AIIRLab/MathDisAM

1MathSE ID	Math Question	Tag
2137352	What are the domain and range of the map f in Devaney's definition of chaos on a metric space?	Unambiguous
32561078	How does the triangle inequality apply to the proof for f-g using epsilon and delta?	Ambiguous
4735461	For a completely positive map $f: M_n \to M_m$ with $f(a) = I_m$ for some $a \in M_n$, prove or disprove that there exists a positive element $b \in M_n$ such that $f(b) = I_m$.	Unambiguous
52461048	What is the cyclic subgroup generated by -1?	Ambiguous
6578647	What is an intuitive explanation of the Fourier transform and how is it used to analyze signals in the frequency domain?	Unambiguous
72430945	"Decide if the statement "" $n^2-1$ is multiple of $4$ if and only if $n-1$ is multiple of $4$"" is true or false."	Unambiguous
81403539	Prove that $\arctan (x) + \arctan(1/x) = \frac{\pi}{2}$ for $x >0$.	Unambiguous
9510991	Prove that every continuous function defined on a compact metric space is uniformly continuous.	Unambiguous
102904744	Prove that $\lim_{x\to 1} f(x) = 3$ where $f(x) = \frac{x^3-1}{x-1}$ for $x \neq 1$.	Unambiguous
11982186	Decompose the expression $\frac{1}{4n^2-1}$ into partial fractions.	Unambiguous
124087621	Between which two consecutive integers does the value of $\sqrt{12}$ lie?	Unambiguous
13697675	How do you multiply a matrix by a row vector from the left?	Unambiguous
141025321	Find the derivative $\frac{dy}{dx}$ using implicit differentiation for the equation $\ln(1+xy) = xy$.	Unambiguous
153313255	What is the definition and standard form of Sum of Products (SOP) in boolean algebra?	Unambiguous
16578647	In the Fourier transform, what does the cross term mean?	Ambiguous
173843282	finding quotient group relationship	Ambiguous
183372191	Asymptotic behavior of $a_{n+1}=a_n + f(a_n)$	Ambiguous
192694508	Find the range of values of $k$ such that both roots of $x^2 - 2kx + k^2 - 1 = 0$ lie in the interval $(-2, 4)$.	Unambiguous
203735022	What is the formula for the curvature of a Frenet curve constrained to lie on a sphere?	Unambiguous
214870757	If I know the dimension, depth, and codimension of $R$ and $I$, can I determine the depth of $I$?	Ambiguous
22166794	Proof that the set of irrational numbers is uncountable	Unambiguous
234127695	Evaluate the sum $\sum_{r=0}^n 2^{n-r} \binom{n+r}{r}$ for non-negative integers $n$ and $r$.	Unambiguous
242949838	Can every basis of a finite dimensional vector space be orthogonal?	Ambiguous
257391	How do I evaluate the definite integral $\int_0^{\pi} \sin(2x)\sin(x) dx$?	Unambiguous
261315088	Evaluate the limit $\lim_{x \to 0} \left( \frac{1}{\sin(x)\arctan(x)} - \frac{1}{\tan(x)\arcsin(x)} \right)$	Unambiguous
2736134	Modern Equivalent of Carr's Synopsis?	Ambiguous
282526693	Evaluate the limit: $\lim_{x \to 0^+} x^{\sqrt{x}}$	Unambiguous
29432999	Prove that the limit of the sequence $\frac{n}{2^n}$ as $n$ approaches infinity is 0.	Unambiguous
301572032	Why is the CDF of $Y_n$ equal to $(\frac{t- \theta}{\theta})$ and what are the restrictions on $t$?	Ambiguous
312186358	What is the correct term for $I^n f(x)$ where $I$ is the integral operator?	Ambiguous
32606172	If the sequence $(x_n)$ of real numbers converges and $\lim_{n \to \infty} (2x_{n+1} - x_n) = x$, is it true that $\lim_{n \to \infty} x_n = x$?	Unambiguous
334468673	If $x$ and $y$ are positive integers such that $5x+3y=100$, what is the maximum value of the product $xy$?	Unambiguous
34166794	I found a proof arguing that the irrationals are countable based on the density of the rationals. I know this is wrong, but where is the mistake in the logic?	Ambiguous
351191610	Prove that the composition of two group homomorphisms is also a group homomorphism.	Unambiguous
364437668	What are some interesting examples of elementary proofs by mathematical induction for students?	Unambiguous
371354331	Does there exist a function that is discontinuous at a point but has a continuous derivative on its domain?	Unambiguous
38970133	Prove that the infinite series $\sum_{n=1}^\infty \frac{n^2(n-1)}{2^n}$ converges to 20.	Unambiguous
39419859	Prove that the graph $y=x^{p/q}$ has a vertical tangent.	Ambiguous
40105575	What is the difference between the soundness and completeness theorems in first-order logic?	Unambiguous
41456595	Prove: $\frac{1}{1^2} +\frac{1}{2^2} + \cdots + \frac{1}{n^2} + \cdots = \sum_{n=1}^\infty \frac{1}{n^2} < 2$	Ambiguous
423075859	What is the next step after showing $(c^k)^t = (id)$ for cycle orders?	Ambiguous
432538727	How to use Cantor's diagonal to prove my space is not sequentially compact?	Ambiguous
44217966	What is the relationship between monomorphisms and pullbacks in category theory?	Unambiguous
453117992	Could different physical systems follow different underlying mathematics?	Ambiguous
46181367	Provide a counterexample to show that a pseudocompact topological space is not necessarily a compact space.	Unambiguous
47854380	How to distinguish between walking on a sphere and walking on a torus?	Ambiguous
483087356	Why is the number 2 included in the definition of prime numbers instead of being excluded as the only even prime?	Unambiguous
494468673	If $x$ and $y$ are positive integers such that $5x+3y=100$, what is the greatest possible value of $xy$?	Ambiguous
504060048	How do I solve integrals of the form integral of (polynomial * exponential) dx using integration by parts?	Unambiguous
511024830	What is the intuitive meaning of a compact set in the context of a metric space or topology?	Unambiguous
523172708	How to efficiently update the inverse of a lower triangular matrix after changing 4 individual elements?	Unambiguous
531133113	How do we define arc length?	Ambiguous
542490690	In how many ways can a payment of $107$ shekels be made using coins/bills of denominations $1$, $10$, and $50$ shekels, assuming an unlimited supply of each?	Unambiguous
551711154	Prove that f is differentiable at x=0	Ambiguous
56551148	Why is $2x^3 + x$, where $x \in \mathbb{N}$, always divisible by 3?	Ambiguous
572304892	Probability of rolling any set of size k on n dice with m sides	Ambiguous
581576678	Can my sine formula with $\pi/6$ be put into Excel somehow to plot the graph?	Ambiguous
592982578	Prove that every finitely generated module over a ring $R$ is a homomorphic image of a free module.	Unambiguous
601619762	What are the equivalent statements of the Generalized Riemann Hypothesis (GRH) expressed in terms of the summatory functions of the Möbius and Liouville functions?	Unambiguous
611882798	Is there a mathematical rule to check if positive numbers $a$ and $b$ satisfy my modulo $P$ equation?	Ambiguous
62289779	I'm looking at the sequence $a_n = 3^n - 2^{n+100}$. How do I handle this?	Ambiguous
631320951	Prove that the intersection of any two cyclic subgroups of a group is also a cyclic subgroup.	Unambiguous
642449459	What's wrong with this fake proof that $2+2=5$?	Ambiguous
651017503	Prove by induction that $\sum_{k=0}^n k\cdot k! = (n+1)! - 1$ for all $n \in \mathbb{N}$.	Unambiguous
663490887	Question on existence of square root function!	Ambiguous
673791238	Why is a curve homeomorphic to a knot?	Ambiguous
682284995	"Expected count of ""double records"""	Ambiguous
693468876	Suppose $Y_1$ and $Y_2$ are independent and $X=Y_1+Y_2$. What is the distribution of $X$?	Ambiguous
70243106	How to apply $\epsilon-\delta$ definition to a 2-valued function?	Ambiguous
712396246	One tank holds more than another by some amount. When the smaller one is about 2/3 full, it holds roughly as much as half the larger one. What is the capacity?	Ambiguous
72303933	Evaluate the limit $\lim_{n \to \infty} \frac{(\log(n))^a}{n^b}$ for positive real constants $a$ and $b$.	Unambiguous
73975836	How to solve linear recurrence relations using the method of repeated substitution?	Unambiguous
74164244	Prove that if a subgroup of index p is the smallest prime dividing the order of a finite group, then it is a normal subgroup.	Unambiguous
752344993	What are some unexpected real-world applications or occurrences of the natural logarithm in mathematics and science?	Unambiguous
76781320	Why does this curve cross the y-axis if $\cos(\sin x)$ is never zero?	Ambiguous
772834788	Is the Euler–Mascheroni constant $\gamma$ known to be irrational?	Unambiguous
783247615	How do I evaluate the polynomial P(x) at x = -2?	Unambiguous
791329195	How to calculate a percentage increase when the result is above 100%?	Unambiguous
802449459	Where is the logical fallacy in the common fake proof that $2+2=5$?	Unambiguous
813017476	Interpreting an agreement function	Ambiguous
821169872	Is there a positional numeral system base $b$ such that every rational number has a terminating representation?	Unambiguous
832923438	How does the RSA cryptosystem work, and what are the steps for key generation, encryption, and decryption?	Unambiguous
843114691	How to find $\lim\limits_{n\to\infty} A^n$?	Ambiguous
851559640	How to find all natural numbers $n$ for quadratic permutations with a perfect square sum?	Ambiguous
863699937	The cyclic subgroups of $p^2$ order non-cyclic group are normal	Ambiguous
87551148	Prove that for every natural number $x$, the expression $2x^3 + x$ is always divisible by 3.	Unambiguous
883537073	Help with proof by induction, please! I may not be understanding how to work with factorials.	Ambiguous
893537073	Can you provide a step-by-step example of how to use proof by induction to prove a summation formula involving factorials?	Unambiguous
901647675	A Quadrilateral and A Triangle in a Trapizium	Ambiguous
912999909	In an arbitrary category, is the hom-set $Hom(X,Y)$ always equipped with an Abelian group structure?	Unambiguous
921466540	Find the $\lim_{x\to \infty} (\ln(x)-x)$	Unambiguous
934247524	Finding the probability that at least 8 houses in a row are of same colour.	Ambiguous
94486600	Given that ai (where i is the imaginary unit) is a root of the polynomial $x^3-bx^2+a^2x-a^2b=0$, prove this fact and find the other two roots in terms of $a$ and $b$.	Unambiguous
95308644	Do I need to include the definition of union in this proof?	Ambiguous
962396246	How do I calculate the volume of water a cylindrical tank can hold given its radius and height?	Unambiguous
973560940	If the weights of three different items are 10g, 25g, and 5g, how do I divide 1kg among them?	Ambiguous
982561078	Prove that if functions $f$ and $g$ are continuous at a point $x$, then their difference $f-g$ is also continuous at $x$.	Unambiguous
991568161	Evaluating $\lim\limits_{x \to \pi} \frac{(\pi -x)\sin x}{1 + \cos x}$	Ambiguous
1001074072	Is there a notion dual to that of a path in a topological space?	Ambiguous
1012664926	Evaluate the complex power $(-\sqrt{3} + i)^9$ using De Moivre's theorem.	Unambiguous
102365782	Prove that $-\int \frac{d^3p}{(2\pi)^3}\, p\,\frac{\partial f(p,t)}{\partial p} = 3F$, where $F = \frac{1}{(2\pi)^3}\int d^3p\, f(p,t)$, using integration by parts.	Unambiguous
1031283817	Given data points $(x_1,y_1),\dots,(x_n,y_n)$ and the model $f(x)=ae^{2x}+be^x+c$, find the transformation of the points $(x_i,y_i)$ needed to fit $f(z)=az^2+bz+c$ via least squares under the substitution $z=e^x$.	Unambiguous
1042539486	How do I solve for the min of $\int_{-1}^1| x^3-a-bx-cx^2|^2 dx$ and the max of $\int_{-1}^1 x^3 g(x) dx$ given the constraints on $g$?	Ambiguous
