MathDisAM_test.tsv

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