Upload 2 files

MathDisAM_test.tsv

@@ -0,0 +1,201 @@
+MathSE ID	Math Question	Tag
+2488771	What is the general formula for the coefficient of $x^k$ in the expansion of the trinomial $(1+x+x^2)^n$?	Unambiguous
+746518	How do I do question 8(i) in this picture? I have $x=\delta\tan(\theta)$, but I'm not sure how to turn this into a PDF.	Ambiguous
+215089	Subset of Power Set of Natural Numbers	Ambiguous
+219343	How can I Prove that $[(p \to\neg q) \wedge q] \to \neg p$ is a tautology?	Ambiguous
+1054529	What is the purpose of mathematical induction in proving statements about integers?	Unambiguous
+1898365	What are the properties of the tensor product of two abelian groups?	Unambiguous
+2274060	If $x$ and $y$ are positive real numbers, what is the minimum value of the expression $(x+y)\left(\frac{1}{x} + \frac{1}{y}\right)$?	Unambiguous
+3198849	Evaluate the limit as $x$ approaches $0$ of $(sin(x) - x) / x^3$ using power series without L'Hospital's Rule.	Unambiguous
+1779361	How to prove $\cos^4x - \sin^4x - \cos^2x + \sin^2x$ is always $0$?	Ambiguous
+2152735	"What does ""open set"" mean in the concept of a topology?"	Ambiguous
+1101966	Provide a step-by-step proof that the definite integral $\int_0^1 \frac{x^3\arctan x}{(3-x^2)^2\sqrt{1-x^2}} \, dx$ equals $\frac{\pi\sqrt{2}}{192}(18-\pi-6\sqrt{3})$.	Unambiguous
+160327	What is the mathematical symbol for the universal quantifier 'for every element' in predicate logic?	Unambiguous
+2596764	If a differentiable function $f: \mathbb{R} \to \mathbb{R}$ satisfies $\lim_{x \to \pm\infty} f(x) = -\infty$, must the function have a global maximum?	Unambiguous
+590489	What are some topologies that work on functions?	Ambiguous
+1761660	How to proceed from the discriminant $1+4y(y+a)\geq 0$ to find the number of integral values of $a$?	Ambiguous
+279233	Is $m^4+4^n\equiv 1\pmod 8$ an iff condition?	Ambiguous
+769619	For which Banach spaces $X$ is the closed unit ball of the dual space $X^*$ strictly convex?	Unambiguous
+3806534	If $\lim_{x\to 0}\left(f(x)+\frac{1}{f(x)}\right)=2,$ show that $\lim_{x\to 0}f(x)=1$.	Ambiguous
+1890614	What are the mathematical definitions of infinity in set theory and calculus?	Unambiguous
+217291	How do I find the mathematical expressions for the red and green lines in my image to put into pgfplots?	Ambiguous
+1726540	How do I find the derivative of the function $f(x) = x^{1/x}$ with respect to $x$?	Unambiguous
+1409828	How to get the center point of the red area where my ellipses intersect?	Ambiguous
+1568161	Evaluate the limit as $x$ approaches $\pi$ of $\frac{(\pi - x) \sin(x)}{1 + \cos(x)}$.	Unambiguous
+884228	For the density f(x) = 1/(b-a) on [a,b], what is R(t)?	Ambiguous
+2170954	Let $p(n)$ denote the productivity of the most productive $n$-state Turing machine, where productivity is the length of the output string if the machine halts in standard position and $0$ otherwise. Prove that $\forall n\, [p(n+11) \ge 2n]$.	Unambiguous
+2657826	Given $P(x) = ax^3 + 3x^2 + bx + 3$ with $P(1) = 0$ and remainder $15$ when divided by $(x-2)$, find $a$ and $b$, and solve the inequality $P(x) \le 0$.	Unambiguous
+2426632	What is the difference between curvature and torsion for a space curve in differential geometry?	Unambiguous
+571232	Show that a natural number $a$ with base-$10$ representation $(r_k r_{k-1}\cdots r_1 r_0)_{10}$ satisfies $4 \mid a$ if and only if $4 \mid (r_1 r_0)_{10}$.	Unambiguous
+2007373	Basic Geometric intuition, context is undergraduate mathematics	Ambiguous
+541081	In the Martingale betting strategy, when is it mathematically optimal to stop doubling down on losses?	Unambiguous
+2323384	equation of common tangent touching circle and the parabola	Ambiguous
+1594324	Find all real numbers x such that $|x|<x$.	Unambiguous
+1902450	What is an easy way to prove that the integral of $\frac{1-y^2}{(1+y^2)^2} \, dy$ equals $\frac{y}{1+y^2} + C$?	Unambiguous
+43945	What is a shortcut method for calculating the probability distribution of a 1D symmetric random walk after $n$ steps?	Unambiguous
+3151361	Prove that $\sqrt{7} + 2\sqrt{5} < 2 + \sqrt{35}$ for positive real numbers.	Unambiguous
+2887105	How do I write $\dfrac{1}{1-i-z}$ as the sum of a series?	Ambiguous
+92962	Suppose $a$ is a number, and for some $n$, $a^n$ and $(a+1)^n$ are both rational. Is $a$ rational?	Ambiguous
+1285663	Area of region inequality	Ambiguous
+1537426	Is it true that sums of several roots are never integers?	Ambiguous
+2007946	Why don't the limits for the angles both go up to $2\pi$ in this integral?	Ambiguous
+1214106	Find all polynomials satisfying a certain functional equation involving $x^2-y^2$.	Ambiguous
+1372825	What is the numerical value of the improper integral $\int_0^{\infty}e^{-t^2 /100} \sin(\pi t) dt$?	Unambiguous
+2540002	Prove that for any $\sigma > 0$ and $N \in \mathbb{N}$, the sum $\sum_{i=0}^{N} \frac{1}{\sqrt{2\pi\sigma^2}} e^{-(i+0.5)^2/(2\sigma^2)} < 0.5$.	Unambiguous
+3984395	How to prove my integral $\Gamma(x)$ decays exponentially bounded by $A_d e^{-|x|/L}$?	Ambiguous
+1524225	Find the least $n$ such that the expression is divisible by $700$.	Ambiguous
+685435	How can I find an upper bound for the remainder term of the Riemann zeta function series for $\zeta(2)$ after $n$ terms?	Unambiguous
+4841809	Why is it a problem that the echelon form isn't unique?	Ambiguous
+612241	Evaluate $\lim_{x\rightarrow 0} \frac{\sin x}{x + \tan x} $ without L'Hopital rule.	Unambiguous
+2617141	Why is the equation of a circle not a function of $x$, and how do I show this algebraically using the vertical line test or solving for $y$?	Unambiguous
+44348	Simplify $||x-5|-10|<\epsilon$.	Ambiguous
+483708	How to prove the identity $\frac{\eta^{14}(q^4)}{\eta^{4}(q^8)}=4\eta^4(q^2)\eta^2(q^4)\eta^4(q^8)+\eta^4(q)\eta^2(q^2)\eta^4(q^4)$ where $\eta(q)$ is the Dedekind eta function?	Unambiguous
+571232	How do I show that 4|$a$ iff 4|($r_{1}$$r_{0}$)$_{10}$?	Ambiguous
+1896418	For $n\ge5$ points in the plane with no four collinear, find a closed-form expression for $f(n)$, the maximum number of points guaranteed such that no three of them are collinear, over all such configurations.	Unambiguous
+3330117	Limit involving prime and composite numbers	Ambiguous
+3390858	Find the range of values of $k$ for which $kx^2 + 8x + k <6$ for all real values of $k$	Ambiguous
+4726367	How do people create trigonometric questions from scratch?	Ambiguous
+1025196	Let $f:[-1,1]\to\mathbb{R}$ be continuous with $f(-1)=f(1)$. Prove that there exists $y\in[0,1]$ such that $f(y)=f(y-1)$.	Unambiguous
+4351550	Intersection of perpendicular bisector and circle in a triangle.	Ambiguous
+3026909	How many presentations are there for $D_8$ using permutations, and can it be done with other groups?	Ambiguous
+4734106	Are there numerical approximations or algorithms for computing the square root of a number that are computationally faster than the standard library function?	Unambiguous
+1730258	What is the formula for the arc length of a parabola given by $y = ax^2$ from $x = 0$ to $x = b$?	Unambiguous
+2092731	Find the radius of convergence of $\sum_{n=1}^\infty \frac{(3n+1)^{10}x^n}{2n^n}$ using the root test.	Unambiguous
+2587259	How many permutations are there for a 9-player baseball batting order?	Unambiguous
+215332	Coefficent of an equation	Ambiguous
+3149675	Open ball in $M=\mathbb{R}\setminus\{-1,1\}$	Ambiguous
+2669535	I can show that $\overline{R(A')} \subseteq N(A)^{\perp}$, but I'm not sure how to prove the converse using the fact that $R(A)$ is Banach.	Ambiguous
+1150363	Is the real number $\pi$ equal to the angle $180^\circ$ in terms of their numerical value?	Unambiguous
+1810374	How many distinct 7-note musical scales can be formed by choosing subsets of notes from the 12-note chromatic scale?	Unambiguous
+1745083	What is the optimal path between $2$ fixed points around an invisible obstructing wall?	Ambiguous
+360449	What does $x \ne y \ne z$ mean?	Ambiguous
+2660495	Probability of Choosing Exactly 1 pair out of 4	Ambiguous
+4083792	Conditional expectation based on two bivariate normal RVs?	Ambiguous
+3017130	How do you factorize the quadratic expression $m^2 + 2ABm + B^2$?	Unambiguous
+348715	Evaluate the limit as $x$ approaches infinity: $\lim_{x \to \infty} \left( \sqrt{x^2+x} + 2\sqrt{x^2+2x} - x \right)$	Unambiguous
+4520880	Why does the distributive law not apply to the expression $2(2.49 \times 10^8)$ when there is only one term inside the parentheses?	Unambiguous
+1114879	How to algorithmically detect if two ellipses in a 2D plane intersect?	Unambiguous
+102524	Finding the formula of an nonlinear function	Ambiguous
+2334855	What are the bounds of $$\frac{x+1}{xy+x+1}+\frac{y+1}{yz+y+1}+\frac{z+1}{zx+z+1}$$?	Ambiguous
+1325994	What is the last digit of 347 raised to the power of 61 in base 10?	Unambiguous
+956087	I found that y' = (1-y)/(x-2). How do I find the tangent line from here?	Ambiguous
+3542957	What is the difference between $-i$ and $j$?	Ambiguous
+68897	What are the properties and definitions of a symmetric positive definite matrix in linear algebra?	Unambiguous
+188945	What are the epistemological and pedagogical disadvantages of relying exclusively on computer algebra systems for mathematical proof and computation compared to manual derivation?	Unambiguous
+2894360	Provide a step-by-step proof to show that the equation $\frac{x + \frac{1}{2}}{\frac{1}{x} + 2} = \frac{x}{2}$ holds for all $x \neq 0, -\frac{1}{2}$.	Unambiguous
+1513096	Prove that the stereographic projection of the upper hemisphere from the North Pole is orientation-preserving if and only if the dimension n of the sphere $S^n$ is even.	Unambiguous
+2140493	Counterintuitive examples in probability	Ambiguous
+3178233	Prove that there is $c \in (0,1)$ such that $e^{f'(c)}f(0)^{f(c)}=f(1)^{f(c)}$	Ambiguous
+1668681	An example that a 2D shape with its centre of mass on its boundary	Ambiguous
+767709	How to show a function with $x^n$ is uniformly convergent on $[0,1]$?	Ambiguous
+1040278	Nobody told me that self teaching could be so damaging...	Ambiguous
+3772635	Find all real solutions to the system of equations $\sqrt{x} + y = 7$ and $x + \sqrt{y} = 11$.	Unambiguous
+2431076	Prove that if $\sigma^r$ is an $n$-cycle for some permutation $\sigma \in S_m$, then $\sigma$ itself must be an $n$-cycle.	Unambiguous
+2867689	For the equation $6x^2+8xy+4y^2+17y-6=0$, use implicit differentiation to find $\frac{dy}{dx}$, isolating all terms containing $\frac{dy}{dx}$ on one side.	Unambiguous
+1901862	For what smooth functions is there a nice closed form for a certain exponential integral, invertible in terms of the upper limit?	Ambiguous
+2445959	find all natural number such that $n-2\mid n^2-2$	Ambiguous
+982093	Proof for $\lim_{n \to \infty}(1+\frac{a_n}{n})^n = e^a$ given that the sequence $a_n$ converges to $a$ as $n \to \infty$.	Unambiguous
+1554069	On a cubic curve $C$ defined by an elliptic curve group law, prove that the tangent at a point $P$ intersects $C$ at $P$ with multiplicity 2 and at the point $4P$, and that the tangent at $5P$ intersects $C$ at $5P$ with multiplicity 2 and at the point $2P$.	Unambiguous
+3804598	In a building with elevators following uniform distributions, if the average wait time with two is 1 minute, what is it with one?	Ambiguous
+128823	What is the limit of $n\int_{0}^{1} x^n f(x) \text{d}x$ as $n\rightarrow \infty$ for a continuous function $f$ on $[0, 1]$?	Unambiguous
+3570162	$\exists b\in S_2\ \forall a\in S_1: p(a,b)$	Ambiguous
+3024780	Given three points forming a triangle in the Cartesian plane, how do you calculate the coordinates of the Fermat point (the point minimizing the sum of distances to the vertices)?	Unambiguous
+2062239	How to construct the polar decomposition for Wegge-Olsen Exercise 1.A.c?	Ambiguous
+2599280	How to find $m$ and point $P$ for a tangent line without using differentiation at all?	Ambiguous
+2538727	Show that the space $X = [0,1]^{[0,1]}$ (the set of all functions from $[0,1]$ to $[0,1]$ equipped with the product topology) is not sequentially compact.	Unambiguous
+574475	Given the points $P=(2,1)$ and $A=(-1,5)$, find the correct parametric equations of the line through $P$ in the direction of $A$.	Unambiguous