1052961334	How to prove a solution to a functional equation is unique?	Ambiguous
106574551	Evaluate $\oint_C P\,dx + Q\,dy$ where $P(x,y)=\frac{y+x}{x^2+y^2}$ and $Q(x,y)=\frac{y-x}{x^2+y^2}$, and $C$ is the circle $x^2+y^2=a$ oriented clockwise, given that $P_y=Q_x$ but $(0,0)$ is a discontinuity.	Unambiguous
1071941590	How do you use Cramer's rule to solve a system of linear equations using determinants?	Unambiguous
1082158739	True or False: Every 3-dimensional linear subspace of the vector space of $2 \times 2$ real matrices $\mathbb{R}^{2 \times 2}$ contains at least one invertible matrix?	Unambiguous
109258546	How to find the parabolic curve equation from length $L$ and area $S$ based on my picture?	Ambiguous
1103328473	How to find the numerical solutions for $2x - \tan(x) = 0$ in the interval $[0, 2\pi]$?	Unambiguous
1114268142	A coin is tossed three times, defining two related random variables from the outcomes. Find their joint pmf and show they're dependent.	Ambiguous
1122881207	How does the irreducibility of a polynomial over a finite field depend on the size of the field?	Unambiguous
1131037122	Prove that for any real number $\lambda$, the integral $\int_0^\infty e^{-x} \sin(\lambda x) \, dx = \frac{\lambda}{1+\lambda^2}$.	Unambiguous
1142166855	How to find the maximum value of the objective function $x + y$ subject to a set of linear constraints?	Unambiguous
1152049971	Prove that the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ cannot be transformed into the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ by any real coordinate transformation, without substituting $b \to ib$.	Unambiguous
1163786586	How to solve an inequality involving the sum of two absolute value expressions, such as $|ax + b| + |cx + d| < k$?	Unambiguous
11741706	What are some practical applications of matrix multiplication in computer science and engineering?	Unambiguous
1183111111	If $X$ is a topological space, does the property still hold for the product of two continuous functions?	Ambiguous
1192022771	In mathematical learning, does the complexity of notation generally decrease as one advances to more abstract levels of study?	Unambiguous
1201980947	Which value is greater, $2^{\sqrt{2}}$ or the mathematical constant $e$?	Unambiguous
121609404	Find all values of $c \in \mathbb{Z}_5$ such that the quotient ring $\mathbb{Z}_5[x]/\langle x^3+2x+c\rangle$ is a field.	Unambiguous
1222539486	(a) Using the best approximation theorem, compute $\min_{a,b,c} \int_{-1}^1 |x^3 - a - bx - cx^2|^2\, dx$. (b) Find $\max \int_{-1}^1 x^3 g(x)\, dx$ subject to $\int_{-1}^1 g(x)\,dx = \int_{-1}^1 xg(x)\,dx = \int_{-1}^1 x^2 g(x)\,dx = 0$ and $\int_{-1}^1 |g(x)|^2\,dx = 1$.	Unambiguous
1233087356	Why is the definition of prime numbers written to include 2	Ambiguous
1243399461	A polynomial defined by two squared terms is divided by two linear factors, giving known remainders. Find the coefficients.	Ambiguous
1254107862	How should I think about the concept of location for a point?	Ambiguous
126744139	Prove by induction that $2\cdot 8^n + 3\cdot 15^n + 2$ is divisible by $7$ for all $n \in \mathbb{N}$.	Unambiguous
1274749159	Equation exercise involving nth-roots in Serge Lang's Basic Mathematics.	Ambiguous
128795	How to prove the theorem of Riemann asked by Kenny?	Ambiguous
129942204	Looking for the logic of a sequence from convolution of probability distributions	Ambiguous
1304835425	Revisiting: Adding an optimizer to a nonlinear set of equations	Ambiguous
1312237243	"What are good ""math habits"" that have improved your mathematical practice?"	Ambiguous
1321406770	Iterated square roots over finite field. When do we hit a nonresidue?	Ambiguous
1333472866	Are $\sigma^2$ and $\mu$ random variables?	Ambiguous
1343806534	If $\lim_{x\to 0}(f(x) + \frac{1}{f(x)}) = 2$ and $\lim_{x\to 0} f(x)$ exists, prove that $\lim_{x\to 0} f(x) = 1$.	Unambiguous
1354325846	How do I formally show that the Zariski tangent space of the intersection of two closed subschemes is the intersection of the tangent spaces?	Ambiguous
1362530549	What are some mental math techniques for performing complex calculations without a calculator?	Unambiguous
137164588	How to find the general coefficient of the Cauchy product of two power series?	Unambiguous
1381838743	In the figure, two circles intersect at $P$ and $Q$...	Ambiguous
139788217	A hammer and a nail cost $1.10, and the hammer costs one dollar more than the nail. How much does the nail cost?	Ambiguous
140216344	Does the equation $ax + by = ab$ have any solutions for $\gcd(a,b)=1$?	Ambiguous
141385679	A matrix satisfying $AB-BA=B$	Ambiguous
142925677	I need the coordinates for an octagon.	Ambiguous
1432159465	Are two polygons homeomorphic?	Ambiguous
144582557	Determine whether $\sum_{n=2}^\infty \frac{1}{n\ln(n^3)}$ converges or diverges.	Unambiguous
1454336	If eigenvalues are positive, is the matrix positive definite?	Ambiguous
146531128	Dimension of a subspace of finite-dimensional product space $V$ equals $\dim V - 1$	Ambiguous
1472388784	What is the intuitive explanation for the formula of variance in probability theory?	Unambiguous
1481165928	What is the general formula for the k-th derivative of $G \circ F$?	Ambiguous
1494768	every permutation is either even or odd,but not both	Ambiguous
1501997424	What is the rigorous definition of the differential of a function in multivariable calculus?	Unambiguous
1512239918	Can someone help me with this: 2^(log(2x,x+2))+3^(log(2,x+3))=sqrt(-1-x)	Ambiguous
152354134	"What is the logical negation of the statement ""$A\subseteq B$"" in set theory?"	Unambiguous
1531871195	Given a function $a(x)$, is $$\max\left(\frac{1}{2}a'(x)\right) > \max\left(\frac{1}{3!}a''(x)\right) > \cdots$$ always true?	Ambiguous
154422707	Finding eigen values of a binary matrix with diagonal elements are all 0s and non-diagonals are 1s	Ambiguous
155747433	What are the standard tests for determining the absolute convergence of an infinite series?	Unambiguous
1563300982	I proved that if a symmetric real matrix commutes with all orthogonal matrices, it is a scalar multiple of the identity. Now how do I do this for orthogonal matrices?	Ambiguous
1571674072	How do you find the greatest integer whose square root differs from another integer's square root by at least one?	Ambiguous
158188703	For $f(x,y,z) = e^{\sqrt{xy}+\sqrt{xz}+\sqrt{zy}}$, determine whether a Taylor series expansion exists at $(0,0,0)$, given that the partial derivatives there take the indeterminate form $0/0$.	Unambiguous
1592072845	Why do different methods of factoring a quadratic trinomial, such as the AC method and trial and error, always yield the same linear factors?	Unambiguous
160730358	Sketch $\tan x$, solve $\cot x=0.15$, and evaluate $\csc x$ given $\cot x=0.15$.	Ambiguous
161853164	Provide an example of a binary relation on a set that is not reflexive, not symmetric, and not transitive.	Unambiguous
1623934173	Can any permutation of elements be represented as a series of rotations in some n-dimensional space?	Ambiguous
163265703	Under what conditions does every element of a monoid occur exactly once in its Cayley table?	Unambiguous
1644067963	What is the sum of $\sum_{n\in M(P)} \frac{1}{n}$ where $M(P)$ contains numbers with prime divisors in $P$?	Ambiguous
1652540359	What is the formal definition of an undecidable problem in computability theory?	Unambiguous
1663519278	$6$ persons enter the train during the journey with $6$ different tickets of the same class. How many different sets of ticket may they have had?	Ambiguous
1673310472	Is the P versus NP problem classified as a topic within the field of Operations Research?	Unambiguous
1681072836	Is every symmetric matrix diagonalizable, and what are the possible eigenvalues of an orthogonal matrix?	Ambiguous
1692096678	Suppose $A^2 + B^2 = AB$ and $AB - BA$ is invertible. What can be said about $n$?	Ambiguous
1701798274	Least value of $a$ for which $4ax^2 + \frac{1}{x} \geq 1$	Ambiguous
171488020	Prove that the set of signed circuits of an oriented matroid is invariant under reorientation of its elements.	Unambiguous
172110579	In how many ways can a batsman score exactly 100 runs using only 6s and 4s, where the order of scoring matters?	Unambiguous
1731249731	Let $X=(X_1,\dotsc,X_n)$ be a random vector with mean $0$. What is $\mathbb{E}[X_1\cdots X_n]$?	Ambiguous
174646238	How do I convince my students that the choice of variable of integration is irrelevant?	Ambiguous
1753125432	Trouble with telescoping algorithm question	Ambiguous
1763285074	Are there other functions that satisfy $|f(x)-f(y)|=|x-y|$?	Ambiguous
177664657	I'm trying to connect $\epsilon$ and $\delta$ for $|f(x) - f(c)| < \epsilon$. How do I do this?	Ambiguous
1782360710	Formula for Natural logarithm of $\pi$	Ambiguous
1791731822	Given some unlabeled 2D data points, how do I construct a k-nearest-neighbour graph and its Laplacian?	Ambiguous
1802146495	For $f(t)=\frac{2}{1+t^2}$, compute the convolution $(f*f)(t) = \int_0^t f(u)f(t-u)\,du$.	Unambiguous
1814341781	The integral of a function along any circle is zero. What does this mean, and what follows from it?	Ambiguous
182227317	Explaining $\cos^\infty$	Ambiguous
1831067717	How do you get $\frac{2\sum_{k=0}^n2^n}{n+1}=2^{n+1}$?	Ambiguous
184108308	Apostol proof divides by zero?	Ambiguous
1852062239	Numerical algorithms for approximating the polar decomposition of a matrix	Unambiguous
186687510	A shape is cut into 3 equal parts. Show one part contains 2 points at distance at least one.	Ambiguous
187330556	$a^{n}$: is it arithmetic or geometric, or neither?	Ambiguous
1883791855	How does $x^2-8x+17=0$ have nonreal solutions?	Ambiguous
189988784	Why is it a common misconception that non-real numbers consist only of square roots of negative numbers?	Unambiguous
1902394502	Under what conditions does a complex number $z=x+iy$ in the third quadrant imply that the ratio $\frac{\bar{z}}{z}$ also lies in the third quadrant?	Unambiguous
1913098928	For a fixed real number $x > 0$, does the sequence $x^{1/n}$ converge as $n$ approaches infinity, where $n$ is a natural number?	Unambiguous
1922823907	 Prove that $\sum_{n=2}^\infty \frac{1}{n\ln(n^3)}$ diverges using the comparison test.	Unambiguous
1933059714	How does one evaluate the multiplication $f(2)\cdot f(3)\cdot f(4)\cdots f(15)$ by formulating?	Ambiguous
1942594206	Prove that there is no $2 \times 2$ real matrix $A$ such that $A^2 = \begin{pmatrix} 0 & 1 \ 0 & 0 \end{pmatrix}$.	Unambiguous
195639890	Minimum value of $\cos(A-B)+\cos(B-C)+\cos(C-A)$?	Ambiguous
196225922	An example for a calculation where imaginary numbers are used but don't occur in the question or the solution.	Ambiguous
197983405	Prove the inequality $\|\nabla f\|_p \le C (\|\nabla \times f\|_p + \|\nabla \cdot f\|_p)$ for vector fields $f$ in $L^p$ spaces on $\mathbb{R}^3$.	Unambiguous
1984693000	Prove that $\frac{1}{\zeta(3)} < 2\gamma - 1$, where $\zeta(s)$ is the Riemann zeta function and $\gamma$ is the Euler-Mascheroni constant.	Unambiguous
199724075	Where can I find the book 'Here's a Cool Problem: A Collection of Short Questions with Clever Solutions'?	Unambiguous
2001302479	I'm having trouble with this function: $$f(x)=\begin{cases}1, & \text{if } x\in \Bbb Q\\ -1, & \text{if } x\in \Bbb R\backslash \Bbb Q. \end{cases}$$	Ambiguous
