{j=1}^{m} d_j$$$ is the minimum possible.",['graphs'],"[0, 0, 1, 0, 0, 0, 0, 0]"994"One important contest will take place on the most famous programming platform (Topforces) very soon!The authors have a pool of $$$n$$$ problems and should choose at most three of them into this contest. The prettiness of the $$$i$$$-th problem is $$$a_i$$$. The authors have to compose the most pretty contest (in other words, the cumulative prettinesses of chosen problems should be maximum possible).But there is one important thing in the contest preparation: because of some superstitions of authors, the prettinesses of problems cannot divide each other. In other words, if the prettinesses of chosen problems are $$$x, y, z$$$, then $$$x$$$ should be divisible by neither $$$y$$$, nor $$$z$$$, $$$y$$$ should be divisible by neither $$$x$$$, nor $$$z$$$ and $$$z$$$ should be divisible by neither $$$x$$$, nor $$$y$$$. If the prettinesses of chosen problems are $$$x$$$ and $$$y$$$ then neither $$$x$$$ should be divisible by $$$y$$$ nor $$$y$$$ should be divisible by $$$x$$$. Any contest composed from one problem is considered good.Your task is to find out the maximum possible total prettiness of the contest composed of at most three problems from the given pool.You have to answer $$$q$$$ independent queries.If you are Python programmer, consider using PyPy instead of Python when you submit your code.",['math'],"[0, 0, 0, 1, 0, 0, 0, 0]"995"  There're $$$n$$$ robots placed on a number line. Initially, $$$i$$$-th of them occupies unit segment $$$[x_i, x_i + 1]$$$. Each robot has a program, consisting of $$$k$$$ instructions numbered from $$$1$$$ to $$$k$$$. The robot performs instructions in a cycle. Each instruction is described by an integer number. Let's denote the number corresponding to the $$$j$$$-th instruction of the $$$i$$$-th robot as $$$f_{i, j}$$$.Initial placement of robots corresponds to the moment of time $$$0$$$. During one second from moment of time $$$t$$$ ($$$0 \le t$$$) until $$$t + 1$$$ the following process occurs:   Each robot performs $$$(t \bmod k + 1)$$$-th instruction from its list of instructions. Robot number $$$i$$$ takes number $$$F = f_{i, (t \bmod k + 1)}$$$. If this number is negative (less than zero), the robot is trying to move to the left with force $$$|F|$$$. If the number is positive (more than zero), the robot is trying to move to the right with force $$$F$$$. Otherwise, the robot does nothing.  Let's imaginary divide robots into groups of consecutive, using the following algorithm:   Initially, each robot belongs to its own group.  Let's sum up numbers corresponding to the instructions of the robots from one group. Note that we are summing numbers without taking them by absolute value. Denote this sum as $$$S$$$. We say that the whole group moves together, and does it with force $$$S$$$ by the same rules as a single robot. That is if $$$S$$$ is negative, the group is trying to move to the left with force $$$|S|$$$. If $$$S$$$ is positive, the group is trying to move to the right with force $$$S$$$. Otherwise, the group does nothing.  If one group is trying to move, and in the direction of movement touches another group, let's unite them. One group is touching another if their outermost robots occupy adjacent unit segments.  Continue this process until groups stop uniting.   Each robot moves by $$$1$$$ in the direction of movement of its group or stays in place if its group isn't moving. But there's one exception.  The exception is if there're two groups of robots, divided by exactly one unit segment, such that the left group is trying to move to the right and the right group is trying to move to the left. Let's denote sum in the left group as $$$S_l$$$ and sum in the right group as $$$S_r$$$. If $$$|S_l| \le |S_r|$$$ only the right group will move. Otherwise, only the left group will move.  Note that robots from one group don't glue together. They may separate in the future. The division into groups is imaginary and is needed only to understand how robots will move during one second $$$[t, t + 1]$$$. An illustration of the process happening during one second:   Rectangles represent robots. Numbers inside rectangles correspond to instructions of robots. The final division into groups is marked with arcs. Below are the positions of the robots after moving. Only the left of the two rightmost groups moved. That's because these two groups tried to move towards each other, and were separated by exactly one unit segment.Look at the examples for a better understanding of the process.You need to answer several questions. What is the position of $$$a_i$$$-th robot at the moment of time $$$t_i$$$?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"996"Ilya is very fond of graphs, especially trees. During his last trip to the forest Ilya found a very interesting tree rooted at vertex 1. There is an integer number written on each vertex of the tree; the number written on vertex i is equal to ai.Ilya believes that the beauty of the vertex x is the greatest common divisor of all numbers written on the vertices on the path from the root to x, including this vertex itself. In addition, Ilya can change the number in one arbitrary vertex to 0 or leave all vertices unchanged. Now for each vertex Ilya wants to know the maximum possible beauty it can have.For each vertex the answer must be considered independently.The beauty of the root equals to number written on it.","['graphs', 'math', 'number theory', 'trees']","[0, 0, 1, 1, 1, 0, 0, 1]"997"A and B are preparing themselves for programming contests.B loves to debug his code. But before he runs the solution and starts debugging, he has to first compile the code.Initially, the compiler displayed n compilation errors, each of them is represented as a positive integer. After some effort, B managed to fix some mistake and then another one mistake.However, despite the fact that B is sure that he corrected the two errors, he can not understand exactly what compilation errors disappeared — the compiler of the language which B uses shows errors in the new order every time! B is sure that unlike many other programming languages, compilation errors for his programming language do not depend on each other, that is, if you correct one error, the set of other error does not change.Can you help B find out exactly what two errors he corrected?",[],"[0, 0, 0, 0, 0, 0, 0, 0]"998