+2490647	What is the asymptotic behavior of the number of involutions on a set of n elements as n approaches infinity?	Unambiguous
+2296759	How to prove $B$ is a symmetric matrix from its inner product?	Ambiguous
+590489	What are the most common topologies used on function spaces, such as the compact-open and product topologies?	Unambiguous
+3906075	Number of non-identity elements of order $2$ in this set. (Related to abelian group of order $34.$)	Ambiguous
+1762325	Is the formula $\neg \neg p \rightarrow p$ a tautology in classical logic, and is it provable in intuitionistic logic?	Unambiguous
+1222768	In the context of mathematical analysis, why are vector-valued functions considered well-defined functions, whereas multivalued functions are not considered functions in the strict set-theoretic sense?	Unambiguous
+547928	What is the minimum value of $a+b+\frac{1}{ab}$ for positive real numbers $a, b$ such that $a^2 + b^2 = 1$?	Unambiguous
+1051513	What is the probability that three of the selected items had values less than or equal to five?	Ambiguous
+1405562	What is the method for calculating the remainder of a large integer when divided by a modulus (modular exponentiation or modular reduction)?	Unambiguous
+1429993	What is the value of an infinite factorial tower?	Ambiguous
+541081	When to stop doubling down?	Ambiguous
+3962287	Find the Maclaurin series expansion of the function $y(t) = (t^2 + t - 2)e^{t^2 + 1}$ centered at $t = 0$.	Unambiguous
+2332969	Find the radius of convergence of the power series $f(x) = \sum_{n=2}^\infty \log(n) x^n$.	Unambiguous
+1695625	Does $h_{a,b}(x) = ((ax + b) \bmod 2^w)/2^{w-M}$ satisfy the Uniform Difference Property?	Ambiguous
+2051197	Can someone explain this formula?	Ambiguous
+602143	Find the area bounded by the curves $y=e^x$, $y=xe^x$, and $x=0$	Ambiguous
+1799824	Determine whether the function $f: \mathbb{R}^2 \to \mathbb{R}^2$ defined by $f(x,y) = (x^2-y^2, 2xy)$ is injective.	Unambiguous
+1663192	Given a linear map with an associated matrix, with respect to which basis is the matrix representation taken?	Ambiguous
+291038	Let $A$ and $B$ be a toeplitz and symmetric positive definite $NxN$ matrices. If $\kappa (A)  > \kappa (B)$, how to show that: $$\kappa (B^{-1}A)  < \kappa (A)$$  where $\kappa $(X) is condition number of matrix $X$?	Unambiguous
+4095882	I have some people each performing repeated trials. How many will get at least a certain number of successes?	Ambiguous
+526574	I need to know if every element is either countable or co-countable.	Ambiguous
+1403539	$\arctan (x) + \arctan(1/x) = \frac{\pi}{2}$	Ambiguous
+4628757	Prove $\frac{ab^2+2}{a+c} +\frac{bc^2+2}{b+a} +\frac{ca^2+2}{c+b} \geq \frac{9}{2}$ for $a,b,c\geq1$	Ambiguous
+546701	Find the number of positive integers whose digits add up to 42	Ambiguous
+1475726	Locating three sets of collinear points	Ambiguous
+2431076	How do I show that if $\sigma^k$ is an $n$-cycle, then $\sigma$ is also an $n$-cycle?	Ambiguous
+536735	Find all functions $f: \mathbb{Q} \to \mathbb{R}$ that satisfy the Cauchy functional equation $f(x+y) = f(x) + f(y)$ for all $x, y \in \mathbb{Q}$.	Unambiguous
+1550929	Determine if a 4-tuple exists	Ambiguous
+840522	Need explanation for simple differential equation	Ambiguous
+2865778	Find $a$ such that $$f(x)=\left\lfloor \frac{1}{3}+2a\sin^3x\right\rfloor$$ satisfies a symmetry condition.	Ambiguous
+1510422	How can I show $f(x)$ is zero if $\int_a^b f(x)g(x) dx=0$ for each continuous $g$?	Ambiguous
+1601141	How to calculate the character of this series $\displaystyle{\sum _{n=0}^{\infty }\left(-1\right)^n\left(\frac{\ln n-1}{n^3+1}\right)}$?	Ambiguous
+2885878	Distance between two stations	Ambiguous
+2512671	I need to find a function in order to satisfy $f(f^{-1}(W)) \subseteq W$.	Ambiguous
+1542105	For a four digit number using {2,3,5,7,9}, is the last digit choice 3 if the remainder is 2 when divided by 3 or 5? Should the answer be (4)(3)(2)(3)?	Ambiguous
+970872	Provide a mathematical proof that the standard solution to the 'Burning Ropes' brain teaser is correct or incorrect, specifically regarding the method of measuring 45 minutes using two non-uniform ropes.	Unambiguous
+1900241	How to find the expected value of a discrete random variable using its probability generating function?	Unambiguous
+254347	Prove that the group isomorphism $\mathbb{Z}^m \cong \mathbb{Z}^n$ implies that $m = n$	Ambiguous
+2775319	For a polynomial $f(x)$, is the root of $f''$ always equal to $\frac13$ of the sum of the roots of $f$?	Ambiguous
+2865778	Find the range of real values for $a$ such that the function $f(x) = \lfloor \frac{1}{3}+2a \sin ^3 x \rfloor$ is an even function.	Unambiguous
+351359	How do I prove my metric space problem using sequential criteria?	Ambiguous
+2093269	Formulas to find bisector coordinates from 3 points without using matrices or vectors?	Ambiguous
+646238	How can I prove to my students that the value of a definite integral is independent of the dummy variable of integration?	Unambiguous
+2634909	How to prove that $e^x > x$ for all real numbers $x$?	Unambiguous
+2619473	Suppose a power series converges at its radius of convergence $R=1$. Does the derivative exist at 1? If not, are there counter-examples?	Ambiguous
+264371	Fun but serious mathematics books to gift advanced undergraduates.	Ambiguous
+2387841	Are the remainders $\beta \mathbb{N} \setminus \mathbb{N}$ and $\beta K_0 \setminus K_0$ homeomorphic?	Ambiguous
+672529	Verify that $-3x+5y-6z=-8$ is the correct equation of the plane consisting of all points equidistant from $(4,3,-2)$ and $(7,-2,4)$.	Unambiguous
+5877	What is an efficient algorithm to find two integers $x$ and $y$ such that $x^2 + y^2 = p$ for a given prime $p \equiv 1 \pmod 4$?	Unambiguous
+772475	Is it true that for a differentiable function $f(x)$, $\lim_{x\to\infty} f(x)=0$ implies $\lim_{x\to\infty} f'(x)=0$? If not, provide a counterexample.	Unambiguous
+4401632	This is a surface on a sphere. What are the equations of the surface in yellow?	Ambiguous
+3348703	How to perform bivariate interpolation to find the function $y(x,z)$ from a given set of scattered data points?	Unambiguous
+4008509	Find the critical points of the multivariable function $f(x,y) = (x-2)(y-2)(x+y-2)$.	Unambiguous
+1822028	Is the closed convex hull of a compact set in a dual space compact?	Ambiguous
+2361362	How is this a bijection?	Ambiguous
+2396015	What is the probability that the sum of three fair six-sided dice is eleven?	Unambiguous
+4253329	For what values of p is the series $\sum_{n=2}^\infty \frac{1}{n(\log n)^p}$ convergent or divergent, and how do I prove it?	Unambiguous
+1779361	How to prove that the trigonometric identity $\cos^4x - \sin^4x - \cos^2x + \sin^2x = 0$ holds for all real numbers $x$?	Unambiguous
+614812	How do you derive the formula $2\cdot(2\cdot(n/4) + \min(n\%4, 2))$?	Ambiguous
+2768349	Evaluate the limit as $n$ approaches infinity of $\left(1 + \frac{a}{n^2}\right)^{n/b}$, assuming $a$ and $b$ are constants with $b \neq 0$.	Unambiguous
+2591017	What is the definition and formula for the nth roots of unity in the complex plane?	Unambiguous
+1987713	If a continuous function $f: \mathbb{R} \to \mathbb{R}$ has a square $f^2$ that is uniformly continuous, must $f$ itself be uniformly continuous?	Unambiguous
+119307	Cauchy Sequence that Does Not Converge	Ambiguous
+1295010	What are statements about the natural numbers where induction is impossible or unnecessary to prove?	Ambiguous
+3178788	How do I determine the type of stationary point without calculating second derivative?	Ambiguous
+2346308	What are the parts of a number in a given base called?	Ambiguous
+2298323	How to merge $C_1, C_2, \dots, C_k$ to form $C_m = C(D)$ for my vertically divided dataset?	Ambiguous
+916805	Prove that for a subset $X$ of a metric space, the distance function $d(a, X) = \inf\{d(x, a) : x \in X\}$ equals $0$ if and only if $a$ is in the closure of $X$.	Unambiguous
+2138866	$\int_{0}^{\frac{\pi}{4}}\frac{\tan^2 x}{1+x^2}\text{d}x$ on 2015 MIT Integration Bee	Ambiguous
+2492396	How do I transform the first-order differential equation $(\sin^2y + x \arctan y)y' = 1$ into the form $M(x,y)dx + N(x,y)dy = 0$ to apply Lagrange's method for finding an integrating factor?	Unambiguous
+218977	For $\Omega = (-1,1)\times(-1,1)$ and the Dirac delta $\delta_0$ defined by $\langle \delta_0, \phi\rangle = \phi(0)$ for all $\phi \in C_0^\infty(\Omega)$, determine the smallest distribution space containing $\delta_0$.	Unambiguous
+2769775	Let $Z=X/(X+Y),$ find the pdf of $Z$.	Ambiguous
+2121133	Given $N$ numbers, how many arrangements are strictly increasing or decreasing?	Ambiguous
+2278214	Let $Y$ be a discrete random variable. What is $H(Y)$?	Ambiguous
+2309119	Prove that the polynomial equation $r^3-7r+1=0$ has no rational roots.	Unambiguous
+2934174	Find all pairs of prime numbers $(p, q)$ that satisfy the equation $pq = 3p + 7q$.	Unambiguous
+1694294	Show that $\pi(p_x^2) \geq 2p_x -2$	Ambiguous
+2093269	What is the formula to find the length of the internal angle bisector segment between two sides of a triangle given the lengths of those two sides and the included angle?	Unambiguous
+249834	If a continuous function goes from $5$ to $7$, why does it have to hit $2x+3$ somewhere?	Ambiguous
+2775319	What is the general formula for the second derivative of a third degree polynomial function?	Unambiguous
+681124	Find all $x$ satisfying $\frac{\tan x}{x}=2$, arising from setting $f'(x)=0$ for $f(x)=\frac{\sin^2x}{x}$.	Unambiguous
+599095	How do I calculate the required sample size to achieve a specific margin of error for a confidence interval of a population mean?	Unambiguous
+963503	If the sum of n vectors in a Euclidean space is zero and all their norms are equal, what can be said about their geometric arrangement?	Unambiguous
+1310890	Compute $\lim_{x\to\infty}\left(\sin{\frac 1x}+\cos{\frac 1x}\right)^x$	Unambiguous
+889570	Prove the trigonometric identity $\cot(\theta) \sec(\theta) = \frac{1}{\sin(\theta)}$.	Unambiguous
+291038	Let $A$, $B$ be matrices with $\kappa(A) > \kappa(B)$. Show $\kappa(B^{-1}A) < \kappa(A)$.	Ambiguous
+1592652	Provide an example of a binary relation on a non-empty set that is symmetric and transitive, but not reflexive.	Unambiguous
+385679	What are the properties of matrices $A$ and $B$ that satisfy the commutation relation $AB-BA=B$?	Unambiguous
+1982237	What is the general solution to the Bessel differential equation of order n?	Unambiguous
+269798	How do you calculate the expected value of a continuous random variable?	Unambiguous
+2175735	For non-negative integers $n, p, k$, find a closed-form expression for $A(p,n,k) = \frac{(pn)!}{(p!)^k(p(n-k))!}$ and for $A(p,n) = \sum_{k=1}^n A(p,n,k)$.	Unambiguous
+2237243	What are effective study habits and problem-solving strategies for improving one's mathematical practice and proficiency?	Unambiguous
+2146495	What are the basic steps for finding the convolution of $(f*f)(t)$ for a fraction?	Ambiguous
+754268	What is the formal statement of the Continuum Hypothesis and its status in ZFC set theory?	Unambiguous
+639890	Prove that the minimum value of $\cos(A-B)+\cos(B-C) +\cos(C-A)$ for any real numbers $A, B, C$ is $-3/2$.	Unambiguous

MathDisAM_train.tsv

@@ -0,0 +1,801 @@
+MathSE ID	Math Question	Tag
+137352	What are the domain and range of the map f in Devaney's definition of chaos on a metric space?	Unambiguous
+2561078	How does the triangle inequality apply to the proof for f-g using epsilon and delta?	Ambiguous
+735461	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
+2461048	What is the cyclic subgroup generated by -1?	Ambiguous
+578647	What is an intuitive explanation of the Fourier transform and how is it used to analyze signals in the frequency domain?	Unambiguous
+2430945	"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
+1403539	Prove that $\arctan (x) + \arctan(1/x) = \frac{\pi}{2}$ for $x >0$.	Unambiguous
+510991	Prove that every continuous function defined on a compact metric space is uniformly continuous.	Unambiguous