20126918	Why is the mathematical constant $\pi$ defined as the ratio of a circle's circumference to its diameter rather than its ratio to the radius?	Unambiguous
202361118	Prove that $4x-x^4 \leq 3, x \in \Bbb R$	Ambiguous
2032006621	How do I calculate the number of subsets of a finite set given its cardinality?	Unambiguous
20462748	What is the difference between two related set-builder notations?	Ambiguous
2052054106	Is it true that pointwise convergence with a uniform bound implies convergence of the integrals?	Ambiguous
206742532	Why is the right endpoint given as $3\frac{i^2}{n^2}$ for the limit $\lim_{n \to \infty} \sum_{i=1}^{n} f(c_{i}) \Delta x_{i}$?	Ambiguous
207105575	What is the difference between Completeness and Soundness in first order logic?	Ambiguous
2081978354	Distribution of $\frac{X+Y}{X-Y}$	Ambiguous
209788217	A hammer and a nail together cost $1.10, and the hammer costs $1.00 more than the nail. What is the cost of the nail?	Unambiguous
2101788328	Find the numerical value of $\sin 10^\circ \sin 50^\circ \sin 70^\circ$.	Ambiguous
2113944628	In the definition of a stopping time, how important are the filtration conditions mentioned earlier?	Ambiguous
2121079395	Inclusion Exclusion Principle (Probability) Problem: What is P(Rain)?	Ambiguous
2131783734	Can you explain the logical basis and validity of the principle of mathematical induction for proving statements over natural numbers?	Unambiguous
2141896418	For a set of $n$ points in the plane where no four are collinear, what is $f(n)$ for the case where no three are collinear?	Ambiguous
2151214106	Find all polynomials $P(X)$ such that $P(x^2-y^2) = P(x-y)P(x+y)$ for all complex numbers $x, y$.	Unambiguous
216700216	Different solutions of $x+y+z=10$ where $x$, $y$, $z$ are all positive integers and $x, y, z \leq 10$	Ambiguous
2173086320	Evaluate $\lim_{x\to 2}\frac{2x-4}{x^2+2x-6}$ and explain why direct substitution giving $0$ differs from the result obtained via L'Hôpital's rule.	Unambiguous
2183289735	Is the arc length of a circle always irrational between two points with rational coordinates?	Unambiguous
2191762325	Is $\sim (\sim p) \rightarrow p$ always true?	Ambiguous
220227317	What is the limit of the sequence defined by the iterated cosine function $x_{n+1} = \cos(x_n)$ as $n$ approaches infinity?	Unambiguous
2214256389	Alternating sum involving reciprocals of powers of primes equals $\ln(\frac{\pi}{4})$	Ambiguous
2221319618	An inequality, comprising of liminf and Sup of a set	Ambiguous
2232498847	Is 1/x a surjective function when the domain and the range are all real numbers?	Ambiguous
2241894173	Does the sum of the reciprocals of all prime numbers converge or diverge?	Unambiguous
2253247615	How to find $P(-2)$ here?	Ambiguous
226602860	How to calculate which point is furthest from the origin in the direction of a normalized vector?	Ambiguous
2273502297	How to show $\log(x)$ loses to powers?	Ambiguous
2281347	How can Euclidean geometry be axiomatized or derived using criteria from hyperbolic geometry?	Unambiguous
22911194	How to solve a linear recurrence relation where the nth term depends on the sum of all previous terms?	Unambiguous
230290435	Find five distinct positive integers whose reciprocals sum to 1.	Unambiguous
2314019674	Prove that the value of $\frac{\arcsin(3/5)}{\pi}$ is an irrational number.	Unambiguous
2322353213	Are there any other Lie subgroups of $\textrm{Gl}(n, \mathbb{R})$ and is there a classification for them?	Ambiguous
2331347	Studying Euclidean geometry using hyperbolic criteria	Ambiguous
2344255331	If $ \lim_{x\to \infty} (f(x+1)-f(x))=1$, then $ \lim_{x\to \infty} \frac{f(x)}{x}=1$?	Ambiguous
2353395798	Definition of a matrix by matching two vectors	Ambiguous
236984271	How does the isomorphism between $\mathbb{K} \otimes M_n(\mathbb{C})$ and $\mathbb{K}$ work?	Ambiguous
2371249731	What is the formula for the first order moment (mean vector) of a multivariate Gaussian random vector?	Unambiguous
2381251548	Provide an example of a set in a metric space that is bounded but not totally bounded.	Unambiguous
2394488057	Can a function be represented by a Laurent series expansion in a neighborhood of a non-isolated singularity?	Unambiguous
240217335	In how many ways can a total of $1.50 be formed using exactly 50 US coins?	Unambiguous
241329838	How to simplify a square root of a number by extracting perfect square factors?	Unambiguous
2421376320	Proving that if $a+b$ and $ab$ are of the same parity, then $a$ and $b$ are even.	Ambiguous
2434383384	How to extract the vector from a rank-$1$ matrix?	Ambiguous
2441895699	I'm stuck on this part of the proof: $\cot C \frac{\sin(A-B)}{\cos A \cos B}$. How do I get to the other side?	Ambiguous
24571874	Can I use my powers for good?	Ambiguous
246395543	What are the formulas for generating all primitive Pythagorean triples?	Unambiguous
247872256	What is the geometric interpretation of finding the $n^{ ext{th}}$ roots of a complex number in the complex plane?	Unambiguous
2482837226	Show that a subset of a complete metric space is complete.	Ambiguous
249538941	If $xy$ and $x+y$ are both even integers (with $x,y$ integers), then $x$ and $y$ are both even integers	Ambiguous
250884228	How do you calculate the reliability function for a continuous uniform distribution given its lower and upper bounds?	Unambiguous
2513035983	How can I find all pairs of integers $(n, m)$ that satisfy the equation $\frac{150 + n}{15 + n} = m$?	Unambiguous
2522880747	Product of Summation for a single table of values.	Ambiguous
2531002480	How to go from talking about line segments to points to prove $X(A)$ is compact?	Ambiguous
254488020	Let $M$ be a matrix, and define $V(M)$ from the signs of vectors in $\ker(M)$. Prove that $V$ is invariant under change of coordinates.	Ambiguous
255859968	Is $X/R$ homeomorphic to $Y$ for the broom spaces?	Ambiguous
2562520461	How to minimize $AP/PB$? It doesn't seem doable for some locations.	Ambiguous
2571496959	What are the necessary and sufficient conditions for a sequence of continuous functions to converge uniformly on a compact set?	Unambiguous
2581279205	Can an irrational number be classified as odd or even under the standard definition of parity for integers?	Unambiguous
2594437668	What are some good, elementary and maybe also interesting proofs by induction?	Ambiguous
2602576976	How do I find the inverse of a $2\times2$ matrix if the determinant is zero?	Unambiguous
2611626298	Prove that $2^{\log_4 n} = n^{\log_4 2}$ using logarithmic identities.	Unambiguous
2623811446	What is the value of the infinite series $\sum_{n=1}^{\infty}\frac{1+2+\cdots +n}{n!}$?	Unambiguous
2631891318	"Number of bit strings of length 8 that do not contain ""$100$""?"	Ambiguous
2641311558	What is the formal definition of a set in Zermelo-Fraenkel set theory and how is it intuitively conceptualized?	Unambiguous
2653960592	Finding the equation of hyperbola that passes through $(2,0)$ and has foci points $(2,3)$ and $(1,0)$	Ambiguous
2663791855	Why does the quadratic equation $x^2-8x+17=0$ have non-real complex solutions?	Unambiguous
2671668681	Provide an example of a 2D planar shape whose center of mass lies on its boundary.	Unambiguous
268219343	How can I prove that the propositional logic formula $[(p \to \neg q) \wedge q] \to \neg p$ is a tautology using a truth table or natural deduction?	Unambiguous
2691141210	I am trying to do a set proof and I don't understand where the part highlighted in green comes from.	Ambiguous
270344268	How do I solve a rational inequality where the variable $x$ appears in the denominator?	Unambiguous
2711501369	How to prove a function is one to one given $f(f(m)+f(n))=m+n$?	Ambiguous
2722207598	Prove or disprove: if $A\times A = B\times B$ then $A=B$.	Ambiguous
273738236	How to calculate the area of a triangle ABC when given three position vectors $a, b$, and $ c$ in 3D?	Ambiguous
2744285591	Does the expression $\frac{1}{1-hL}$ have an upper bound for $h > 0$ and $L > 0$ such that $hL < 1$?	Unambiguous
2754080901	Using a block matrix to show that $A$ and $B$ commute	Ambiguous
2763582088	Solving for polynomial $f(x)$ satisfying $f(x)f(y)=f(x)+f(y)+f(xy)-2 \forall x\in \mathbb{R}, f(1)\ne 1, f(3)=10$	Ambiguous
2773729349	If a series $\sum a_n$ converges and the sequence of its terms $(a_n)$ is monotonically increasing, why must the terms be non-positive?	Unambiguous
2782494260	List mathematical theorems, conjectures, or concepts that are named after geographic locations.	Unambiguous
2793265216	How to perform horizontal and vertical rescaling of a function $f(x)$ in algebra?	Unambiguous
280114560	Sum of normal bundle and tangent bundle.	Ambiguous
2811001101	Let $f:\mathbb{R}\to\mathbb{R}$ be an arbitrary function (not necessarily continuous). Prove there exists an interval $(a,b)$ and $c>0$ such that for every $x\in(a,b)$, there is a sequence $x_n\to x$ with $|f(x_n)|\le c$.	Unambiguous
2824564068	What is the formula for 0 + 1 + 3 + 6 + 10 + 15 ... ?	Ambiguous
283227211	Is there a real-valued function $f: \mathbb{R} \to \mathbb{R}$ such that $f(f(x)) = f(x)$ for all $x$ and $f(0) = 1$?	Unambiguous
2841381335	How to tell which function fits my series to make it a Riemann Sum?	Ambiguous
285217315	How do you tell the difference between the minus signs?	Ambiguous
2861432379	What is the minimum of $F(x,y,z)=\frac{1}{x+y}+\frac{1}{x+z}-\frac{1}{x+y+z}$?	Ambiguous
287217315	In algebra, what is the difference between the unary minus operator used to indicate a negative number and the binary minus operator used for subtraction?	Unambiguous
2883711681	How to compute the matrix derivative of the scalar function $f(X) = \operatorname{tr}(B X X^T A)$ with respect to the matrix $X$?	Unambiguous
2891435003	How to find $P(X_i=1)$ to get the expected number of empty boxes?	Ambiguous
2901562459	Prove that the sequence $(1 + n^{-2})^n$ converges to 1 as $n$ approaches infinity.	Unambiguous
2911304823	Can someone explain the parts of this equation?	Ambiguous
2922657952	"What's the relationship between mathematical logic and just plain old ""logic""? The notation differs, do the concepts? The rules?"	Ambiguous
293982093	Proof for $\lim_{n \to \infty}\left(1+\frac{a_n}{n}\right)^n = e^a$?	Ambiguous
294251033	What are the search terms for the largest subset with no 3 element progressions?	Ambiguous
2952766082	How do you simplify the algebraic expression $\frac{10x\Delta x + 5(\Delta x)^2}{\Delta x}$ to get $10x + 5\Delta x$, assuming $\Delta x \neq 0$?	Unambiguous
296608477	What can you tell me about $4^{n+1}+5^{2n-1}$ and 21?	Ambiguous
2971329254	Solve the ODE $\frac{d^2y}{dt^2}+10\frac{dy}{dt} = -5t^2+t^4\cosh(t)$ using the Fast Fourier Transform method.	Unambiguous
2983422194	Compute the sum of the infinite series $\sum_{k=1}^{\infty} \frac{1}{k(k+2)}$ using partial fraction decomposition.	Unambiguous
299244424	If rank x < rank y, does it follow that x is an element of y?	Ambiguous
300687510	Provide an example of a Pigeonhole Principle problem involving points placed inside a square area.	Unambiguous
3012863657	Simple Ratios Shortcut - Why does it work?	Ambiguous
302721784	What is the absolute value of the complex exponential function $e^{iz}$ where $z$ is a real number?	Unambiguous