+2904744	Prove that $\lim_{x\to 1} f(x) = 3$ where $f(x) = \frac{x^3-1}{x-1}$ for $x \neq 1$.	Unambiguous
+982186	Decompose the expression $\frac{1}{4n^2-1}$ into partial fractions.	Unambiguous
+4087621	Between which two consecutive integers does the value of $\sqrt{12}$ lie?	Unambiguous
+697675	How do you multiply a matrix by a row vector from the left?	Unambiguous
+1025321	Find the derivative $\frac{dy}{dx}$ using implicit differentiation for the equation $\ln(1+xy) = xy$.	Unambiguous
+3313255	What is the definition and standard form of Sum of Products (SOP) in boolean algebra?	Unambiguous
+578647	In the Fourier transform, what does the cross term mean?	Ambiguous
+3843282	finding quotient group relationship	Ambiguous
+3372191	Asymptotic behavior of $a_{n+1}=a_n + f(a_n)$	Ambiguous
+2694508	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
+3735022	What is the formula for the curvature of a Frenet curve constrained to lie on a sphere?	Unambiguous
+4870757	If I know the dimension, depth, and codimension of $R$ and $I$, can I determine the depth of $I$?	Ambiguous
+166794	Proof that the set of irrational numbers is uncountable	Unambiguous
+4127695	Evaluate the sum $\sum_{r=0}^n 2^{n-r} \binom{n+r}{r}$ for non-negative integers $n$ and $r$.	Unambiguous
+2949838	Can every basis of a finite dimensional vector space be orthogonal?	Ambiguous
+7391	How do I evaluate the definite integral $\int_0^{\pi} \sin(2x)\sin(x) dx$?	Unambiguous
+1315088	Evaluate the limit $\lim_{x \to 0} \left( \frac{1}{\sin(x)\arctan(x)} - \frac{1}{\tan(x)\arcsin(x)} \right)$	Unambiguous
+36134	Modern Equivalent of Carr's Synopsis?	Ambiguous
+2526693	Evaluate the limit: $\lim_{x \to 0^+} x^{\sqrt{x}}$	Unambiguous
+432999	Prove that the limit of the sequence $\frac{n}{2^n}$ as $n$ approaches infinity is 0.	Unambiguous
+1572032	Why is the CDF of $Y_n$ equal to $(\frac{t- \theta}{\theta})$ and what are the restrictions on $t$?	Ambiguous
+2186358	What is the correct term for $I^n f(x)$ where $I$ is the integral operator?	Ambiguous
+606172	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
+4468673	If $x$ and $y$ are positive integers such that $5x+3y=100$, what is the maximum value of the product $xy$?	Unambiguous
+166794	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
+1191610	Prove that the composition of two group homomorphisms is also a group homomorphism.	Unambiguous
+4437668	What are some interesting examples of elementary proofs by mathematical induction for students?	Unambiguous
+1354331	Does there exist a function that is discontinuous at a point but has a continuous derivative on its domain?	Unambiguous
+970133	Prove that the infinite series $\sum_{n=1}^\infty \frac{n^2(n-1)}{2^n}$ converges to 20.	Unambiguous
+419859	Prove that the graph $y=x^{p/q}$ has a vertical tangent.	Ambiguous
+105575	What is the difference between the soundness and completeness theorems in first-order logic?	Unambiguous
+456595	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
+3075859	What is the next step after showing $(c^k)^t = (id)$ for cycle orders?	Ambiguous
+2538727	How to use Cantor's diagonal to prove my space is not sequentially compact?	Ambiguous
+217966	What is the relationship between monomorphisms and pullbacks in category theory?	Unambiguous
+3117992	Could different physical systems follow different underlying mathematics?	Ambiguous
+181367	Provide a counterexample to show that a pseudocompact topological space is not necessarily a compact space.	Unambiguous
+854380	How to distinguish between walking on a sphere and walking on a torus?	Ambiguous
+3087356	Why is the number 2 included in the definition of prime numbers instead of being excluded as the only even prime?	Unambiguous
+4468673	If $x$ and $y$ are positive integers such that $5x+3y=100$, what is the greatest possible value of $xy$?	Ambiguous
+4060048	How do I solve integrals of the form integral of (polynomial * exponential) dx using integration by parts?	Unambiguous
+1024830	What is the intuitive meaning of a compact set in the context of a metric space or topology?	Unambiguous
+3172708	How to efficiently update the inverse of a lower triangular matrix after changing 4 individual elements?	Unambiguous
+1133113	How do we define arc length?	Ambiguous
+2490690	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
+1711154	Prove that f is differentiable at x=0	Ambiguous
+551148	Why is $2x^3 + x$, where $x \in \mathbb{N}$, always divisible by 3?	Ambiguous
+2304892	Probability of rolling any set of size k on n dice with m sides	Ambiguous
+1576678	Can my sine formula with $\pi/6$ be put into Excel somehow to plot the graph?	Ambiguous
+2982578	Prove that every finitely generated module over a ring $R$ is a homomorphic image of a free module.	Unambiguous
+1619762	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
+1882798	Is there a mathematical rule to check if positive numbers $a$ and $b$ satisfy my modulo $P$ equation?	Ambiguous
+289779	I'm looking at the sequence $a_n = 3^n - 2^{n+100}$. How do I handle this?	Ambiguous
+1320951	Prove that the intersection of any two cyclic subgroups of a group is also a cyclic subgroup.	Unambiguous
+2449459	What's wrong with this fake proof that $2+2=5$?	Ambiguous
+1017503	Prove by induction that $\sum_{k=0}^n k\cdot k! = (n+1)! - 1$ for all $n \in \mathbb{N}$.	Unambiguous
+3490887	Question on existence of square root function!	Ambiguous
+3791238	Why is a curve homeomorphic to a knot?	Ambiguous
+2284995	"Expected count of ""double records"""	Ambiguous
+3468876	Suppose $Y_1$ and $Y_2$ are independent and $X=Y_1+Y_2$. What is the distribution of $X$?	Ambiguous
+243106	How to apply $\epsilon-\delta$ definition to a 2-valued function?	Ambiguous
+2396246	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
+303933	Evaluate the limit $\lim_{n \to \infty} \frac{(\log(n))^a}{n^b}$ for positive real constants $a$ and $b$.	Unambiguous
+975836	How to solve linear recurrence relations using the method of repeated substitution?	Unambiguous
+164244	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
+2344993	What are some unexpected real-world applications or occurrences of the natural logarithm in mathematics and science?	Unambiguous
+781320	Why does this curve cross the y-axis if $\cos(\sin x)$ is never zero?	Ambiguous
+2834788	Is the Euler–Mascheroni constant $\gamma$ known to be irrational?	Unambiguous
+3247615	How do I evaluate the polynomial P(x) at x = -2?	Unambiguous
+1329195	How to calculate a percentage increase when the result is above 100%?	Unambiguous
+2449459	Where is the logical fallacy in the common fake proof that $2+2=5$?	Unambiguous
+3017476	Interpreting an agreement function	Ambiguous
+1169872	Is there a positional numeral system base $b$ such that every rational number has a terminating representation?	Unambiguous
+2923438	How does the RSA cryptosystem work, and what are the steps for key generation, encryption, and decryption?	Unambiguous
+3114691	How to find $\lim\limits_{n\to\infty} A^n$?	Ambiguous
+1559640	How to find all natural numbers $n$ for quadratic permutations with a perfect square sum?	Ambiguous
+3699937	The cyclic subgroups of $p^2$ order non-cyclic group are normal	Ambiguous
+551148	Prove that for every natural number $x$, the expression $2x^3 + x$ is always divisible by 3.	Unambiguous
+3537073	Help with proof by induction, please! I may not be understanding how to work with factorials.	Ambiguous
+3537073	Can you provide a step-by-step example of how to use proof by induction to prove a summation formula involving factorials?	Unambiguous
+1647675	A Quadrilateral and A Triangle in a Trapizium	Ambiguous
+2999909	In an arbitrary category, is the hom-set $Hom(X,Y)$ always equipped with an Abelian group structure?	Unambiguous
+1466540	Find the $\lim_{x\to \infty} (\ln(x)-x)$	Unambiguous
+4247524	Finding the probability that at least 8 houses in a row are of same colour.	Ambiguous
+486600	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
+308644	Do I need to include the definition of union in this proof?	Ambiguous
+2396246	How do I calculate the volume of water a cylindrical tank can hold given its radius and height?	Unambiguous
+3560940	If the weights of three different items are 10g, 25g, and 5g, how do I divide 1kg among them?	Ambiguous
+2561078	Prove that if functions $f$ and $g$ are continuous at a point $x$, then their difference $f-g$ is also continuous at $x$.	Unambiguous
+1568161	Evaluating $\lim\limits_{x \to \pi} \frac{(\pi -x)\sin x}{1 + \cos x}$	Ambiguous
+1074072	Is there a notion dual to that of a path in a topological space?	Ambiguous
+2664926	Evaluate the complex power $(-\sqrt{3} + i)^9$ using De Moivre's theorem.	Unambiguous
+365782	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
+1283817	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
+2539486	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
+2961334	How to prove a solution to a functional equation is unique?	Ambiguous
+574551	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
+1941590	How do you use Cramer's rule to solve a system of linear equations using determinants?	Unambiguous
+2158739	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
+258546	How to find the parabolic curve equation from length $L$ and area $S$ based on my picture?	Ambiguous
+3328473	How to find the numerical solutions for $2x - \tan(x) = 0$ in the interval $[0, 2\pi]$?	Unambiguous
+4268142	A coin is tossed three times, defining two related random variables from the outcomes. Find their joint pmf and show they're dependent.	Ambiguous
+2881207	How does the irreducibility of a polynomial over a finite field depend on the size of the field?	Unambiguous
+1037122	Prove that for any real number $\lambda$, the integral $\int_0^\infty e^{-x} \sin(\lambda x) \, dx = \frac{\lambda}{1+\lambda^2}$.	Unambiguous
+2166855	How to find the maximum value of the objective function $x + y$ subject to a set of linear constraints?	Unambiguous
+2049971	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
+3786586	How to solve an inequality involving the sum of two absolute value expressions, such as $|ax + b| + |cx + d| < k$?	Unambiguous
+41706	What are some practical applications of matrix multiplication in computer science and engineering?	Unambiguous
+3111111	If $X$ is a topological space, does the property still hold for the product of two continuous functions?	Ambiguous
+2022771	In mathematical learning, does the complexity of notation generally decrease as one advances to more abstract levels of study?	Unambiguous
+1980947	Which value is greater, $2^{\sqrt{2}}$ or the mathematical constant $e$?	Unambiguous
+609404	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
+2539486	(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
+3087356	Why is the definition of prime numbers written to include 2	Ambiguous
+3399461	A polynomial defined by two squared terms is divided by two linear factors, giving known remainders. Find the coefficients.	Ambiguous
+4107862	How should I think about the concept of location for a point?	Ambiguous
+744139	Prove by induction that $2\cdot 8^n + 3\cdot 15^n + 2$ is divisible by $7$ for all $n \in \mathbb{N}$.	Unambiguous
+4749159	Equation exercise involving nth-roots in Serge Lang's Basic Mathematics.	Ambiguous
+795	How to prove the theorem of Riemann asked by Kenny?	Ambiguous
+942204	Looking for the logic of a sequence from convolution of probability distributions	Ambiguous
+4835425	Revisiting: Adding an optimizer to a nonlinear set of equations	Ambiguous
+2237243	"What are good ""math habits"" that have improved your mathematical practice?"	Ambiguous
+1406770	Iterated square roots over finite field. When do we hit a nonresidue?	Ambiguous
+3472866	Are $\sigma^2$ and $\mu$ random variables?	Ambiguous
+3806534	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
+4325846	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
+2530549	What are some mental math techniques for performing complex calculations without a calculator?	Unambiguous
+164588	How to find the general coefficient of the Cauchy product of two power series?	Unambiguous
+1838743	In the figure, two circles intersect at $P$ and $Q$...	Ambiguous
+788217	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
+216344	Does the equation $ax + by = ab$ have any solutions for $\gcd(a,b)=1$?	Ambiguous
+385679	A matrix satisfying $AB-BA=B$	Ambiguous
+925677	I need the coordinates for an octagon.	Ambiguous
+2159465	Are two polygons homeomorphic?	Ambiguous
+582557	Determine whether $\sum_{n=2}^\infty \frac{1}{n\ln(n^3)}$ converges or diverges.	Unambiguous
+4336	If eigenvalues are positive, is the matrix positive definite?	Ambiguous
+531128	Dimension of a subspace of finite-dimensional product space $V$ equals $\dim V - 1$	Ambiguous
+2388784	What is the intuitive explanation for the formula of variance in probability theory?	Unambiguous
+1165928	What is the general formula for the k-th derivative of $G \circ F$?	Ambiguous
+4768	every permutation is either even or odd,but not both	Ambiguous
+1997424	What is the rigorous definition of the differential of a function in multivariable calculus?	Unambiguous
+2239918	Can someone help me with this: 2^(log(2x,x+2))+3^(log(2,x+3))=sqrt(-1-x)	Ambiguous
+354134	"What is the logical negation of the statement ""$A\subseteq B$"" in set theory?"	Unambiguous
+1871195	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
+422707	Finding eigen values of a binary matrix with diagonal elements are all 0s and non-diagonals are 1s	Ambiguous
+747433	What are the standard tests for determining the absolute convergence of an infinite series?	Unambiguous
+3300982	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
+1674072	How do you find the greatest integer whose square root differs from another integer's square root by at least one?	Ambiguous
+188703	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
+2072845	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
+730358	Sketch $\tan x$, solve $\cot x=0.15$, and evaluate $\csc x$ given $\cot x=0.15$.	Ambiguous
+853164	Provide an example of a binary relation on a set that is not reflexive, not symmetric, and not transitive.	Unambiguous
+3934173	Can any permutation of elements be represented as a series of rotations in some n-dimensional space?	Ambiguous
+265703	Under what conditions does every element of a monoid occur exactly once in its Cayley table?	Unambiguous
+4067963	What is the sum of $\sum_{n\in M(P)} \frac{1}{n}$ where $M(P)$ contains numbers with prime divisors in $P$?	Ambiguous
+2540359	What is the formal definition of an undecidable problem in computability theory?	Unambiguous
+3519278	$6$ 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
+3310472	Is the P versus NP problem classified as a topic within the field of Operations Research?	Unambiguous
+1072836	Is every symmetric matrix diagonalizable, and what are the possible eigenvalues of an orthogonal matrix?	Ambiguous
+2096678	Suppose $A^2 + B^2 = AB$ and $AB - BA$ is invertible. What can be said about $n$?	Ambiguous
+1798274	Least value of $a$ for which $4ax^2 + \frac{1}{x} \geq 1$	Ambiguous
+488020	Prove that the set of signed circuits of an oriented matroid is invariant under reorientation of its elements.	Unambiguous
+110579	In how many ways can a batsman score exactly 100 runs using only 6s and 4s, where the order of scoring matters?	Unambiguous
+1249731	Let $X=(X_1,\dotsc,X_n)$ be a random vector with mean $0$. What is $\mathbb{E}[X_1\cdots X_n]$?	Ambiguous
+646238	How do I convince my students that the choice of variable of integration is irrelevant?	Ambiguous
+3125432	Trouble with telescoping algorithm question	Ambiguous
+3285074	Are there other functions that satisfy $|f(x)-f(y)|=|x-y|$?	Ambiguous
+664657	I'm trying to connect $\epsilon$ and $\delta$ for $|f(x) - f(c)| < \epsilon$. How do I do this?	Ambiguous
+2360710	Formula for Natural logarithm of $\pi$	Ambiguous
+1731822	Given some unlabeled 2D data points, how do I construct a k-nearest-neighbour graph and its Laplacian?	Ambiguous
+2146495	For $f(t)=\frac{2}{1+t^2}$, compute the convolution $(f*f)(t) = \int_0^t f(u)f(t-u)\,du$.	Unambiguous
+4341781	The integral of a function along any circle is zero. What does this mean, and what follows from it?	Ambiguous
+227317	Explaining $\cos^\infty$	Ambiguous
+1067717	How do you get $\frac{2\sum_{k=0}^n2^n}{n+1}=2^{n+1}$?	Ambiguous
+108308	Apostol proof divides by zero?	Ambiguous
+2062239	Numerical algorithms for approximating the polar decomposition of a matrix	Unambiguous
+687510	A shape is cut into 3 equal parts. Show one part contains 2 points at distance at least one.	Ambiguous
+330556	$a^{n}$: is it arithmetic or geometric, or neither?	Ambiguous
+3791855	How does $x^2-8x+17=0$ have nonreal solutions?	Ambiguous
+988784	Why is it a common misconception that non-real numbers consist only of square roots of negative numbers?	Unambiguous
+2394502	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
+3098928	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
+2823907	 Prove that $\sum_{n=2}^\infty \frac{1}{n\ln(n^3)}$ diverges using the comparison test.	Unambiguous
+3059714	How does one evaluate the multiplication $f(2)\cdot f(3)\cdot f(4)\cdots f(15)$ by formulating?	Ambiguous
+2594206	Prove that there is no $2 \times 2$ real matrix $A$ such that $A^2 = \begin{pmatrix} 0 & 1 \ 0 & 0 \end{pmatrix}$.	Unambiguous
+639890	Minimum value of $\cos(A-B)+\cos(B-C)+\cos(C-A)$?	Ambiguous
+225922	An example for a calculation where imaginary numbers are used but don't occur in the question or the solution.	Ambiguous
+983405	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
+4693000	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
+724075	Where can I find the book 'Here's a Cool Problem: A Collection of Short Questions with Clever Solutions'?	Unambiguous
+1302479	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
+26918	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
+361118	Prove that $4x-x^4 \leq 3, x \in \Bbb R$	Ambiguous
+2006621	How do I calculate the number of subsets of a finite set given its cardinality?	Unambiguous
+62748	What is the difference between two related set-builder notations?	Ambiguous
+2054106	Is it true that pointwise convergence with a uniform bound implies convergence of the integrals?	Ambiguous
+742532	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
+105575	What is the difference between Completeness and Soundness in first order logic?	Ambiguous
+1978354	Distribution of $\frac{X+Y}{X-Y}$	Ambiguous
+788217	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
+1788328	Find the numerical value of $\sin 10^\circ \sin 50^\circ \sin 70^\circ$.	Ambiguous
+3944628	In the definition of a stopping time, how important are the filtration conditions mentioned earlier?	Ambiguous
+1079395	Inclusion Exclusion Principle (Probability) Problem: What is P(Rain)?	Ambiguous
+1783734	Can you explain the logical basis and validity of the principle of mathematical induction for proving statements over natural numbers?	Unambiguous
+1896418	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
+1214106	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
+700216	Different solutions of $x+y+z=10$ where $x$, $y$, $z$ are all positive integers and $x, y, z \leq 10$	Ambiguous
+3086320	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
+3289735	Is the arc length of a circle always irrational between two points with rational coordinates?	Unambiguous
+1762325	Is $\sim (\sim p) \rightarrow p$ always true?	Ambiguous
+227317	What is the limit of the sequence defined by the iterated cosine function $x_{n+1} = \cos(x_n)$ as $n$ approaches infinity?	Unambiguous
+4256389	Alternating sum involving reciprocals of powers of primes equals $\ln(\frac{\pi}{4})$	Ambiguous
+1319618	An inequality, comprising of liminf and Sup of a set	Ambiguous
+2498847	Is 1/x a surjective function when the domain and the range are all real numbers?	Ambiguous
+1894173	Does the sum of the reciprocals of all prime numbers converge or diverge?	Unambiguous
+3247615	How to find $P(-2)$ here?	Ambiguous
+602860	How to calculate which point is furthest from the origin in the direction of a normalized vector?	Ambiguous
+3502297	How to show $\log(x)$ loses to powers?	Ambiguous
+1347	How can Euclidean geometry be axiomatized or derived using criteria from hyperbolic geometry?	Unambiguous
+11194	How to solve a linear recurrence relation where the nth term depends on the sum of all previous terms?	Unambiguous
+290435	Find five distinct positive integers whose reciprocals sum to 1.	Unambiguous
+4019674	Prove that the value of $\frac{\arcsin(3/5)}{\pi}$ is an irrational number.	Unambiguous
+2353213	Are there any other Lie subgroups of $\textrm{Gl}(n, \mathbb{R})$ and is there a classification for them?	Ambiguous
+1347	Studying Euclidean geometry using hyperbolic criteria	Ambiguous
+4255331	If $ \lim_{x\to \infty} (f(x+1)-f(x))=1$, then $ \lim_{x\to \infty} \frac{f(x)}{x}=1$?	Ambiguous
+3395798	Definition of a matrix by matching two vectors	Ambiguous
+984271	How does the isomorphism between $\mathbb{K} \otimes M_n(\mathbb{C})$ and $\mathbb{K}$ work?	Ambiguous
+1249731	What is the formula for the first order moment (mean vector) of a multivariate Gaussian random vector?	Unambiguous
+1251548	Provide an example of a set in a metric space that is bounded but not totally bounded.	Unambiguous
+4488057	Can a function be represented by a Laurent series expansion in a neighborhood of a non-isolated singularity?	Unambiguous
+217335	In how many ways can a total of $1.50 be formed using exactly 50 US coins?	Unambiguous
+329838	How to simplify a square root of a number by extracting perfect square factors?	Unambiguous
+1376320	Proving that if $a+b$ and $ab$ are of the same parity, then $a$ and $b$ are even.	Ambiguous
+4383384	How to extract the vector from a rank-$1$ matrix?	Ambiguous
+1895699	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
+71874	Can I use my powers for good?	Ambiguous
+395543	What are the formulas for generating all primitive Pythagorean triples?	Unambiguous
+872256	What is the geometric interpretation of finding the $n^{ ext{th}}$ roots of a complex number in the complex plane?	Unambiguous
+2837226	Show that a subset of a complete metric space is complete.	Ambiguous
+538941	If $xy$ and $x+y$ are both even integers (with $x,y$ integers), then $x$ and $y$ are both even integers	Ambiguous
+884228	How do you calculate the reliability function for a continuous uniform distribution given its lower and upper bounds?	Unambiguous
+3035983	How can I find all pairs of integers $(n, m)$ that satisfy the equation $\frac{150 + n}{15 + n} = m$?	Unambiguous
+2880747	Product of Summation for a single table of values.	Ambiguous
+1002480	How to go from talking about line segments to points to prove $X(A)$ is compact?	Ambiguous
+488020	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
+859968	Is $X/R$ homeomorphic to $Y$ for the broom spaces?	Ambiguous
+2520461	How to minimize $AP/PB$? It doesn't seem doable for some locations.	Ambiguous
+1496959	What are the necessary and sufficient conditions for a sequence of continuous functions to converge uniformly on a compact set?	Unambiguous
+1279205	Can an irrational number be classified as odd or even under the standard definition of parity for integers?	Unambiguous
+4437668	What are some good, elementary and maybe also interesting proofs by induction?	Ambiguous
+2576976	How do I find the inverse of a $2\times2$ matrix if the determinant is zero?	Unambiguous
+1626298	Prove that $2^{\log_4 n} = n^{\log_4 2}$ using logarithmic identities.	Unambiguous
+3811446	What is the value of the infinite series $\sum_{n=1}^{\infty}\frac{1+2+\cdots +n}{n!}$?	Unambiguous
+1891318	"Number of bit strings of length 8 that do not contain ""$100$""?"	Ambiguous
+1311558	What is the formal definition of a set in Zermelo-Fraenkel set theory and how is it intuitively conceptualized?	Unambiguous
+3960592	Finding the equation of hyperbola that passes through $(2,0)$ and has foci points $(2,3)$ and $(1,0)$	Ambiguous
+3791855	Why does the quadratic equation $x^2-8x+17=0$ have non-real complex solutions?	Unambiguous
+1668681	Provide an example of a 2D planar shape whose center of mass lies on its boundary.	Unambiguous
+219343	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