3032581976	How do I find the limit of a function as x approaches a specific value?	Unambiguous
3041645747	Compute the derivative of $x$ with respect to $\frac{1}{x}$.	Unambiguous
305997152	Why does the General Leibniz Rule for the nth derivative of a product have the same binomial coefficient structure as the Binomial Theorem for raising a sum to the nth power?	Unambiguous
3061475726	How to find three sets of three collinear points in a given 2D point set?	Unambiguous
307981781	$$f(x)=x^3-x^2-14x+24$$	Ambiguous
3081040128	How do I evaluate the improper integral $\int_1^\infty \frac{dx}{x^2\sqrt{x^2-1}}$ using trigonometric substitution?	Unambiguous
3092278214	Calculate the Shannon entropy of the random variable representing the number of heads obtained when flipping a fair coin ten times.	Unambiguous
3102751401	If $a\mid b$ and $a>0$ then $(a,b)=a$	Ambiguous
3114057665	Are these topologies the same?	Ambiguous
3121466559	How to find the angle between two vectors using the dot product formula derived from the law of cosines?	Unambiguous
3131602977	What's an example of an infinitesimal?	Ambiguous
3141899905	How can physical quantities that are modeled by irrational numbers be realized or measured in the physical world?	Unambiguous
3152604608	Prove that the constant function $f(x) = 2a + 3b$ is continuous for all x in its domain, where a and b are real constants.	Unambiguous
3161761660	Find the value of $a$ such that the range of the function $f(x) = ax(1-x)$ contains the interval $[0,1]$.	Unambiguous
3171695878	Let $A$ and $B$ be square matrices of order n so that $AB = BA$. Prove $\mathrm{rank}(A+B) \leq \mathrm{rank}(A) + \rm{rank}(B) - \mathrm{rank}(AB)$.	Ambiguous
3184269673	Is there a notation to indicate that a variable in this equation is constant?	Ambiguous
319127376	In the context of the extended real number system, why is the limit of $\frac{1}{x}$ as $x$ approaches $0$ from the right defined as $\infty$, while the limit from the left is $-\infty$?	Unambiguous
3203905420	Would a certain cyclic or dihedral group be an example? How do we find its subgroups?	Ambiguous
3212138866	What is the value of the definite integral $\int_{0}^{\pi/4} \frac{\tan^2 x}{1+x^2} \text{d}x$ from the 2015 MIT Integration Bee?	Unambiguous
3221688323	How to find the coefficients of a quadratic polynomial given three coordinate points using the method of undetermined coefficients or differentiation?	Unambiguous
32313582	Prove that every continuous function from a closed interval $[a, b]$ to itself has a fixed point.	Unambiguous
324254561	Provide a proof that $\arctan(x) = \arcsin(x/\sqrt{1+x^2})$ for all real numbers $x$.	Unambiguous
3251222768	Why are vector valued functions 'well-defined' when multivalued functions aren't?	Ambiguous
326738236	How do I calculate the area of a triangle with vertices defined by the position vectors $a$, $b$, and $c$ in 3D space using the cross product?	Unambiguous
3272597743	Can a certain type of integral be solved using the gamma function method?	Ambiguous
3282152735	What is the definition of an open set in the context of a general topological space?	Unambiguous
3292820510	How to find the rotation quaternion from the angles between coordinate axes $x_1, y_1, z_1$ and $x_2, y_2, z_2$?	Ambiguous
3302365371	Prove $ \frac{1}{x^{2n}}+x^{2n}$ is an integer for all $n$.	Ambiguous
331401477	Let $A$ be a commutative free algebra of rank $2$ over a ring. Is $A$ isomorphic to a quotient of a polynomial ring by an irreducible element?	Ambiguous
3322717459	What is the formula for the expected value of a weighted sum of positive random variables?	Unambiguous
3332639179	Am I wrong about the Monty Hall Problem?	Ambiguous
3341176215	Prove by induction that $3^{3n+1} + 2^{n+1}$ is divisible by 5 for all non-negative integers $n$.	Unambiguous
3351271168	What is the difference between the square of a logarithm, $(\log x)^2$, and the logarithm of a square, $\log(x^2)$?	Unambiguous
3361376320	Prove that for integers $a$ and $b$, if $a+b$ and $ab$ have the same parity, then both $a$ and $b$ must be even.	Unambiguous
337620187	What do these symbols mean?	Ambiguous
3382401656	Compute the value of ${34}^{429} \mod 431$.	Unambiguous
3391290374	Can you provide an intuitive explanation and a proof for both parts of the Fundamental Theorem of Calculus?	Unambiguous
34019236	Efficient numerical algorithms to compute the Euler gamma function for real and complex arguments	Unambiguous
3411194644	Is every integral extension of commutative rings a finite extension?	Unambiguous
3422619473	Does the one-sided derivative of a real power series always exist at the boundary of its interval of convergence if the series converges there?	Unambiguous
343846706	What strategies can I use to find a conformal map from a lens shape to the first quadrant?	Ambiguous
344547928	What is the minimum value of $a+b+\frac{1}{ab}$ if $a^2 + b^2 = 1$?	Ambiguous
3452273478	"Translate ""There is exactly one book that no students read"" into first-order predicate logic using predicates $\text{book}(y)$, $\text{student}(x)$, and $\text{reads}(x,y)$."	Unambiguous
3461435003	How do I calculate the total number of empty bins in the balls and bins problem given n balls and m bins?	Unambiguous
347973713	What are the common notation styles and conventions for the second-derivative symbol in calculus?	Unambiguous
3482489152	Why is the bracket always zero?	Ambiguous
3493005021	Proof Verification: Differentiability implies continuity.	Ambiguous
3502096678	Provide a set of practice exercises on basic matrix arithmetic, including addition, subtraction, and multiplication.	Unambiguous
3512618534	Compute the indefinite integral of $\int \frac {1+\sin (x)}{1+\cos (x)} dx$ with respect to $x$.	Unambiguous
352159505	Find the value of $x$ that solves the linear congruence $3x + 7 \equiv 6 \pmod{13}$.	Unambiguous
35312139	Number of relations that are both symmetric and reflexive	Ambiguous
354114107	Determine the rotation necessary to transform one point on a sphere to another.	Ambiguous
3551899905	How irrational quantities physically exist in nature?	Ambiguous
3561894173	Sum of reciprocal of primes	Ambiguous
3573628	Find a general formula for the n-th derivative of the function $f(x)=\frac {x} {x^2-1}$.	Unambiguous
3581314138	How do I prove my integral limit is $0$ using the substitution $xt=u$?	Ambiguous
359332632	What is the probability that a person which has a positive test result in AIDS, is indeed infected with AIDS?	Ambiguous
3601383373	Provide the solution and explanation for Question 55 from the GRE Mathematics Subject Test form 1268.	Unambiguous
3612452316	Evaluating a contour integral using Cauchy's generalised integral formula	Ambiguous
3622705618	Evaluate $\lim_{n\to\infty} \frac{e^{1/n} + e^{2/n} + \cdots + e^{n/n}}{n}$.	Unambiguous
3633021170	Provide an example of a sequence of Lebesgue integrable functions on $R$ that converges pointwise to a limit function, but where the integral of the limit is not equal to the limit of the integrals.	Unambiguous
3644569632	In $\triangle ABC$, $BD$ is a median, $\angle DAB=15$ and $\angle ABD=30$. Find $\angle ACB$.	Unambiguous
3653303672	Line intersecting circle from origin	Ambiguous
36692765	Find all complex roots of the polynomial equation $z^4 + 2z^3 + 6z - 9 = 0$	Unambiguous
3672138847	If $f$ is a holomorphic function on the unit disk $\mathbb{D}$ such that $|f(z)| \le 1$ for all $z \in \mathbb{D}$ and $f(1/2) + f(-1/2) = 0$, prove that $|f(0)| \le 1/4$ and specify if a Möbius transformation is required for the proof.	Unambiguous
368685435	Trying to get a bound on the tail of the series for $\zeta(2)$	Ambiguous
369820424	What are the standard methods and laws for simplifying Boolean functions in digital logic?	Unambiguous
3703117241	Complex function to Taylor and Laurent series	Ambiguous
3711516096	When does $A^TA=0$ imply $A=0$?	Ambiguous
3721950367	Does equality of the complexifications imply equality of the algebras?	Ambiguous
373551350	Find all complex solutions to the equation $x^4+24x^3+18x^2-27=0$.	Unambiguous
3741735792	Prove that for a homogeneous polynomial $P$ of degree $k$ in $n$ complex variables, $$P(z_1,\dots,z_n)=\frac{1}{k!}\sum_{\epsilon_1,\dots,\epsilon_k\in\{0,1\}}(-1)^{k-\sum_{j=1}^k \epsilon_j}P\left(a+\left(\sum_{j=1}^k\epsilon_j\right)z\right)$$ for any $a\in\mathbb{C}^n$.	Unambiguous
3751145215	Consider a matrix $A$ that is full rank. Is $A(A^\top A)^{-1}A^\top = I$?	Ambiguous
3762987994	Provide a list of examples of definite integrals solvable using the Feynman trick (Leibniz integral rule).	Unambiguous
377462723	What are some examples of mathematical games suitable for undergraduate students that can be used for teaching discrete mathematics or game theory?	Unambiguous
378184593	Is it necessary for the angle $\theta$ to be measured in radians for the limit $\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1$ to hold?	Unambiguous
3792186187	How do you calculate the first Fourier coefficient $a_0$ for a periodic function in its Fourier series representation?	Unambiguous
380574465	How do you find the complex roots of the polynomial equation $x^4 - 16 = 0$ to show they are $2$, $-2$, $2i$, and $-2i$?	Unambiguous
3812289082	Compute the definite integral $\int_1^3\frac{\ln x}{x^2+3}\ dx$	Unambiguous
382269798	Is it true that $E(X^a) = \int x^a g(x)\,dx$?	Ambiguous
3834683394	What is the category-theoretic concept that unifies affine hull, convex hull, closure, and interior as intersections of all sets with a given property containing a set?	Unambiguous
3844398943	What does 'abstract' mean in this context?	Ambiguous
3854367689	Does there exist an ordered field extension of the rational numbers $\mathbb{Q}$ in which $\mathbb{Q}$ is not a dense subset?	Unambiguous
3861867732	Gradient and Hessian of function on matrix domain	Ambiguous
387791508	Let $A, B, X$ be subgroups of $G$ with $A \subset X \subset AB$. Prove that $A(B\cap X) = X$.	Unambiguous
3881950039	What is the identity property of multiplication by one in arithmetic?	Unambiguous
3894073406	Infinite subset of natural numbers	Ambiguous
390823613	What is the time complexity in $\Theta$-notation in terms of $n$ for the Merge Sort algorithm?	Unambiguous
3911896195	Let $M$ be a module over a ring. If $\operatorname{Hom}(M,M)$ is free, what can we conclude about $M$?	Ambiguous
3923300982	Prove that if a real orthogonal matrix A commutes with all real orthogonal matrices of the same dimension, then A must be a scalar multiple of the identity matrix.	Unambiguous
3933178788	What are the methods to determine the type of stationary point of a single-variable function without calculating the second derivative?	Unambiguous
3941587079	How do I prove the map $(i,j) \mapsto (i+j, ij_0 - i_0 j)$ is surjective?	Ambiguous
3953764582	Is there a convenient implementation in R to compute my incomplete gamma function integral?	Ambiguous
3962918597	Find angle of which a cannon of fixed length has to fire	Ambiguous
3971651728	Prove that the union and intersection of two recursively enumerable sets are also recursively enumerable.	Unambiguous
3983487811	Evaluate the limit $\lim_{x \to 0} {1-x\cot x\over x^2}$	Unambiguous
3991302479	Prove that the limit of the Dirichlet function does not exist at any point in its domain.	Unambiguous