+1141210	I am trying to do a set proof and I don't understand where the part highlighted in green comes from.	Ambiguous
+344268	How do I solve a rational inequality where the variable $x$ appears in the denominator?	Unambiguous
+1501369	How to prove a function is one to one given $f(f(m)+f(n))=m+n$?	Ambiguous
+2207598	Prove or disprove: if $A\times A = B\times B$ then $A=B$.	Ambiguous
+738236	How to calculate the area of a triangle ABC when given three position vectors $a, b$, and $ c$ in 3D?	Ambiguous
+4285591	Does the expression $\frac{1}{1-hL}$ have an upper bound for $h > 0$ and $L > 0$ such that $hL < 1$?	Unambiguous
+4080901	Using a block matrix to show that $A$ and $B$ commute	Ambiguous
+3582088	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
+3729349	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
+2494260	List mathematical theorems, conjectures, or concepts that are named after geographic locations.	Unambiguous
+3265216	How to perform horizontal and vertical rescaling of a function $f(x)$ in algebra?	Unambiguous
+114560	Sum of normal bundle and tangent bundle.	Ambiguous
+1001101	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
+4564068	What is the formula for 0 + 1 + 3 + 6 + 10 + 15 ... ?	Ambiguous
+227211	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
+1381335	How to tell which function fits my series to make it a Riemann Sum?	Ambiguous
+217315	How do you tell the difference between the minus signs?	Ambiguous
+1432379	What is the minimum of $F(x,y,z)=\frac{1}{x+y}+\frac{1}{x+z}-\frac{1}{x+y+z}$?	Ambiguous
+217315	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
+3711681	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
+1435003	How to find $P(X_i=1)$ to get the expected number of empty boxes?	Ambiguous
+1562459	Prove that the sequence $(1 + n^{-2})^n$ converges to 1 as $n$ approaches infinity.	Unambiguous
+1304823	Can someone explain the parts of this equation?	Ambiguous
+2657952	"What's the relationship between mathematical logic and just plain old ""logic""? The notation differs, do the concepts? The rules?"	Ambiguous
+982093	Proof for $\lim_{n \to \infty}\left(1+\frac{a_n}{n}\right)^n = e^a$?	Ambiguous
+251033	What are the search terms for the largest subset with no 3 element progressions?	Ambiguous
+2766082	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
+608477	What can you tell me about $4^{n+1}+5^{2n-1}$ and 21?	Ambiguous
+1329254	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
+3422194	Compute the sum of the infinite series $\sum_{k=1}^{\infty} \frac{1}{k(k+2)}$ using partial fraction decomposition.	Unambiguous
+244424	If rank x < rank y, does it follow that x is an element of y?	Ambiguous
+687510	Provide an example of a Pigeonhole Principle problem involving points placed inside a square area.	Unambiguous
+2863657	Simple Ratios Shortcut - Why does it work?	Ambiguous
+721784	What is the absolute value of the complex exponential function $e^{iz}$ where $z$ is a real number?	Unambiguous
+2581976	How do I find the limit of a function as x approaches a specific value?	Unambiguous
+1645747	Compute the derivative of $x$ with respect to $\frac{1}{x}$.	Unambiguous
+997152	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
+1475726	How to find three sets of three collinear points in a given 2D point set?	Unambiguous
+981781	$$f(x)=x^3-x^2-14x+24$$	Ambiguous
+1040128	How do I evaluate the improper integral $\int_1^\infty \frac{dx}{x^2\sqrt{x^2-1}}$ using trigonometric substitution?	Unambiguous
+2278214	Calculate the Shannon entropy of the random variable representing the number of heads obtained when flipping a fair coin ten times.	Unambiguous
+2751401	If $a\mid b$ and $a>0$ then $(a,b)=a$	Ambiguous
+4057665	Are these topologies the same?	Ambiguous
+1466559	How to find the angle between two vectors using the dot product formula derived from the law of cosines?	Unambiguous
+1602977	What's an example of an infinitesimal?	Ambiguous
+1899905	How can physical quantities that are modeled by irrational numbers be realized or measured in the physical world?	Unambiguous
+2604608	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
+1761660	Find the value of $a$ such that the range of the function $f(x) = ax(1-x)$ contains the interval $[0,1]$.	Unambiguous
+1695878	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
+4269673	Is there a notation to indicate that a variable in this equation is constant?	Ambiguous
+127376	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
+3905420	Would a certain cyclic or dihedral group be an example? How do we find its subgroups?	Ambiguous
+2138866	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
+1688323	How to find the coefficients of a quadratic polynomial given three coordinate points using the method of undetermined coefficients or differentiation?	Unambiguous
+13582	Prove that every continuous function from a closed interval $[a, b]$ to itself has a fixed point.	Unambiguous
+254561	Provide a proof that $\arctan(x) = \arcsin(x/\sqrt{1+x^2})$ for all real numbers $x$.	Unambiguous
+1222768	Why are vector valued functions 'well-defined' when multivalued functions aren't?	Ambiguous
+738236	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
+2597743	Can a certain type of integral be solved using the gamma function method?	Ambiguous
+2152735	What is the definition of an open set in the context of a general topological space?	Unambiguous
+2820510	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
+2365371	Prove $ \frac{1}{x^{2n}}+x^{2n}$ is an integer for all $n$.	Ambiguous
+401477	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
+2717459	What is the formula for the expected value of a weighted sum of positive random variables?	Unambiguous
+2639179	Am I wrong about the Monty Hall Problem?	Ambiguous
+1176215	Prove by induction that $3^{3n+1} + 2^{n+1}$ is divisible by 5 for all non-negative integers $n$.	Unambiguous
+1271168	What is the difference between the square of a logarithm, $(\log x)^2$, and the logarithm of a square, $\log(x^2)$?	Unambiguous
+1376320	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
+620187	What do these symbols mean?	Ambiguous
+2401656	Compute the value of ${34}^{429} \mod 431$.	Unambiguous
+1290374	Can you provide an intuitive explanation and a proof for both parts of the Fundamental Theorem of Calculus?	Unambiguous
+19236	Efficient numerical algorithms to compute the Euler gamma function for real and complex arguments	Unambiguous
+1194644	Is every integral extension of commutative rings a finite extension?	Unambiguous
+2619473	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
+846706	What strategies can I use to find a conformal map from a lens shape to the first quadrant?	Ambiguous
+547928	What is the minimum value of $a+b+\frac{1}{ab}$ if $a^2 + b^2 = 1$?	Ambiguous
+2273478	"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
+1435003	How do I calculate the total number of empty bins in the balls and bins problem given n balls and m bins?	Unambiguous
+973713	What are the common notation styles and conventions for the second-derivative symbol in calculus?	Unambiguous
+2489152	Why is the bracket always zero?	Ambiguous
+3005021	Proof Verification: Differentiability implies continuity.	Ambiguous
+2096678	Provide a set of practice exercises on basic matrix arithmetic, including addition, subtraction, and multiplication.	Unambiguous
+2618534	Compute the indefinite integral of $\int \frac {1+\sin (x)}{1+\cos (x)} dx$ with respect to $x$.	Unambiguous
+159505	Find the value of $x$ that solves the linear congruence $3x + 7 \equiv 6 \pmod{13}$.	Unambiguous
+12139	Number of relations that are both symmetric and reflexive	Ambiguous
+114107	Determine the rotation necessary to transform one point on a sphere to another.	Ambiguous
+1899905	How irrational quantities physically exist in nature?	Ambiguous
+1894173	Sum of reciprocal of primes	Ambiguous
+3628	Find a general formula for the n-th derivative of the function $f(x)=\frac {x} {x^2-1}$.	Unambiguous
+1314138	How do I prove my integral limit is $0$ using the substitution $xt=u$?	Ambiguous
+332632	What is the probability that a person which has a positive test result in AIDS, is indeed infected with AIDS?	Ambiguous
+1383373	Provide the solution and explanation for Question 55 from the GRE Mathematics Subject Test form 1268.	Unambiguous
+2452316	Evaluating a contour integral using Cauchy's generalised integral formula	Ambiguous
+2705618	Evaluate $\lim_{n\to\infty} \frac{e^{1/n} + e^{2/n} + \cdots + e^{n/n}}{n}$.	Unambiguous
+3021170	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
+4569632	In $\triangle ABC$, $BD$ is a median, $\angle DAB=15$ and $\angle ABD=30$. Find $\angle ACB$.	Unambiguous
+3303672	Line intersecting circle from origin	Ambiguous
+92765	Find all complex roots of the polynomial equation $z^4 + 2z^3 + 6z - 9 = 0$	Unambiguous
+2138847	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
+685435	Trying to get a bound on the tail of the series for $\zeta(2)$	Ambiguous
+820424	What are the standard methods and laws for simplifying Boolean functions in digital logic?	Unambiguous
+3117241	Complex function to Taylor and Laurent series	Ambiguous
+1516096	When does $A^TA=0$ imply $A=0$?	Ambiguous
+1950367	Does equality of the complexifications imply equality of the algebras?	Ambiguous
+551350	Find all complex solutions to the equation $x^4+24x^3+18x^2-27=0$.	Unambiguous
+1735792	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
+1145215	Consider a matrix $A$ that is full rank. Is $A(A^\top A)^{-1}A^\top = I$?	Ambiguous
+2987994	Provide a list of examples of definite integrals solvable using the Feynman trick (Leibniz integral rule).	Unambiguous
+462723	What are some examples of mathematical games suitable for undergraduate students that can be used for teaching discrete mathematics or game theory?	Unambiguous
+184593	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
+2186187	How do you calculate the first Fourier coefficient $a_0$ for a periodic function in its Fourier series representation?	Unambiguous
+574465	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
+2289082	Compute the definite integral $\int_1^3\frac{\ln x}{x^2+3}\ dx$	Unambiguous
+269798	Is it true that $E(X^a) = \int x^a g(x)\,dx$?	Ambiguous
+4683394	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
+4398943	What does 'abstract' mean in this context?	Ambiguous
+4367689	Does there exist an ordered field extension of the rational numbers $\mathbb{Q}$ in which $\mathbb{Q}$ is not a dense subset?	Unambiguous
+1867732	Gradient and Hessian of function on matrix domain	Ambiguous
+791508	Let $A, B, X$ be subgroups of $G$ with $A \subset X \subset AB$. Prove that $A(B\cap X) = X$.	Unambiguous
+1950039	What is the identity property of multiplication by one in arithmetic?	Unambiguous
+4073406	Infinite subset of natural numbers	Ambiguous
+823613	What is the time complexity in $\Theta$-notation in terms of $n$ for the Merge Sort algorithm?	Unambiguous
+1896195	Let $M$ be a module over a ring. If $\operatorname{Hom}(M,M)$ is free, what can we conclude about $M$?	Ambiguous
+3300982	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
+3178788	What are the methods to determine the type of stationary point of a single-variable function without calculating the second derivative?	Unambiguous
+1587079	How do I prove the map $(i,j) \mapsto (i+j, ij_0 - i_0 j)$ is surjective?	Ambiguous
+3764582	Is there a convenient implementation in R to compute my incomplete gamma function integral?	Ambiguous
+2918597	Find angle of which a cannon of fixed length has to fire	Ambiguous
+1651728	Prove that the union and intersection of two recursively enumerable sets are also recursively enumerable.	Unambiguous
+3487811	Evaluate the limit $\lim_{x \to 0} {1-x\cot x\over x^2}$	Unambiguous
+1302479	Prove that the limit of the Dirichlet function does not exist at any point in its domain.	Unambiguous
+3493088	$\mathbb{C} \cup \{ \infty \}$ and $\mathbb{R} \cup \{ -\infty, +\infty \}$	Ambiguous
+2520262	Show that $a_{n+1}= 1 + \frac{1}{a_n}$ converges.	Ambiguous
+377705	How many repetitions do I need to find a prime between n and 2n?	Ambiguous
+726827	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
+3564289	What is the expected number of returns to the origin for a random walk?	Ambiguous
+1232385	How do I prove $\mathbb{Z}[w]/(a)$ is isomorphic to $\mathbb{Z}_{N(a)}$ using the norm?	Ambiguous
+700216	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
+3533343	Counter examples in measures	Ambiguous
+1612241	What is the logarithmic identity for the sum of logarithms of multiple terms?	Unambiguous
+3743564	Another coins weighing puzzle	Ambiguous
+4336	If all eigenvalues of a Hermitian matrix are positive, is the matrix positive definite?	Unambiguous
+166553	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
+532830	How can I get more intuition about the complex numbers (C) and their multiplication and exponentiation?	Unambiguous
+2054106	What are some common generalizations or variations of the Lebesgue Dominated Convergence Theorem in measure theory?	Unambiguous
+29758	Prove that every injective entire function from the complex plane to itself is a linear polynomial.	Unambiguous
+20813	How does a number system without irrational numbers handle circles?	Ambiguous
+2730855	Water is poured into a container at a given rate. How fast does the level rise at a certain height?	Ambiguous
+4194118	Area between square and circumscribed circle is less than half the area of the circle.	Ambiguous
+2614579	How do I evaluate the limit $\lim_{x\to1} \frac{\sqrt{2x-1} -1}{x^2-1}$ step-by-step?	Unambiguous
+1483533	How do I solve a linear equation for a real variable $x$?	Unambiguous
+1145215	What is the definition of a matrix inverse in linear algebra and how does it relate to the identity matrix?	Unambiguous
+218977	Let $\Omega$ be a rectangle. What is the smallest space to which $\delta_0$ belongs?	Ambiguous
+2361340	What is the limit as $x$ approaches infinity of $\frac{1}{x^{n-1}}$ for $n > 1$?	Unambiguous
+2770657	Find the Maclaurin series for $\ln(1-x)$ and state its interval of convergence using sigma notation.	Unambiguous
+2344095	Find coefficient of $x^2$ in a complicated expansion	Ambiguous
+1725118	What are the axioms needed to state the Fixed Point Theorem? Does it apply to ZF, Robinson arithmetic, or PA?	Ambiguous
+2732822	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
+2805228	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
+20096	Basic addition and subtraction word problems for preschool and kindergarten children.	Unambiguous
+2824634	Prove that $\sin(x/2) + 2\sin(x/4) \leq 3\sqrt{3}/2$ for all $x \in [0, 2\pi]$.	Unambiguous
+292425	How do I solve a piecewise function if it's not defined on $(-4,-1)$?	Ambiguous
+2059626	choose 1 song to be played 3 times and the rest only once	Ambiguous
+494832	What is the definition of a ring that is complete with respect to the I-adic topology defined by an ideal $I$?	Unambiguous
+1613067	subtracting the eigenvalue matrix from the original transformation of a 2x2 hit's the other eigenvalue proof	Ambiguous
+546701	Find the number of positive integers with exactly 5 digits whose digits add up to 42	Unambiguous
+3440597	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
+2512671	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
+62621	Is there at least one irrational number with the property that it cannot be defined by a finite string of information?	Ambiguous
+4494965	How many ways I can make \$5 in 50c, \$1 and \$2 coins?	Ambiguous
+2112893	How to calculate the 100th term of an arithmetic and a geometric series?	Unambiguous
+2108420	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
+2806957	Under what conditions does a line intersect a circle in a 2D plane?	Unambiguous
+311580	Coordinate-free proof of $\operatorname{Tr}(AB)=\operatorname{Tr}(BA)$?	Ambiguous
+1726540	How to differentiate $x^{\frac1x}$?	Ambiguous
+1432379	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
+2999909	Does Hom(X, Y) always form an abelian group?	Ambiguous
+1891318	How many binary strings of length 8 do not contain the substring '100'?	Unambiguous
+11194	Solving the recurrence relation that contains summation of nth term	Ambiguous
+942204	How to derive the general term of a sequence formed by the repeated convolution of a probability distribution with itself?	Unambiguous
+3254331	Prove that the definite integral $\int_0^\infty(\arctan(1/x))^2 \mathrm{d}x$ converges to $\pi\ln 2$.	Unambiguous
+2326677	What is the signifigance of a saddle point?	Ambiguous
+2732822	What do you call a system where $a+b=1$ and $ab=-1$?	Ambiguous
+451799	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
+1258718	"The ""Smooth Factor"" in a number sequence"	Ambiguous
+846706	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
+3103052	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
+1491490	What is the group generated by $M_\pi$?	Ambiguous
+720688	What is the average power of this signal?	Ambiguous
+2932046	Is this formula correct: $\frac{|(p-a) \times d|}{|d|}$?	Ambiguous
+2675393	How to find the derivative of a power series?	Unambiguous
+1075740	What happens to the angles of an isosceles triangle if one vertex is at infinity?	Ambiguous
+808565	What is the standard mathematical notation for the dimension of a vector space $V$?	Unambiguous
+1194644	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
+973713	Problem in the second-derivative symbol.	Ambiguous
+3328149	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
+3481798	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
+3472866	Are the sample mean and sample variance considered random variables?	Unambiguous
+345024	What are the applications of Catalan numbers in computer programming and algorithm design?	Unambiguous
+553280	What is the number of banded $(0,1)$-matrices with exactly $k$ ones and prescribed row/column sums?	Ambiguous
+1531029	If an abelian group $G$ has property $N$, then $G$ is finitely generated	Ambiguous
+2901457	Determine for which values of $k$ and $a$ the series $\sum_{n=1}^\infty \frac{\ln^k(n)}{n^a}$ converges.	Unambiguous
+3255482	Evaluate $\lim_{x \to 0} \frac{1}{x} \int_{x}^{2x} e^{-t^{2}}\,dt$	Unambiguous
+3002761	How do I find the maximum value of a variable $x$ when it is subject to a given equality constraint?	Unambiguous
+298648	Is there a common symbol for concatenating two (finite) sequences?	Ambiguous
+1720548	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
+4768	Prove that every permutation of a finite set is either even or odd, but not both.	Unambiguous
+314408	What are the conditional probabilities for my simple OR-gate where $P(A)=0$ and $P(B)=0$?	Ambiguous
+4221432	What are the general integer solutions to the Diophantine equation $a^2+b^2=c^2-1$ for positive integers $a, b, c$?	Unambiguous
+744139	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
+2368762	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
+1673403	A function is never zero, takes both signs, and is continuous. How is that possible?	Ambiguous
+378970	For a right triangle with a hypotenuse of 12 cm, what measures of the acute angles would produce the maximum area?	Unambiguous
+736346	What are the geometric methods to fold a rectangular piece of A4 paper into three equal parallel strips?	Unambiguous
+3673715	I need help with the heat equation in a 3D ball.	Ambiguous
+1494248	Evaluate the indefinite integral $\int \frac{\sin(4x)}{\sin(x)} \, dx$.	Unambiguous
+185267	How do I prove that a subset of a set with Lebesgue outer measure zero also has outer measure zero?	Unambiguous
+2492396	An equation is first-order nonlinear. How do I transform it into standard linear form?	Ambiguous
+289779	Determine whether the sequence $a_n = 3^n - 2^{n+100}$ for $n \in \mathbb{N}$ converges or diverges.	Unambiguous
+330556	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
+3487811	$\lim_{x \to 0} {1-x\cot x\over x^2}$	Ambiguous
+366529	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
+387437	How can you prove that a number is an algebraic integer?	Ambiguous
+2030790	3 variable diophantine equation word problem: coins	Ambiguous
+464426	Evaluate the limit $\lim_{x \to 1} \frac{x^{1/5}-1}{x^{1/3}-1}$ for real-valued $x$.	Unambiguous
+3601099	For a symmetric matrix and a unit vector, why does a certain quartic inequality involving inner products hold?	Ambiguous
+1455761	What is the relationship between the determinant of a square matrix and its inverse?	Unambiguous
+2887105	Express $\frac{1}{1-i-z}$ as a power series in $z$ valid for $|-z-i|<1$.	Unambiguous
+2614579	How to solve $\lim_\limits{x\to1} \frac{\sqrt{2x-1} -1}{x^2-1}$?	Ambiguous
+531128	What are the subspaces of a finite-dimensional vector space $V$ that have dimension $\dim V - 1$, and what are they called?	Unambiguous
+735461	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
+1861189	$PC$ bisects $\angle APB$. Prove $XY \parallel AB$.	Ambiguous
+2369950	Is the statement $|x|=\pm x$ for $x>0$ a correct definition of absolute value, and why is it confusing?	Unambiguous
+736044	A number divided by $q$ leaves remainder $r$. Find $r+q$.	Ambiguous
+2339487	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
+4044721	Find the limit of $\lim_{x\to 0}\left(\frac{\cos x}{\cos 2x}\right)^\frac{1}{x^2}$	Ambiguous
+2666204	How do I optimize $(\pi R_1 R_2)h$ when I don't know what the surface area should be?	Ambiguous
+1071811	Why is the Zariski topology on an algebraic variety over an algebraically closed field not Hausdorff?	Unambiguous
+2260	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
+128823	limit of integral $n\int_{0}^{1} x^n f(x) \text{d}x$ as $n\rightarrow \infty$	Ambiguous
+2006377	Average integral increasing	Ambiguous
+20813	What is the intuitionistic construction of the unit circle in constructive mathematics?	Unambiguous
+3367250	For this equation, does it matter which variable we treat as independent?	Ambiguous
+4236641	What is the best way to find the closest vector in a set to a given one?	Ambiguous
+721784	Absolute value of complex exponential	Ambiguous
+2844637	I need to calculate $\cos(\pi/20)$ with an error less than $10^{-5}$ using a series. Is my answer correct?	Ambiguous
+835454	$f: SL_2(\mathbb{R}) \to GL_4(\mathbb{R})$ show that $Im(f)=SL_4(\mathbb{R})$	Ambiguous
+568493	How do you find the number of common codes for [1,5], [4,8] and [3,7], [2,6]?	Ambiguous
+1768578	How to find the relation between $a$, $c$ and $b$ to make my density function integral $1$?	Ambiguous
+3815586	If a function is differentiable almost everywhere, what is its regularity?	Ambiguous
+2931443	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
+1283678	Prove $||\lambda x_1 + (1-\lambda) x_2 - y|| \leq ||x_1 - y||$	Ambiguous
+1942246	I need to prove trigonometric equation	Ambiguous
+2620801	Proof Group Homomorphism: $\varphi(e)= e'$ & $\varphi(\hat{a})= \widehat{\varphi(a)}$	Ambiguous
+960632	Evaluating product of exponent and polynomial	Ambiguous
+697675	multiplying a matrix by a row vector	Ambiguous
+2360710	What is the numerical value or a series expansion formula for the natural logarithm of $\pi$, $\ln(\pi)$?	Unambiguous
+587560	What is the purpose of the base case in a proof by mathematical induction?	Unambiguous
+2038540	Permutations with men and women alternating	Ambiguous
+1663674	Does my genome exist in the digits of pi?	Ambiguous
+2315737	Is checking a norm on a function space just the triangle inequality and the other two rules?	Ambiguous
+3969765	Evaluate the integral $\int_0^\infty \frac{e^{-tx^2}-e^{-x^2}}{x}dx$ for $t > 0$.	Unambiguous
+2205890	In what mathematical context or under what non-standard ordering could the inequality $\frac{1}{2} < \frac{1}{4}$ be true?	Unambiguous
+1072741	Is an open $n$-ball homeomorphic to $\mathbb{R}^{n}$?	Ambiguous
+62473	What is the definition of a quasifinite morphism of schemes?	Unambiguous
+824000	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
+2165873	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
+264371	Recommend challenging but engaging mathematics books suitable for advanced undergraduate students that cover topics beyond the standard curriculum.	Unambiguous
+3111260	Error evaluating $ \lim_{x\to 0}\frac{x-\tan x}{x^3} $	Ambiguous