4003493088	$\mathbb{C} \cup \{ \infty \}$ and $\mathbb{R} \cup \{ -\infty, +\infty \}$	Ambiguous
4012520262	Show that $a_{n+1}= 1 + \frac{1}{a_n}$ converges.	Ambiguous
402377705	How many repetitions do I need to find a prime between n and 2n?	Ambiguous
403726827	Showing a Mapping Between $\left\langle a,b \mid abab^{-1}\right\rangle$ and $\left\langle c,d \mid c^2 d^2 \right\rangle$ is Surjective	Ambiguous
4043564289	What is the expected number of returns to the origin for a random walk?	Ambiguous
4051232385	How do I prove $\mathbb{Z}[w]/(a)$ is isomorphic to $\mathbb{Z}_{N(a)}$ using the norm?	Ambiguous
406700216	How many distinct integer solutions are there for the equation $x + y + z = 10$, where $x$, $y$, and $z$ are all positive integers such that $x, y, z \leq 10$?	Unambiguous
4073533343	Counter examples in measures	Ambiguous
4081612241	What is the logarithmic identity for the sum of logarithms of multiple terms?	Unambiguous
4093743564	Another coins weighing puzzle	Ambiguous
4104336	If all eigenvalues of a Hermitian matrix are positive, is the matrix positive definite?	Unambiguous
411166553	Can you provide a proof for the definite integral identity $\int_a^b f(x) dx = \int_a^b f(a+b-x) dx$?	Unambiguous
412532830	How can I get more intuition about the complex numbers (C) and their multiplication and exponentiation?	Unambiguous
4132054106	What are some common generalizations or variations of the Lebesgue Dominated Convergence Theorem in measure theory?	Unambiguous
41429758	Prove that every injective entire function from the complex plane to itself is a linear polynomial.	Unambiguous
41520813	How does a number system without irrational numbers handle circles?	Ambiguous
4162730855	Water is poured into a container at a given rate. How fast does the level rise at a certain height?	Ambiguous
4174194118	Area between square and circumscribed circle is less than half the area of the circle.	Ambiguous
4182614579	How do I evaluate the limit $\lim_{x\to1} \frac{\sqrt{2x-1} -1}{x^2-1}$ step-by-step?	Unambiguous
4191483533	How do I solve a linear equation for a real variable $x$?	Unambiguous
4201145215	What is the definition of a matrix inverse in linear algebra and how does it relate to the identity matrix?	Unambiguous
421218977	Let $\Omega$ be a rectangle. What is the smallest space to which $\delta_0$ belongs?	Ambiguous
4222361340	What is the limit as $x$ approaches infinity of $\frac{1}{x^{n-1}}$ for $n > 1$?	Unambiguous
4232770657	Find the Maclaurin series for $\ln(1-x)$ and state its interval of convergence using sigma notation.	Unambiguous
4242344095	Find coefficient of $x^2$ in a complicated expansion	Ambiguous
4251725118	What are the axioms needed to state the Fixed Point Theorem? Does it apply to ZF, Robinson arithmetic, or PA?	Ambiguous
4262732822	How do I solve a system of two equations with two variables where one equation is additive $(x + y = a)$ and the other is multiplicative $(xy = b)$?	Unambiguous
4272805228	Let $X_1,\dots,X_{20}$ be independent Poisson random variables with mean $1$. Using Markov's Inequality, find and verify the bound for $P(X_1+\cdots+X_{20} > 15)$.	Unambiguous
42820096	Basic addition and subtraction word problems for preschool and kindergarten children.	Unambiguous
4292824634	Prove that $\sin(x/2) + 2\sin(x/4) \leq 3\sqrt{3}/2$ for all $x \in [0, 2\pi]$.	Unambiguous
430292425	How do I solve a piecewise function if it's not defined on $(-4,-1)$?	Ambiguous
4312059626	choose 1 song to be played 3 times and the rest only once	Ambiguous
432494832	What is the definition of a ring that is complete with respect to the I-adic topology defined by an ideal $I$?	Unambiguous
4331613067	subtracting the eigenvalue matrix from the original transformation of a 2x2 hit's the other eigenvalue proof	Ambiguous
434546701	Find the number of positive integers with exactly 5 digits whose digits add up to 42	Unambiguous
4353440597	What is the maximum order of an element in the symmetric group $S_n$ if all elements of the permutation's cycle decomposition must have odd lengths?	Unambiguous
4362512671	Find an example of a function $f$ and a set $W$ such that $f(f^{-1}(W)) \neq W$, where $f^{-1}(W)$ denotes the preimage of $W$.	Unambiguous
43762621	Is there at least one irrational number with the property that it cannot be defined by a finite string of information?	Ambiguous
4384494965	How many ways I can make \$5 in 50c, \$1 and \$2 coins?	Ambiguous
4392112893	How to calculate the 100th term of an arithmetic and a geometric series?	Unambiguous
4402108420	How to prove that the definite integral $\int_{0}^{\pi}(1+2x)\frac{\sin^3(x)}{1+\cos^2(x)} dx = (\pi+1)(\pi-2)$?	Unambiguous
4412806957	Under what conditions does a line intersect a circle in a 2D plane?	Unambiguous
442311580	Coordinate-free proof of $\operatorname{Tr}(AB)=\operatorname{Tr}(BA)$?	Ambiguous
4431726540	How to differentiate $x^{\frac1x}$?	Ambiguous
4441432379	Find the minimum value of the expression $\frac{1}{x+y} + \frac{1}{x+z} - \frac{1}{x+y+z}$ for positive real numbers $x, y, z$ subject to a fixed sum $x+y+z=S$.	Unambiguous
4452999909	Does Hom(X, Y) always form an abelian group?	Ambiguous
4461891318	How many binary strings of length 8 do not contain the substring '100'?	Unambiguous
44711194	Solving the recurrence relation that contains summation of nth term	Ambiguous
448942204	How to derive the general term of a sequence formed by the repeated convolution of a probability distribution with itself?	Unambiguous
4493254331	Prove that the definite integral $\int_0^\infty(\arctan(1/x))^2 \mathrm{d}x$ converges to $\pi\ln 2$.	Unambiguous
4502326677	What is the signifigance of a saddle point?	Ambiguous
4512732822	What do you call a system where $a+b=1$ and $ab=-1$?	Ambiguous
452451799	How to prove that $\lim_{x \to 1} \frac{(\sum_{k=1}^n x^k) - n}{x - 1} = \frac{n(n + 1)}{2}$ for a fixed positive integer $n$?	Unambiguous
4531258718	"The ""Smooth Factor"" in a number sequence"	Ambiguous
454846706	Find a conformal mapping from the lens-shaped intersection of the unit disks centered at $0$ and $1$ in $\mathbb{C}$ to the first quadrant.	Unambiguous
4553103052	Compute the limit as $n$ approaches infinity of $n^2\left(\left(1 + \frac{1}{n+1}\right)^{n+1} - \left(1 + \frac{1}{n}\right)^n\right)$	Unambiguous
4561491490	What is the group generated by $M_\pi$?	Ambiguous
457720688	What is the average power of this signal?	Ambiguous
4582932046	Is this formula correct: $\frac{|(p-a) \times d|}{|d|}$?	Ambiguous
4592675393	How to find the derivative of a power series?	Unambiguous
4601075740	What happens to the angles of an isosceles triangle if one vertex is at infinity?	Ambiguous
461808565	What is the standard mathematical notation for the dimension of a vector space $V$?	Unambiguous
4621194644	I'm having trouble coming up with a concrete example of an integral infinite ring extension where $R$ is infinitely generated but $S[R]$ is finitely generated.	Ambiguous
463973713	Problem in the second-derivative symbol.	Ambiguous
4643328149	Tangents to parabola $y^2=-4(x-1)$ with slopes in a certain range determine chords, bisected by $x=1$, of a circle. Find the equation of the circle.	Ambiguous
4653481798	Prove that $\int_0^t\frac{(B^1_s)^2}{\big((B_s^1)^2+(B_s^2)^2+(B_s^3)^2\big)^3}ds<\infty $ a.s.	Ambiguous
4663472866	Are the sample mean and sample variance considered random variables?	Unambiguous
467345024	What are the applications of Catalan numbers in computer programming and algorithm design?	Unambiguous
468553280	What is the number of banded $(0,1)$-matrices with exactly $k$ ones and prescribed row/column sums?	Ambiguous
4691531029	If an abelian group $G$ has property $N$, then $G$ is finitely generated	Ambiguous
4702901457	Determine for which values of $k$ and $a$ the series $\sum_{n=1}^\infty \frac{\ln^k(n)}{n^a}$ converges.	Unambiguous
4713255482	Evaluate $\lim_{x \to 0} \frac{1}{x} \int_{x}^{2x} e^{-t^{2}}\,dt$	Unambiguous
4723002761	How do I find the maximum value of a variable $x$ when it is subject to a given equality constraint?	Unambiguous
473298648	Is there a common symbol for concatenating two (finite) sequences?	Ambiguous
4741720548	Given a vector $v$, is it possible to find another vector $w$ such that $v$ is the result of the cross product of $w$ and some third vector $u$?	Unambiguous
4754768	Prove that every permutation of a finite set is either even or odd, but not both.	Unambiguous
476314408	What are the conditional probabilities for my simple OR-gate where $P(A)=0$ and $P(B)=0$?	Ambiguous
4774221432	What are the general integer solutions to the Diophantine equation $a^2+b^2=c^2-1$ for positive integers $a, b, c$?	Unambiguous
478744139	I'm having trouble with the expression 2*8^n + 3*15^n + 2. I tried using k+1 but I can't find a solution.	Ambiguous
4792368762	Given that $\frac{\sin(x)}{a}=\frac{\cos(x)}{b}$ for some constants $a$ and $b$, what is the value of $a\sin(2x)+b\cos(2x)$ in terms of $a$ and $b$?	Unambiguous
4801673403	A function is never zero, takes both signs, and is continuous. How is that possible?	Ambiguous
481378970	For a right triangle with a hypotenuse of 12 cm, what measures of the acute angles would produce the maximum area?	Unambiguous
482736346	What are the geometric methods to fold a rectangular piece of A4 paper into three equal parallel strips?	Unambiguous
4833673715	I need help with the heat equation in a 3D ball.	Ambiguous
4841494248	Evaluate the indefinite integral $\int \frac{\sin(4x)}{\sin(x)} \, dx$.	Unambiguous
485185267	How do I prove that a subset of a set with Lebesgue outer measure zero also has outer measure zero?	Unambiguous
4862492396	An equation is first-order nonlinear. How do I transform it into standard linear form?	Ambiguous
487289779	Determine whether the sequence $a_n = 3^n - 2^{n+100}$ for $n \in \mathbb{N}$ converges or diverges.	Unambiguous
488330556	Is the sequence defined by $a_n = 3^{2n}$ a geometric progression? If so, find its common ratio and the sum of the first n terms.	Unambiguous
4893487811	$\lim_{x \to 0} {1-x\cot x\over x^2}$	Ambiguous
490366529	Is it valid to define $\int_{-a}^a f(x)\,dx = \int_{-a}^0 f(t)\,dt + \int_0^a f(x)\,dx$ for a real-valued function $f$?	Unambiguous
491387437	How can you prove that a number is an algebraic integer?	Ambiguous
4922030790	3 variable diophantine equation word problem: coins	Ambiguous
493464426	Evaluate the limit $\lim_{x \to 1} \frac{x^{1/5}-1}{x^{1/3}-1}$ for real-valued $x$.	Unambiguous
4943601099	For a symmetric matrix and a unit vector, why does a certain quartic inequality involving inner products hold?	Ambiguous
4951455761	What is the relationship between the determinant of a square matrix and its inverse?	Unambiguous
4962887105	Express $\frac{1}{1-i-z}$ as a power series in $z$ valid for $|-z-i|<1$.	Unambiguous
4972614579	How to solve $\lim_\limits{x\to1} \frac{\sqrt{2x-1} -1}{x^2-1}$?	Ambiguous
498531128	What are the subspaces of a finite-dimensional vector space $V$ that have dimension $\dim V - 1$, and what are they called?	Unambiguous
499735461	Suppose $f\colon M_n \to M_m$ with $f(a) = I_m$. Is there a positive element $b\in M_n$ such that $f(b)=I_m$?	Ambiguous
5001861189	$PC$ bisects $\angle APB$. Prove $XY \parallel AB$.	Ambiguous
5012369950	Is the statement $|x|=\pm x$ for $x>0$ a correct definition of absolute value, and why is it confusing?	Unambiguous
502736044	A number divided by $q$ leaves remainder $r$. Find $r+q$.	Ambiguous
5032339487	rove that whenever there exists an $[N,K,D]$-linear code over $\mathbb{F}_{q^{n-1}}$, there exists an $[nN,(n-1)K,2D]$-linear code over $\mathbb{F}_q$.	Unambiguous
5044044721	Find the limit of $\lim_{x\to 0}\left(\frac{\cos x}{\cos 2x}\right)^\frac{1}{x^2}$	Ambiguous
5052666204	How do I optimize $(\pi R_1 R_2)h$ when I don't know what the surface area should be?	Ambiguous
5061071811	Why is the Zariski topology on an algebraic variety over an algebraically closed field not Hausdorff?	Unambiguous
5072260	Prove by mathematical induction that the sum of the first $n$ positive integers is $1+2+3+4+\cdots+n = \frac{n\times(n+1)}2$.	Unambiguous
508128823	limit of integral $n\int_{0}^{1} x^n f(x) \text{d}x$ as $n\rightarrow \infty$	Ambiguous
5092006377	Average integral increasing	Ambiguous
51020813	What is the intuitionistic construction of the unit circle in constructive mathematics?	Unambiguous
5113367250	For this equation, does it matter which variable we treat as independent?	Ambiguous
5124236641	What is the best way to find the closest vector in a set to a given one?	Ambiguous
513721784	Absolute value of complex exponential	Ambiguous
5142844637	I need to calculate $\cos(\pi/20)$ with an error less than $10^{-5}$ using a series. Is my answer correct?	Ambiguous
515835454	$f: SL_2(\mathbb{R}) \to GL_4(\mathbb{R})$ show that $Im(f)=SL_4(\mathbb{R})$	Ambiguous
516568493	How do you find the number of common codes for [1,5], [4,8] and [3,7], [2,6]?	Ambiguous
5171768578	How to find the relation between $a$, $c$ and $b$ to make my density function integral $1$?	Ambiguous
5183815586	If a function is differentiable almost everywhere, what is its regularity?	Ambiguous
5192931443	If $\sum c_n x^n = (\sum a_n x^n)(\sum b_n x^n)$, does it follow that $\sum c_n = (\sum a_n)(\sum b_n)$?	Ambiguous
5201283678	Prove $||\lambda x_1 + (1-\lambda) x_2 - y|| \leq ||x_1 - y||$	Ambiguous
5211942246	I need to prove trigonometric equation	Ambiguous
5222620801	Proof Group Homomorphism: $\varphi(e)= e'$ & $\varphi(\hat{a})= \widehat{\varphi(a)}$	Ambiguous
523960632	Evaluating product of exponent and polynomial	Ambiguous
524697675	multiplying a matrix by a row vector	Ambiguous
5252360710	What is the numerical value or a series expansion formula for the natural logarithm of $\pi$, $\ln(\pi)$?	Unambiguous
526587560	What is the purpose of the base case in a proof by mathematical induction?	Unambiguous
5272038540	Permutations with men and women alternating	Ambiguous
5281663674	Does my genome exist in the digits of pi?	Ambiguous
5292315737	Is checking a norm on a function space just the triangle inequality and the other two rules?	Ambiguous
5303969765	Evaluate the integral $\int_0^\infty \frac{e^{-tx^2}-e^{-x^2}}{x}dx$ for $t > 0$.	Unambiguous
5312205890	In what mathematical context or under what non-standard ordering could the inequality $\frac{1}{2} < \frac{1}{4}$ be true?	Unambiguous
5321072741	Is an open $n$-ball homeomorphic to $\mathbb{R}^{n}$?	Ambiguous
53362473	What is the definition of a quasifinite morphism of schemes?	Unambiguous
534824000	If a function $f: X \to X$ satisfies $f(f(x)) = x$ for all $x \in X$, does it follow that $f$ is bijective?	Unambiguous
5352165873	For $f(x)=\begin{cases}2,&x>2\\3,&x<2\end{cases}$ and $g(x)=\begin{cases}-2,&x\ge2\\2,&x<2\end{cases}$, find $g(x)-f(x)$, addressing whether the result is defined at $x=2$.	Unambiguous
536264371	Recommend challenging but engaging mathematics books suitable for advanced undergraduate students that cover topics beyond the standard curriculum.	Unambiguous
5373111260	Error evaluating $ \lim_{x\to 0}\frac{x-\tan x}{x^3} $	Ambiguous
5381176875	Find all integer solutions $(a, b, c)$ to the equation $a + b + c = abc$.	Unambiguous
5394094225	Given point $K$ in a triangle, is it possible to construct a line through $K$ that bisects the area of the triangle?	Ambiguous
54013582	Show that a continuous function has a fixed point	Ambiguous
5413735022	Does knowing the Gauss curvature is $1/R^2$ help prove the curve's curvature is $\geq 1/R$?	Ambiguous
542587560	What is the purpose of the first test in an inductive proof?	Ambiguous
5431108298	Why is the canonical map $M \otimes_A N \to M \otimes_{A/I} N$ an isomorphism?	Ambiguous
5443979952	Are $\sum_{i=1}^{n} \xi_i / i$ and $\max\xi_n$ equally distributed?	Ambiguous
545665341	Using only a compass fixed at a radius of $5$ inches and a straightedge, construct a circle of radius $4$ inches.	Unambiguous
5464067963	Prove that the sum of the reciprocals of the integers in the set $M(P)$, where $M(P)$ consists of all positive integers whose prime factors are in the set $P$, converges and find its sum in terms of the prime zeta function.	Unambiguous
5471048	Compare the different definitions of trigonometric functions using right triangles, the unit circle, and power series.	Unambiguous
548675332	On the set $S = \mathbb{R}\setminus\{-1\}$ with operation $a * b = a+b+ab$, solve $2*x*3 = 7$ for $x$.	Unambiguous
5492605427	Prove that for any connected graph $G$, there exists a partition of the vertex set into two sets $V_1$ and $V_2$ such that the induced subgraphs $G[V_1]$ and $G[V_2]$ are both connected.	Unambiguous
5503729113	What is the definition and region of convergence for a Dirichlet series of a complex variable?	Unambiguous
551290435	Find five positive integers whose reciprocals sum to $1$	Ambiguous
552261680	Find integer values of $a,b$ such that $k=a^3-b^2$ is a perfect square.	Ambiguous
5532042807	Construct a homeomorphism between the unit circle $S^1$ and the boundary of a square in $R^2$.	Unambiguous
554288432	Let a power series have radius of convergence 1 and diverge at $x=1$. Does it diverge as $x\to 1$?	Ambiguous
5554812185	How to prove what the Wikipedia page says about the entry wise norm limit being the maximum norm?	Ambiguous
5561490059	What is the definition of the curl of a 3D vector field in vector calculus?	Unambiguous
5573468876	Calculate the mean, variance, and marginal distribution of the random variable X = Y1 + Y2, where Y1 and Y2 are independent standard normal random variables.	Unambiguous
558331345	How do I find the solutions for the equation $2x^2 - 5x + 3 = 0$?	Unambiguous
559295318	Are all algebraic vector bundles on the affine line over a principal ideal domain (PID) trivial?	Unambiguous
560219774	Finding the periods of functions...	Ambiguous
561172468	Why does MatLab generate polynomials for cyclic codes for all $n,k$ I try dividing $x^n-1$?	Ambiguous
5624117494	Volume of tetrahedron Solid	Ambiguous
5632007373	What are the fundamental geometric intuitions used to understand concepts in undergraduate linear algebra and multivariable calculus?	Unambiguous
5642897989	Prove that the dilation operator on $L^2(\mathbb{R})$ is a bounded linear operator.	Unambiguous
5654567071	How can you show the results from substituting $x=a\cos{\theta}$ and $x=a\sin{\theta}$ are the same when finding $\int{\frac{16}{\sqrt{9-x^2}}}$?	Ambiguous
566730358	How do I graph a general sine function of the form y = a sin(bx - c) + d?	Unambiguous
5672394582	Show that $f(x) = x^3 - x$ is an odd function.	Unambiguous
5683839814	Probability that $\text{det}(A)$ is an even number.	Ambiguous
5692160552	Do elementary row operations change the eigenvalues of a square matrix?	Unambiguous
5704016729	Find all real values of theta such that $(\sin(\theta) - \cos(\theta))(2+\sin(\theta) \cos(\theta)) \leq 2$.	Unambiguous
571315869	Sequence of sets and neighborhoods	Ambiguous
572742878	Modeling with Exponential and Logarithmic Functions	Ambiguous
5731496959	How to show $f_n$ converges uniformly to $f'$ if the derivative is bounded by $3|x-y|$?	Ambiguous
5741745083	How do you find the shortest path between two fixed points in a 2D plane that avoids a given linear obstacle?	Unambiguous
575124287	Is the map $f(a+bi)=a$ a homomorphism of rings?	Ambiguous
5764221432	Near-Pythagorean triplets: What are the general solutions to $a^2+b^2=c^2-1$?	Ambiguous
5772540002	Let $$S=\sum_{i=0}^N \frac{1}{\sqrt{2\pi\sigma^2}}e^{-(i+0.5)^2/(2\sigma^2)}$$. Any ideas?	Ambiguous
578765044	Challenge: Demonstrate a Contradiction in Leibniz' differential notation	Ambiguous
5791896195	Under what conditions is the endomorphism ring of a module free as a module over the base ring?	Unambiguous
5801882798	Given constants $X, Y, P$, determine whether there exist positive integers $a, b$ such that $(aX+bY) \bmod P = 0$.	Unambiguous
5812540359	Does the fact that both examples and counterexamples exist mean a property is undecidable from the axioms?	Ambiguous
5823516041	Third order Taylor polynomial of $f:(0, \infty) \times \mathbb{R} \to \mathbb{R}$ given by $f(x,y) = x^y$	Ambiguous
5832128278	How do I compute the indefinite integral of the hyperbolic sine function $\sinh(x)$?	Unambiguous
5844087621	In which of the intervals is $\sqrt{12}$	Ambiguous
585458261	How do I rewrite a triple integral with different limits for $x+y-2z=2$?	Ambiguous
5864044721	Find the limit of $\lim_{x\to 0}\left(\frac{\cos x}{\cos 2x}\right)^\frac{1}{x^2}$	Unambiguous
5871310890	What is $\lim_{x\to\infty}\left(\sin{\frac 1x}+\cos{\frac 1x}\right)^x$?	Ambiguous
588195504	Prove that for a linear functional, the kernel has codimension one.	Ambiguous
5891438319	Prove that $\lim_{x\to 0} \frac{e^x-1}{x}=1$ using the definition of the exponential function or power series, without using L'Hôpital's rule or derivatives.	Unambiguous
5902597743	How do I compute the indefinite integral of a function of the form $f(x) = x^n e^{-ax}$?	Unambiguous
5912093882	Problem in solving a question related to finite difference.	Ambiguous
5922555463	How do I find the coordinates of a point in a 2D plane that is equidistant from two given points and a given line?	Unambiguous
5932339487	Does putting $n-1$ copies of a codeword after itself prove there is a linear code?	Ambiguous
59412139	How many binary relations on a set of size n are both symmetric and reflexive?	Unambiguous
5953084662	How to convert a complex number $z = a + bi$ to polar form (calculate the magnitude r and the argument theta).	Unambiguous
5961007952	For a random variable $X$ with pdf $f(x)=\frac12 e^{-|x|}$, what can we say about it?	Ambiguous
597608477	What is the general method for a proof by mathematical induction, and can you provide a step-by-step example?	Unambiguous
5984569632	In $\triangle ABC$, $BD$ is a median, $\angle DAB=15$ and $\angle ABD=30$. Find $\angle ACB$.	Ambiguous
5991002480	Is the Minkowski sum of a finite collection of compact line segments in Euclidean space compact?	Unambiguous
6002782880	Proof check: AB not invertible	Ambiguous
60125648	Prove that if two integers $a$ and $b$ are congruent modulo $n$, then $\gcd(a, n) = \gcd(b, n)$.	Unambiguous
6022548701	Find the integer part of the value of the infinite nested radical $x = \sqrt[3]{24+\sqrt[3]{24+\sqrt[3]{24+\cdots}}}$	Unambiguous
6033017595	Dimension of subspace 10x10	Ambiguous
6042932095	Find $a$ so that a line is tangent, secant or external from a sphere	Ambiguous
605176625	Prove that there do not exist positive integers $x$ and $y$ with $x^2 - y^2 = n$	Ambiguous
6062035929	How to prove bounds of area integral?	Ambiguous