+1176875	Find all integer solutions $(a, b, c)$ to the equation $a + b + c = abc$.	Unambiguous
+4094225	Given point $K$ in a triangle, is it possible to construct a line through $K$ that bisects the area of the triangle?	Ambiguous
+13582	Show that a continuous function has a fixed point	Ambiguous
+3735022	Does knowing the Gauss curvature is $1/R^2$ help prove the curve's curvature is $\geq 1/R$?	Ambiguous
+587560	What is the purpose of the first test in an inductive proof?	Ambiguous
+1108298	Why is the canonical map $M \otimes_A N \to M \otimes_{A/I} N$ an isomorphism?	Ambiguous
+3979952	Are $\sum_{i=1}^{n} \xi_i / i$ and $\max\xi_n$ equally distributed?	Ambiguous
+665341	Using only a compass fixed at a radius of $5$ inches and a straightedge, construct a circle of radius $4$ inches.	Unambiguous
+4067963	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
+1048	Compare the different definitions of trigonometric functions using right triangles, the unit circle, and power series.	Unambiguous
+675332	On the set $S = \mathbb{R}\setminus\{-1\}$ with operation $a * b = a+b+ab$, solve $2*x*3 = 7$ for $x$.	Unambiguous
+2605427	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
+3729113	What is the definition and region of convergence for a Dirichlet series of a complex variable?	Unambiguous
+290435	Find five positive integers whose reciprocals sum to $1$	Ambiguous
+261680	Find integer values of $a,b$ such that $k=a^3-b^2$ is a perfect square.	Ambiguous
+2042807	Construct a homeomorphism between the unit circle $S^1$ and the boundary of a square in $R^2$.	Unambiguous
+288432	Let a power series have radius of convergence 1 and diverge at $x=1$. Does it diverge as $x\to 1$?	Ambiguous
+4812185	How to prove what the Wikipedia page says about the entry wise norm limit being the maximum norm?	Ambiguous
+1490059	What is the definition of the curl of a 3D vector field in vector calculus?	Unambiguous
+3468876	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
+331345	How do I find the solutions for the equation $2x^2 - 5x + 3 = 0$?	Unambiguous
+295318	Are all algebraic vector bundles on the affine line over a principal ideal domain (PID) trivial?	Unambiguous
+219774	Finding the periods of functions...	Ambiguous
+172468	Why does MatLab generate polynomials for cyclic codes for all $n,k$ I try dividing $x^n-1$?	Ambiguous
+4117494	Volume of tetrahedron Solid	Ambiguous
+2007373	What are the fundamental geometric intuitions used to understand concepts in undergraduate linear algebra and multivariable calculus?	Unambiguous
+2897989	Prove that the dilation operator on $L^2(\mathbb{R})$ is a bounded linear operator.	Unambiguous
+4567071	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
+730358	How do I graph a general sine function of the form y = a sin(bx - c) + d?	Unambiguous
+2394582	Show that $f(x) = x^3 - x$ is an odd function.	Unambiguous
+3839814	Probability that $\text{det}(A)$ is an even number.	Ambiguous
+2160552	Do elementary row operations change the eigenvalues of a square matrix?	Unambiguous
+4016729	Find all real values of theta such that $(\sin(\theta) - \cos(\theta))(2+\sin(\theta) \cos(\theta)) \leq 2$.	Unambiguous
+315869	Sequence of sets and neighborhoods	Ambiguous
+742878	Modeling with Exponential and Logarithmic Functions	Ambiguous
+1496959	How to show $f_n$ converges uniformly to $f'$ if the derivative is bounded by $3|x-y|$?	Ambiguous
+1745083	How do you find the shortest path between two fixed points in a 2D plane that avoids a given linear obstacle?	Unambiguous
+124287	Is the map $f(a+bi)=a$ a homomorphism of rings?	Ambiguous
+4221432	Near-Pythagorean triplets: What are the general solutions to $a^2+b^2=c^2-1$?	Ambiguous
+2540002	Let $$S=\sum_{i=0}^N \frac{1}{\sqrt{2\pi\sigma^2}}e^{-(i+0.5)^2/(2\sigma^2)}$$. Any ideas?	Ambiguous
+765044	Challenge: Demonstrate a Contradiction in Leibniz' differential notation	Ambiguous
+1896195	Under what conditions is the endomorphism ring of a module free as a module over the base ring?	Unambiguous
+1882798	Given constants $X, Y, P$, determine whether there exist positive integers $a, b$ such that $(aX+bY) \bmod P = 0$.	Unambiguous
+2540359	Does the fact that both examples and counterexamples exist mean a property is undecidable from the axioms?	Ambiguous
+3516041	Third order Taylor polynomial of $f:(0, \infty) \times \mathbb{R} \to \mathbb{R}$ given by $f(x,y) = x^y$	Ambiguous
+2128278	How do I compute the indefinite integral of the hyperbolic sine function $\sinh(x)$?	Unambiguous
+4087621	In which of the intervals is $\sqrt{12}$	Ambiguous
+458261	How do I rewrite a triple integral with different limits for $x+y-2z=2$?	Ambiguous
+4044721	Find the limit of $\lim_{x\to 0}\left(\frac{\cos x}{\cos 2x}\right)^\frac{1}{x^2}$	Unambiguous
+1310890	What is $\lim_{x\to\infty}\left(\sin{\frac 1x}+\cos{\frac 1x}\right)^x$?	Ambiguous
+195504	Prove that for a linear functional, the kernel has codimension one.	Ambiguous
+1438319	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
+2597743	How do I compute the indefinite integral of a function of the form $f(x) = x^n e^{-ax}$?	Unambiguous
+2093882	Problem in solving a question related to finite difference.	Ambiguous
+2555463	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
+2339487	Does putting $n-1$ copies of a codeword after itself prove there is a linear code?	Ambiguous
+12139	How many binary relations on a set of size n are both symmetric and reflexive?	Unambiguous
+3084662	How to convert a complex number $z = a + bi$ to polar form (calculate the magnitude r and the argument theta).	Unambiguous
+1007952	For a random variable $X$ with pdf $f(x)=\frac12 e^{-|x|}$, what can we say about it?	Ambiguous
+608477	What is the general method for a proof by mathematical induction, and can you provide a step-by-step example?	Unambiguous
+4569632	In $\triangle ABC$, $BD$ is a median, $\angle DAB=15$ and $\angle ABD=30$. Find $\angle ACB$.	Ambiguous
+1002480	Is the Minkowski sum of a finite collection of compact line segments in Euclidean space compact?	Unambiguous
+2782880	Proof check: AB not invertible	Ambiguous
+25648	Prove that if two integers $a$ and $b$ are congruent modulo $n$, then $\gcd(a, n) = \gcd(b, n)$.	Unambiguous
+2548701	Find the integer part of the value of the infinite nested radical $x = \sqrt[3]{24+\sqrt[3]{24+\sqrt[3]{24+\cdots}}}$	Unambiguous
+3017595	Dimension of subspace 10x10	Ambiguous
+2932095	Find $a$ so that a line is tangent, secant or external from a sphere	Ambiguous
+176625	Prove that there do not exist positive integers $x$ and $y$ with $x^2 - y^2 = n$	Ambiguous
+2035929	How to prove bounds of area integral?	Ambiguous
+1024146	Is a group divisible if and only if it has no maximal subgroup?	Ambiguous
+2140493	What are some classic counterintuitive examples or paradoxes in probability theory, such as the Monty Hall problem or the Birthday paradox?	Unambiguous
+2657826	How to find values of $a$ and $b$ if the rest is $15$ when $P(x)$ is divided by $(x-2)$?	Ambiguous
+2604608	How do I prove $f(x)$ is continuous when it only has constants $a$ and $b$?	Ambiguous
+795652	How to find nth moment?	Ambiguous
+1391080	Can I express the theorem about $\sqrt{2}$ as $p\Rightarrow q$?	Ambiguous
+83035	How to prove $(1+1/x)^x$ is increasing when $x>0$?	Ambiguous
+36545	What are some examples of mathematically simple puzzles or conjectures that remained unsolved or were counterintuitive to professional mathematicians for a long time?	Unambiguous
+2044396	Can this be proven for all groups?	Ambiguous
+184593	Is it necessary to have $\theta$ in radians to obtain $\frac{\sin{\theta}}{\theta} \to 1$ as $\theta\to 0$?	Ambiguous
+3083195	How to solve a linear first-order ODE for various initial conditions?	Unambiguous
+3219211	Can a first-order homogeneous ordinary differential equation have a singular solution that is not a member of the general solution family?	Unambiguous
+225922	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
+3172708	I have the inverse of a sparse matrix, and only 4 entries change. I need a quick way to get the new inverse.	Ambiguous
+2457906	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
+289461	Is the set of midpoints of the open intervals that make up the complement of the Cantor ternary set countable?	Unambiguous
+3123246	What is the closed-form value of the definite integral $\int_0^\pi \ln(1+\sin^2 t) dt$?	Unambiguous
+19236	Algorithm to compute Gamma function	Ambiguous
+3415231	How do I prove the universal generalization theorem in my discrete math image?	Ambiguous
+262629	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
+1386774	What is the definition and convergence criteria for a power series representation of a function?	Unambiguous
+1890614	Mathematical definition of infinity	Ambiguous
+3762261	What is the geometric intuition behind the spectral theorem for Hermitian operators on a finite-dimensional complex Hilbert space?	Unambiguous
+314809	Super or subscript notation on the left hand side of a symbol?	Ambiguous
+4814	Is there a name for graphs with 2 connections per node?	Ambiguous
+1824176	Why is the graph of the function $f(x) = \sin(\arccos(x))$ a semicircle with radius $1$ centered at the origin?	Unambiguous
+262371	How do you solve the area of a trapezoid using diagonals	Ambiguous
+3524864	Solve the first-order linear partial differential equation $-x^2u_x+u_y=1$ with the initial condition $u(e^y,y)=y$.	Unambiguous
+1291623	Direct Proof on Divisibilty	Ambiguous
+2445959	Find all natural numbers $n$ such that $n - 2$ divides $n^2 - 2$.	Unambiguous
+1176875	Find all integral solutions to $a+b+c=abc$.	Ambiguous
+67407	Prove that every group of order 15 is abelian.	Unambiguous
+457969	What is the definition and property of the minimum of a function on a closed interval?	Unambiguous
+125846	L'Hopital's rule, why is this incorrect?	Ambiguous
+2344095	How do I find the coefficient of $x^2$ in the binomial expansion of $(1 + 2x + 3x^2)^10$?	Unambiguous
+71874	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
+2338603	What is the formula and definition for the binomial coefficient ${n\choose k}$?	Unambiguous
+1327420	Solve the system of equations $x + xy + y = 11$ and $yx^2 + xy^2 = 30$ for real numbers $x$ and $y$.	Unambiguous
+1831188	How to compute the 2006th power of a given square matrix $M$?	Unambiguous
+1067717	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
+3976203	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
+456595	Prove that the infinite series $\sum_{n=1}^\infty \frac{1}{n^2}$ converges and that its sum is less than 2.	Unambiguous
+3005021	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
+1951021	Why in calculus the angles are measured in radians?	Ambiguous
+1831188	Calculating the matrix $M^{2006}$	Ambiguous
+3832187	Distributing 3 objects in 5 boxes	Ambiguous
+2903468	Prove that the definite integral $\int_0^1\frac{\ln(x)}{\sqrt{1-x^2}}dx = -\frac{\pi}{2}\ln(2)$.	Unambiguous
+110579	In how many ways can a batsman score a century in all $6$'s and $4$'s?	Ambiguous
+2246596	What is the max function exactly and how do I examine its contour maps?	Ambiguous
+510991	If every continuous function on $X$ is uniformly continuous, is $X$ compact?	Ambiguous
+614812	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
+1290291	Identification of a quadrilateral as a trapezoid, rectangle, or square	Ambiguous
+3483693	Find the smallest integer $n > 1$ such that $n^2 \equiv n \pmod{1000}$.	Unambiguous
+1153474	Is the function $f(n) = 2^{\log_3(n) + 1}$ in $\Theta(n)$ for asymptotic running time analysis?	Unambiguous
+2206714	Evaluate the indefinite integral $\int \sin(2x) \frac{e^{\cos^2 x}}{e^{\sin^2 x}} dx$.	Unambiguous
+2470166	What are the conditions for the eigenvectors of this matrix to be to be real and orthogonal?	Ambiguous
+2307642	Proof of the spectral theorem for normal operators on a finite-dimensional complex inner product space	Unambiguous
+1054529	the purpose of induction	Ambiguous
+2669751	How do you count all unique groups that can be made with items	Ambiguous
+2049971	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