6071024146	Is a group divisible if and only if it has no maximal subgroup?	Ambiguous
6082140493	What are some classic counterintuitive examples or paradoxes in probability theory, such as the Monty Hall problem or the Birthday paradox?	Unambiguous
6092657826	How to find values of $a$ and $b$ if the rest is $15$ when $P(x)$ is divided by $(x-2)$?	Ambiguous
6102604608	How do I prove $f(x)$ is continuous when it only has constants $a$ and $b$?	Ambiguous
611795652	How to find nth moment?	Ambiguous
6121391080	Can I express the theorem about $\sqrt{2}$ as $p\Rightarrow q$?	Ambiguous
61383035	How to prove $(1+1/x)^x$ is increasing when $x>0$?	Ambiguous
61436545	What are some examples of mathematically simple puzzles or conjectures that remained unsolved or were counterintuitive to professional mathematicians for a long time?	Unambiguous
6152044396	Can this be proven for all groups?	Ambiguous
616184593	Is it necessary to have $\theta$ in radians to obtain $\frac{\sin{\theta}}{\theta} \to 1$ as $\theta\to 0$?	Ambiguous
6173083195	How to solve a linear first-order ODE for various initial conditions?	Unambiguous
6183219211	Can a first-order homogeneous ordinary differential equation have a singular solution that is not a member of the general solution family?	Unambiguous
619225922	Provide an example of a problem in real-variable calculus or algebra where the solution is real, but the most efficient method of calculation requires the use of complex numbers as an intermediate step.	Unambiguous
6203172708	I have the inverse of a sparse matrix, and only 4 entries change. I need a quick way to get the new inverse.	Ambiguous
6212457906	Evaluate the double integral in polar coordinates $\int_{0}^{3\pi/2}\int_{1}^{4}(4-r)(2+\sin\theta)r dr d\theta$.	Unambiguous
622289461	Is the set of midpoints of the open intervals that make up the complement of the Cantor ternary set countable?	Unambiguous
6233123246	What is the closed-form value of the definite integral $\int_0^\pi \ln(1+\sin^2 t) dt$?	Unambiguous
62419236	Algorithm to compute Gamma function	Ambiguous
6253415231	How do I prove the universal generalization theorem in my discrete math image?	Ambiguous
626262629	What is the probability of a specific set of group stage pairings occurring in a UEFA Champions League draw, assuming all valid permutations are equally likely?	Unambiguous
6271386774	What is the definition and convergence criteria for a power series representation of a function?	Unambiguous
6281890614	Mathematical definition of infinity	Ambiguous
6293762261	What is the geometric intuition behind the spectral theorem for Hermitian operators on a finite-dimensional complex Hilbert space?	Unambiguous
630314809	Super or subscript notation on the left hand side of a symbol?	Ambiguous
6314814	Is there a name for graphs with 2 connections per node?	Ambiguous
6321824176	Why is the graph of the function $f(x) = \sin(\arccos(x))$ a semicircle with radius $1$ centered at the origin?	Unambiguous
633262371	How do you solve the area of a trapezoid using diagonals	Ambiguous
6343524864	Solve the first-order linear partial differential equation $-x^2u_x+u_y=1$ with the initial condition $u(e^y,y)=y$.	Unambiguous
6351291623	Direct Proof on Divisibilty	Ambiguous
6362445959	Find all natural numbers $n$ such that $n - 2$ divides $n^2 - 2$.	Unambiguous
6371176875	Find all integral solutions to $a+b+c=abc$.	Ambiguous
63867407	Prove that every group of order 15 is abelian.	Unambiguous
639457969	What is the definition and property of the minimum of a function on a closed interval?	Unambiguous
640125846	L'Hopital's rule, why is this incorrect?	Ambiguous
6412344095	How do I find the coefficient of $x^2$ in the binomial expansion of $(1 + 2x + 3x^2)^10$?	Unambiguous
64271874	In the context of power sets in set theory, can I apply the properties of power sets to prove the existence of larger cardinalities?	Unambiguous
6432338603	What is the formula and definition for the binomial coefficient ${n\choose k}$?	Unambiguous
6441327420	Solve the system of equations $x + xy + y = 11$ and $yx^2 + xy^2 = 30$ for real numbers $x$ and $y$.	Unambiguous
6451831188	How to compute the 2006th power of a given square matrix $M$?	Unambiguous
6461067717	How do I prove that the summation $\frac{2\sum_{k=0}^n 2^k}{n+1} = 2^{n+1}$ is false, or what is the correct simplification of $\frac{2\sum_{k=0}^n 2^k}{n+1}$?	Unambiguous
6473976203	What do I do with the extra variables in this equation: $$0 = -i+(iq-ix+1)\ln(1+i)-(1-d)^xln(1+i)-(1-d)^xln(1-d)+t (1-d)^xln(1+i)+t \ln(1-d)^x$$	Ambiguous
648456595	Prove that the infinite series $\sum_{n=1}^\infty \frac{1}{n^2}$ converges and that its sum is less than 2.	Unambiguous
6493005021	Provide a formal proof that if a function $f: \mathbb{R} \to \mathbb{R}$ is differentiable at a point $x$, then it must also be continuous at that point.	Unambiguous
6501951021	Why in calculus the angles are measured in radians?	Ambiguous
6511831188	Calculating the matrix $M^{2006}$	Ambiguous
6523832187	Distributing 3 objects in 5 boxes	Ambiguous
6532903468	Prove that the definite integral $\int_0^1\frac{\ln(x)}{\sqrt{1-x^2}}dx = -\frac{\pi}{2}\ln(2)$.	Unambiguous
654110579	In how many ways can a batsman score a century in all $6$'s and $4$'s?	Ambiguous
6552246596	What is the max function exactly and how do I examine its contour maps?	Ambiguous
656510991	If every continuous function on $X$ is uniformly continuous, is $X$ compact?	Ambiguous
657614812	What is the maximum number of knights that can be placed on a $2 \times n$ chessboard such that no two knights attack each other?	Unambiguous
6581290291	Identification of a quadrilateral as a trapezoid, rectangle, or square	Ambiguous
6593483693	Find the smallest integer $n > 1$ such that $n^2 \equiv n \pmod{1000}$.	Unambiguous
6601153474	Is the function $f(n) = 2^{\log_3(n) + 1}$ in $\Theta(n)$ for asymptotic running time analysis?	Unambiguous
6612206714	Evaluate the indefinite integral $\int \sin(2x) \frac{e^{\cos^2 x}}{e^{\sin^2 x}} dx$.	Unambiguous
6622470166	What are the conditions for the eigenvectors of this matrix to be to be real and orthogonal?	Ambiguous
6632307642	Proof of the spectral theorem for normal operators on a finite-dimensional complex inner product space	Unambiguous
6641054529	the purpose of induction	Ambiguous
6652669751	How do you count all unique groups that can be made with items	Ambiguous
6662049971	Is it possible to transform $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ into $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$?	Ambiguous
6671731227	Find he local maximum and minimum value and saddle points of the function?	Ambiguous
6684343109	For a graph $(G, V)$, $\# \text{edges}$ is given by the rank of $H_1(G, V)$.	Ambiguous
6691271168	How do I square a logarithm?	Ambiguous
670292425	For $f(x)=\begin{cases}\dfrac{x^2-1}{x+1},&x<-4\\[4pt] ax^2+2x+b,&-1\le x<0\\[4pt] |x+a+2|,&x\ge0\end{cases}$, find $a$ and $b$ such that $f$ is continuous everywhere it is defined, noting that $f$ has no definition on $(-4,-1)$.	Unambiguous
6712897989	Let $D$ be defined by $D(a)=f_a$ for $f\in L^2(\mathbb{R})$. Show that $D$ is continuous.	Ambiguous
672241316	Does the infinite series $\sum_{k=1}^{\infty} \frac{\sqrt{k}}{(k+1)^2}$ converge?	Unambiguous
6731961686	If $X$ is a topological space, is it true that for any two points $x, y \in X$, the subspaces $X \setminus \{x\}$ and $X \setminus \{y\}$ are homeomorphic?	Unambiguous
6741438319	Prove that $\lim_\limits{x\to 0}{\frac{e^x-1}{x}}=1$ without derivatives	Ambiguous
67533296	How do you prove that a differentiable function is monotonic given information about the sign of its derivative?	Unambiguous
6764317139	Bianchi classification of solvable Lie groups and cocompact subgroups	Ambiguous
677820492	Are there algorithms to simulate the continuous deformation of a cup into a donut?	Ambiguous
678521814	I have this picture and I know a height. How do I find its other proportions using perspective projection?	Ambiguous
679709004	Evaluate $\lim_{x\to\infty}\left(\frac{x^2+2x-1}{2x^3-3x-2}\right)^{1/x}$.	Unambiguous
680866327	Prove that $\lim_{x\to 1} \frac{x^n-1}{x-1} = n$ for all positive integers $n$ without using L'Hôpital's rule.	Unambiguous
6812296759	What is the relationship between a symmetric matrix and the inner product it defines on a vector space?	Unambiguous
6823538836	How is this inequality proved for the squeeze theorem if it doesn't hold for all $x$?	Ambiguous
6831900241	What is the expected value of $X$, given its generating function?	Ambiguous
6844595703	If parallel chords of a circle subtend central angles of $72^\circ$ and $144^\circ$, then the perpendicular distance between them is half the radius	Ambiguous
6852537775	Evaluate $\prod_{r=1}^{10}(1+\tan r^\circ)\cdot\prod_{r=46}^{55}(1+\cot r^\circ)$.	Unambiguous
6862249320	Find a closed-form expression for $g(x) = \cos x + \cos 3x + \cos 5x + \cdots + \cos((2n-1)x)$.	Unambiguous
687972189	Provide a proof that the limit as $n$ approaches infinity of $\left(\frac{n - 1}{n}\right)^n$ is equal to $\frac{1}{e}$.	Unambiguous
688298648	What is the standard notation for the concatenation of two finite sequences in mathematics and computer science?	Unambiguous
6892820510	How are quaternions used to represent 3D rotations in geometry?	Unambiguous
6901812294	How to find $P(B\cup C)$ when $A$ and $B$ are independent and the union of $A,B,C$ is $1$?	Ambiguous
6911111222	Is a function that's unbounded near a boundary point still integrable if we redefine its value there?	Ambiguous
692986013	What should I do with the fact that my Laurent series converges in $z=1+i$?	Ambiguous
693108308	Does Tom Apostol's proof of the Fundamental Theorem of Calculus involve a division by zero?	Unambiguous
6942697247	What are the standard integral representations of the Bessel function of the first kind $J_n(x)$?	Unambiguous
6952296602	Simplify the propositional logic expression $\neg ((P \to (Q \to R)) \to (Q \to (P \to R)))$ to its simplest form.	Unambiguous
6963944628	I'm reading a book and, in its section on the definition of a stopping time(continuous), the author declares at the start that for the whole section every filtration will be complete and right-continuous. So, in the definition of a Stopping Time, how important are these conditions? Why would they matter?	Unambiguous
697548732	In set theory, is it true that $A\cup(B\triangle C)=(A\cup B)\triangle(A\cup C)$ for all sets A, B, and C?	Unambiguous
6981045011	How to find the limit $\lim_{n\to \infty}\frac{n}{2^n-1}$?	Ambiguous
6991258955	Convergence of the integral $\int_0^{\pi/2}\ln(\cos(x))dx$	Ambiguous
7002427685	Calculate the limit $\lim_{n \to \infty} \sum_{k=1}^{n} \frac{(-1)^{k-1}}{k^2 \binom{n}{k}}$.	Unambiguous
701124986	Is the function $\alpha : \mathbb{Z}_{11} \rightarrow \mathbb{Z}_{11}$ defined by $\alpha(x) = 4x^2 - 3x^7 \pmod{11}$ a permutation, and if so, is it a single cycle?	Unambiguous
7023289735	Is the arc length always irrational between two rational points?	Ambiguous
7033564289	What is the expected number of times a simple symmetric random walk on a 1D integer lattice returns to the origin?	Unambiguous
7042381284	For $f_{a,c}(x)=\int_0^\infty\left[t^a(1+t)^{c-a-1}e^{-xt}-e^{-t}+e^{-xt}\right]\frac{dt}{t}$ with $a>0,x>0,c\in\mathbb{R}$, determine the sign of $f_{a,c}(x)$, given that $f_{a,c}'(x)<0$.	Unambiguous