+1731227	Find he local maximum and minimum value and saddle points of the function?	Ambiguous
+4343109	For a graph $(G, V)$, $\# \text{edges}$ is given by the rank of $H_1(G, V)$.	Ambiguous
+1271168	How do I square a logarithm?	Ambiguous
+292425	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
+2897989	Let $D$ be defined by $D(a)=f_a$ for $f\in L^2(\mathbb{R})$. Show that $D$ is continuous.	Ambiguous
+241316	Does the infinite series $\sum_{k=1}^{\infty} \frac{\sqrt{k}}{(k+1)^2}$ converge?	Unambiguous
+1961686	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
+1438319	Prove that $\lim_\limits{x\to 0}{\frac{e^x-1}{x}}=1$ without derivatives	Ambiguous
+33296	How do you prove that a differentiable function is monotonic given information about the sign of its derivative?	Unambiguous
+4317139	Bianchi classification of solvable Lie groups and cocompact subgroups	Ambiguous
+820492	Are there algorithms to simulate the continuous deformation of a cup into a donut?	Ambiguous
+521814	I have this picture and I know a height. How do I find its other proportions using perspective projection?	Ambiguous
+709004	Evaluate $\lim_{x\to\infty}\left(\frac{x^2+2x-1}{2x^3-3x-2}\right)^{1/x}$.	Unambiguous
+866327	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
+2296759	What is the relationship between a symmetric matrix and the inner product it defines on a vector space?	Unambiguous
+3538836	How is this inequality proved for the squeeze theorem if it doesn't hold for all $x$?	Ambiguous
+1900241	What is the expected value of $X$, given its generating function?	Ambiguous
+4595703	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
+2537775	Evaluate $\prod_{r=1}^{10}(1+\tan r^\circ)\cdot\prod_{r=46}^{55}(1+\cot r^\circ)$.	Unambiguous
+2249320	Find a closed-form expression for $g(x) = \cos x + \cos 3x + \cos 5x + \cdots + \cos((2n-1)x)$.	Unambiguous
+972189	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
+298648	What is the standard notation for the concatenation of two finite sequences in mathematics and computer science?	Unambiguous
+2820510	How are quaternions used to represent 3D rotations in geometry?	Unambiguous
+1812294	How to find $P(B\cup C)$ when $A$ and $B$ are independent and the union of $A,B,C$ is $1$?	Ambiguous
+1111222	Is a function that's unbounded near a boundary point still integrable if we redefine its value there?	Ambiguous
+986013	What should I do with the fact that my Laurent series converges in $z=1+i$?	Ambiguous
+108308	Does Tom Apostol's proof of the Fundamental Theorem of Calculus involve a division by zero?	Unambiguous
+2697247	What are the standard integral representations of the Bessel function of the first kind $J_n(x)$?	Unambiguous
+2296602	Simplify the propositional logic expression $\neg ((P \to (Q \to R)) \to (Q \to (P \to R)))$ to its simplest form.	Unambiguous
+3944628	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
+548732	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
+1045011	How to find the limit $\lim_{n\to \infty}\frac{n}{2^n-1}$?	Ambiguous
+1258955	Convergence of the integral $\int_0^{\pi/2}\ln(\cos(x))dx$	Ambiguous
+2427685	Calculate the limit $\lim_{n \to \infty} \sum_{k=1}^{n} \frac{(-1)^{k-1}}{k^2 \binom{n}{k}}$.	Unambiguous
+124986	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
+3289735	Is the arc length always irrational between two rational points?	Ambiguous
+3564289	What is the expected number of times a simple symmetric random walk on a 1D integer lattice returns to the origin?	Unambiguous
+2381284	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
+1592652	Example of a relation that is symmetric and transitive, but not reflexive	Ambiguous
+984271	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
+3762872	Zeno's Achilles & Tortoise - Where exactly is the proof wrong?	Ambiguous
+1489146	How is the imaginary unit $i$ formally defined in the context of complex numbers, and why is $i$ not equal to $-i$?	Unambiguous
+1987599	It's known that a certain function behaves like $1/u$ near $u=0$. Are there sharper inequalities describing this behavior?	Ambiguous
+859968	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
+1204425	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
+1059293	Can someone explain how the proof for f(x) works?	Ambiguous
+956087	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
+3023583	Does the infinite series $\sum_{n=1}^\infty \frac{n^2(-1)^n}{1+n^2}$ converge or diverge?	Unambiguous
+759112	Given that $Y - X^2$ is an irreducible polynomial, prove that the variety $V(Y-X^2)$ is irreducible.	Unambiguous
+959137	Finding missing coordinates in a non-linear graph?	Ambiguous
+2715973	$$ \frac{n}{n+2}\sum_{k=1}^n\frac{k}{k+3}$$	Ambiguous
+801923	Prove that for all integers $x$ and $y$, if $x + y + xy = 0$, then $(x, y) = (0, 0)$ or $(x, y) = (-2, -2)$.	Unambiguous
+1503414	Why is the natural logarithm of 1 equal to 0?	Unambiguous
+3035983	How can I find integers which satisfy $\frac{150+n}{15+n}=m$?	Ambiguous
+476186	Is there a category whose internal logic is paraconsistent?	Ambiguous
+108191	Why won't sine functions work as a counter example for my infinite integral proof?	Ambiguous
+401477	What is the definition of a $\mathbb{Z}/N\mathbb{Z}$-algebra?	Unambiguous
+1135959	What is the standard way to represent a complex number as a $2 \times 2$ real matrix?	Unambiguous
+802940	How to calculate the numerical value of the sum from $k=1$ to $50$ of $k^{1/3}$?	Unambiguous
+1035200	Let $p:\mathbb{N}\to\mathbb{N}$. What is $\lim p(n)$?	Ambiguous
+685681	Prove that a finite-dimensional vector space has a finite basis.	Unambiguous
+1893124	If 16 players are divided into eight pairs, what is the probability that only one of S1 or S2 advances?	Ambiguous
+158469	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
+1564864	What does the symbol # mean?	Ambiguous
+2564222	Find the equations of the tangent lines drawn from the point $(11, 3)$ to the circle defined by $x^2 + y^2 = 65$.	Unambiguous
+1295010	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
+3743564	Examples of classic coin weighing puzzles and their optimal solutions using a balance scale	Unambiguous
+783173	Evaluate the definite integral $\int_0^1 \frac{dx}{\sqrt{x(1-x)}}$	Unambiguous
+4814	What are the names of directed graphs where every vertex has an in-degree of 2 and an out-degree of 2?	Unambiguous
+2448533	How to evaluate a limit with arctan(x)	Ambiguous
+2365371	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
+1232385	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
+1528935	How to prove a set is linearly independent without using a basis or FTA?	Ambiguous
+2274060	If $x$ and $y$ are positive numbers, what is the minimum possible value of $(x+y)\left(\frac1x + \frac1y\right)?$	Ambiguous
+1788328	Calculate the numerical value of the product $\sin(10^\circ) \sin(50^\circ) \sin(70^\circ)$.	Unambiguous
+311665	Proof a graph is bipartite if and only if it contains no odd cycles	Ambiguous
+176625	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
+1883279	What are examples of continuous functions that cannot be uniformly approximated by polynomials on a given interval?	Unambiguous
+405972	A basis for a subspace of $\mathbb R^3$	Ambiguous
+4734106	Is there an equation similar to square root, but faster for a computer to compute?	Ambiguous
+3226462	On an infinitely sided die, is it possible to roll and get the same number twice?	Ambiguous
+529215	Is it true that if a sequence {a_n} converges, then the set of its values has exactly one limit point?	Ambiguous
+2297557	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
+1796173	I tried to expand the bracket $[X,Y]=XY-YX=Y$, how do I proceed to find $[Y,Z]$?	Ambiguous
+3154407	Applying a function on any multiset of real numbers	Ambiguous
+2591017	Unity roots that sum to 1	Ambiguous
+1382185	How many divisors of the combination of numbers?	Ambiguous
+151810	What is the probability of getting exactly 3 heads in 10 independent flips of a fair coin?	Unambiguous
+807589	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
+927248	Maximum value of $a+b$ given that $\frac{1}{a} + \frac{1}{b} = \frac{1}{20}$	Ambiguous
+1286651	How do I compute the limit of a function of two variables as it approaches a point $(x, y)$?	Unambiguous
+2666326	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
+2322481	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
+4093371	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
+1565903	What are some simple-to-prove, counter-intuitive theorems in set theory that depend on the Axiom of Choice?	Unambiguous
+2893411	Evaluate the sum $\sum\limits_{k=1}^n {\binom{n}{k} f(k)}$.	Ambiguous
+2555463	Find the point equidistant from two points and a line	Ambiguous
+728599	How to evaluate the triple integral for region $E$ bounded by the spheres?	Ambiguous
+5877	Efficiently finding two squares which sum to a prime	Ambiguous
+3111260	How do I evaluate $\lim_{x\to 0}\frac{x-\tan x}{x^3} $?	Unambiguous
+2307642	How to find an orthonormal $T$ eigenbasis without matrices?	Ambiguous
+1774294	Explain why $y^2 = 2x^2 + C$ is not equivalent to $y = \sqrt{2x^2} + C$.	Unambiguous
+3322357	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
+449507	Compute the indefinite integral of $\frac{\sqrt{x^2-4}}{x^4}$ with respect to $x$.	Unambiguous
+1037426	Find $\lim\limits_{x \to \infty}{e^{-x}\sqrt{x}}$	Ambiguous
+1822028	In a normed vector space, is the closed convex hull of a weak*-compact subset of the dual space also weak*-compact?	Unambiguous
+2959054	Can I use stars and bars for this?	Ambiguous
+398593	Calculate the length of curve $f(x)=\arcsin(e^x)$, check solution, please.	Ambiguous
+199447	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
+43118	How can I explain to a layperson why the 'staircase' proof that $\pi = 4$ is mathematically incorrect?	Unambiguous
+1562459	Prove that $( 1 + n^{-2}) ^n \to 1$.	Ambiguous
+2116037	Does $\sum_{n=1}^{\infty} \frac{\sin^2(nx)}{n}$ converge uniformly?	Ambiguous
+3258160	Calculate $\iint_\Omega \left(\sqrt{\frac xa}+\sqrt{\frac yb}\right)\,dxdy$	Ambiguous
+4564068	Is there a closed-form formula for the sum of the first $n$ squares, $\sum_{k=1}^{n} k^2$?	Unambiguous
+1086318	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
+68897	Let $A$ be positive definite and $B$ be full rank. Show that $BAB'$ is positive definite.	Ambiguous
+611256	How do you convert a decimal number to 8-bit two's complement binary representation?	Unambiguous
+1410163	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
+1982170	Probability of an event in relationship with a certain geometrical representation	Ambiguous
+763272	Which is greater: the value of $\sqrt{1001} - \sqrt{1000}$ or the fraction $\frac{1}{10}$?	Unambiguous
+1372825	Evaluating numerically $\int_0^{\infty}e^{-t^2 /100} \sin \pi t $	Ambiguous
+1117045	What are the standard methods and techniques for evaluating definite integrals given specific upper and lower limits of integration?	Unambiguous
+1762215	Diamond and Shurman page 73 says $j'(\tau)$ doesn't vanish, how do I prove this?	Ambiguous
+405972	How do I find a basis for a subspace of $\mathbb{R}^3$ defined by a system of linear equations?	Unambiguous
+2490690	In how many ways can you make 107 using 1, 10, and 50?	Ambiguous
+2247820	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
+1602977	What is an example of an infinitesimal in the context of non-standard analysis?	Unambiguous
+611256	What are the least and greatest decimal integers that can be stored in an 8-bit register?	Ambiguous
+1502938	How to prove the sum equals $n/(n+1)$ using induction?	Ambiguous
+811451	Evaluate $\sqrt{2+\sqrt[3]{3+\sqrt[4]{4+\cdots}}}$.	Unambiguous
+4566790	Can someone help me understand eigenvalues?	Ambiguous
+1008311	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
+262371	How do you calculate the area of a trapezoid given the lengths of its diagonals and the angle between them?	Unambiguous
+3248549	Let $f$ be a continuous map then which of the following is correct	Ambiguous
+119307	Give an example of a Cauchy sequence that does not converge in the space of rational numbers $Q$.	Unambiguous