7051592652	Example of a relation that is symmetric and transitive, but not reflexive	Ambiguous
706984271	Provide an explicit isomorphism showing that $\mathbb{K} \otimes M_n(\mathbb{C})$ is isomorphic to $\mathbb{K}$, where $\mathbb{K}$ denotes the compact operators on a separable Hilbert space.	Unambiguous
7073762872	Zeno's Achilles & Tortoise - Where exactly is the proof wrong?	Ambiguous
7081489146	How is the imaginary unit $i$ formally defined in the context of complex numbers, and why is $i$ not equal to $-i$?	Unambiguous
7091987599	It's known that a certain function behaves like $1/u$ near $u=0$. Are there sharper inequalities describing this behavior?	Ambiguous
710859968	Let $Y = \{(x, x/n) \in \mathbb{R}\times\mathbb{R} : x \in [0,1], n \in \mathbb{N}\}$ and $X = \bigcup_{n\in\mathbb{N}} [0,1]\times\{n\}$ with equivalence relation $(0,n) \sim (0,m)$ for all $n,m \in \mathbb{N}$. Prove or disprove that the quotient space $X/{\sim}$ is homeomorphic to $Y$.	Unambiguous
7111204425	Solve the nonlinear ODE $y' + y^2 = 0$ using a power series solution $y = \sum_{n=0}^\infty a_n x^n$, given that substitution leads to $\sum_{n=1}^\infty a_n n x^{n-1} + \sum_{n=0}^\infty\left(\sum_{k=0}^n a_k a_{n-k}\right)x^n = 0$.	Unambiguous
7121059293	Can someone explain how the proof for f(x) works?	Ambiguous
713956087	Provide an example problem and a step-by-step solution involving implicit differentiation of a curve defined by an equation in x and y.	Unambiguous
7143023583	Does the infinite series $\sum_{n=1}^\infty \frac{n^2(-1)^n}{1+n^2}$ converge or diverge?	Unambiguous
715759112	Given that $Y - X^2$ is an irreducible polynomial, prove that the variety $V(Y-X^2)$ is irreducible.	Unambiguous
716959137	Finding missing coordinates in a non-linear graph?	Ambiguous
7172715973	$$ \frac{n}{n+2}\sum_{k=1}^n\frac{k}{k+3}$$	Ambiguous
718801923	Prove that for all integers $x$ and $y$, if $x + y + xy = 0$, then $(x, y) = (0, 0)$ or $(x, y) = (-2, -2)$.	Unambiguous
7191503414	Why is the natural logarithm of 1 equal to 0?	Unambiguous
7203035983	How can I find integers which satisfy $\frac{150+n}{15+n}=m$?	Ambiguous
721476186	Is there a category whose internal logic is paraconsistent?	Ambiguous
722108191	Why won't sine functions work as a counter example for my infinite integral proof?	Ambiguous
723401477	What is the definition of a $\mathbb{Z}/N\mathbb{Z}$-algebra?	Unambiguous
7241135959	What is the standard way to represent a complex number as a $2 \times 2$ real matrix?	Unambiguous
725802940	How to calculate the numerical value of the sum from $k=1$ to $50$ of $k^{1/3}$?	Unambiguous
7261035200	Let $p:\mathbb{N}\to\mathbb{N}$. What is $\lim p(n)$?	Ambiguous
727685681	Prove that a finite-dimensional vector space has a finite basis.	Unambiguous
7281893124	If 16 players are divided into eight pairs, what is the probability that only one of S1 or S2 advances?	Ambiguous
729158469	Prove that if a linear endomorphism $\alpha$ on a finite-dimensional complex inner product space satisfies $\alpha^* = -\alpha$, then its eigenvalues are purely imaginary.	Unambiguous
7301564864	What does the symbol # mean?	Ambiguous
7312564222	Find the equations of the tangent lines drawn from the point $(11, 3)$ to the circle defined by $x^2 + y^2 = 65$.	Unambiguous
7321295010	What are examples of true statements about natural numbers that can be proven using non-inductive methods, or that are independent of Peano arithmetic and thus cannot be proven by induction?	Unambiguous
7333743564	Examples of classic coin weighing puzzles and their optimal solutions using a balance scale	Unambiguous
734783173	Evaluate the definite integral $\int_0^1 \frac{dx}{\sqrt{x(1-x)}}$	Unambiguous
7354814	What are the names of directed graphs where every vertex has an in-degree of 2 and an out-degree of 2?	Unambiguous
7362448533	How to evaluate a limit with arctan(x)	Ambiguous
7372365371	If $x + frac{1}{x}$ is an integer, prove that $x^{2n} + frac{1}{x^{2n}}$ is an integer for all positive integers $n$.	Unambiguous
7381232385	Let $S=\{\sqrt{D}: D\equiv 2,3\pmod4\}$ and $T=\left\{\frac{1-\sqrt{D}}{2}: D\equiv 1\pmod 4\right\}$. For $w\in S\cup T$ and the norm map $N:\mathbb{Z}[w]\to\mathbb{Z}$, prove that $\mathbb{Z}[w]/(a) \cong \mathbb{Z}_{N(a)}$ for $a\in\mathbb{Z}[w]$.	Unambiguous
7391528935	How to prove a set is linearly independent without using a basis or FTA?	Ambiguous
7402274060	If $x$ and $y$ are positive numbers, what is the minimum possible value of $(x+y)\left(\frac1x + \frac1y\right)?$	Ambiguous
7411788328	Calculate the numerical value of the product $\sin(10^\circ) \sin(50^\circ) \sin(70^\circ)$.	Unambiguous
742311665	Proof a graph is bipartite if and only if it contains no odd cycles	Ambiguous
743176625	For which positive integers $n$ is it true that there do not exist positive integers $x$ and $y$ such that $x^2 - y^2 = n$?	Unambiguous
7441883279	What are examples of continuous functions that cannot be uniformly approximated by polynomials on a given interval?	Unambiguous
745405972	A basis for a subspace of $\mathbb R^3$	Ambiguous
7464734106	Is there an equation similar to square root, but faster for a computer to compute?	Ambiguous
7473226462	On an infinitely sided die, is it possible to roll and get the same number twice?	Ambiguous
748529215	Is it true that if a sequence {a_n} converges, then the set of its values has exactly one limit point?	Ambiguous
7492297557	Given three points $A, B, M$ in the plane, determine whether there is a unique point $R$ satisfying $d(R,A)\cdot d(R,B) = d(R,M)^2$, or find the curve formed by all such points $R$.	Unambiguous
7501796173	I tried to expand the bracket $[X,Y]=XY-YX=Y$, how do I proceed to find $[Y,Z]$?	Ambiguous
7513154407	Applying a function on any multiset of real numbers	Ambiguous
7522591017	Unity roots that sum to 1	Ambiguous
7531382185	How many divisors of the combination of numbers?	Ambiguous
754151810	What is the probability of getting exactly 3 heads in 10 independent flips of a fair coin?	Unambiguous
755807589	For a pure tensor $v_1\otimes\cdots\otimes v_n \in V^{\otimes n}$, what is the standard terminology for the component vector $v_j$?	Unambiguous
756927248	Maximum value of $a+b$ given that $\frac{1}{a} + \frac{1}{b} = \frac{1}{20}$	Ambiguous
7571286651	How do I compute the limit of a function of two variables as it approaches a point $(x, y)$?	Unambiguous
7582666326	For a square matrix $A$ over the complex numbers, is it true that if $A^4 = I$, then $A^2 = I$ or $A^2 = -I$?	Unambiguous
7592322481	Does the limit as $k$ approaches infinity of the ratio $\frac{\sum_{n=1}^{k} 2^{2 \cdot 3^n}}{2^{2 \cdot 3^k}}$ equal $1$?	Unambiguous
7604093371	Show that a graph that is connected but not complete has vertices u,v and w such that uv and vw are edges but not uw.	Ambiguous
7611565903	What are some simple-to-prove, counter-intuitive theorems in set theory that depend on the Axiom of Choice?	Unambiguous
7622893411	Evaluate the sum $\sum\limits_{k=1}^n {\binom{n}{k} f(k)}$.	Ambiguous
7632555463	Find the point equidistant from two points and a line	Ambiguous
764728599	How to evaluate the triple integral for region $E$ bounded by the spheres?	Ambiguous
7655877	Efficiently finding two squares which sum to a prime	Ambiguous
7663111260	How do I evaluate $\lim_{x\to 0}\frac{x-\tan x}{x^3} $?	Unambiguous
7672307642	How to find an orthonormal $T$ eigenbasis without matrices?	Ambiguous
7681774294	Explain why $y^2 = 2x^2 + C$ is not equivalent to $y = \sqrt{2x^2} + C$.	Unambiguous
7693322357	How is the matrix identity $\det\begin{pmatrix}A&B\\B&A\end{pmatrix}=\det(A+B)\det(A-B)$ proved for square matrices $A$ and $B$ of the same size?	Unambiguous
770449507	Compute the indefinite integral of $\frac{\sqrt{x^2-4}}{x^4}$ with respect to $x$.	Unambiguous
7711037426	Find $\lim\limits_{x \to \infty}{e^{-x}\sqrt{x}}$	Ambiguous
7721822028	In a normed vector space, is the closed convex hull of a weak*-compact subset of the dual space also weak*-compact?	Unambiguous
7732959054	Can I use stars and bars for this?	Ambiguous
774398593	Calculate the length of curve $f(x)=\arcsin(e^x)$, check solution, please.	Ambiguous
775199447	Provide a proof that the derivative of the function $f(x) = e^x$ is $f'(x) = e^x$ using the limit definition of the derivative and without using the chain rule.	Unambiguous
77643118	How can I explain to a layperson why the 'staircase' proof that $\pi = 4$ is mathematically incorrect?	Unambiguous
7771562459	Prove that $( 1 + n^{-2}) ^n \to 1$.	Ambiguous
7782116037	Does $\sum_{n=1}^{\infty} \frac{\sin^2(nx)}{n}$ converge uniformly?	Ambiguous
7793258160	Calculate $\iint_\Omega \left(\sqrt{\frac xa}+\sqrt{\frac yb}\right)\,dxdy$	Ambiguous
7804564068	Is there a closed-form formula for the sum of the first $n$ squares, $\sum_{k=1}^{n} k^2$?	Unambiguous
7811086318	I'm looking for an explanation regarding an exponential function and a Taylor series approximation. Could you elaborate on how these relate in the context of a measurement?	Ambiguous
78268897	Let $A$ be positive definite and $B$ be full rank. Show that $BAB'$ is positive definite.	Ambiguous
783611256	How do you convert a decimal number to 8-bit two's complement binary representation?	Unambiguous
7841410163	Show that $\lim_{(x,y)\to(0,0)} \frac{xy^2}{x^2+y^4}$ does not exist by evaluating the limit along the paths $y=x$ and $y=-x$.	Unambiguous
7851982170	Probability of an event in relationship with a certain geometrical representation	Ambiguous
786763272	Which is greater: the value of $\sqrt{1001} - \sqrt{1000}$ or the fraction $\frac{1}{10}$?	Unambiguous
7871372825	Evaluating numerically $\int_0^{\infty}e^{-t^2 /100} \sin \pi t $	Ambiguous
7881117045	What are the standard methods and techniques for evaluating definite integrals given specific upper and lower limits of integration?	Unambiguous
7891762215	Diamond and Shurman page 73 says $j'(\tau)$ doesn't vanish, how do I prove this?	Ambiguous
790405972	How do I find a basis for a subspace of $\mathbb{R}^3$ defined by a system of linear equations?	Unambiguous
7912490690	In how many ways can you make 107 using 1, 10, and 50?	Ambiguous
7922247820	For some general variable $c$, why is the limit always positive infinity except when $c=1$? What happens if $c=0$ or $c=-1$?	Ambiguous
7931602977	What is an example of an infinitesimal in the context of non-standard analysis?	Unambiguous
794611256	What are the least and greatest decimal integers that can be stored in an 8-bit register?	Ambiguous
7951502938	How to prove the sum equals $n/(n+1)$ using induction?	Ambiguous
796811451	Evaluate $\sqrt{2+\sqrt[3]{3+\sqrt[4]{4+\cdots}}}$.	Unambiguous
7974566790	Can someone help me understand eigenvalues?	Ambiguous
7981008311	Prove that for a bounded linear operator $L$ on a Hilbert space $H$, the null space of $L^*$ is equal to the null space of $L L^*$.	Unambiguous
799262371	How do you calculate the area of a trapezoid given the lengths of its diagonals and the angle between them?	Unambiguous
8003248549	Let $f$ be a continuous map then which of the following is correct	Ambiguous
801119307	Give an example of a Cauchy sequence that does not converge in the space of rational numbers $Q$.	Unambiguous
802