MathSE ID int64 795 4.87M | Math Question stringlengths 23 305 | Tag stringclasses 2
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|---|---|---|
2,548,701 | Find the integer part of the value of the infinite nested radical $x = \sqrt[3]{24+\sqrt[3]{24+\sqrt[3]{24+\cdots}}}$ | Unambiguous |
3,017,595 | Dimension of subspace 10x10 | Ambiguous |
2,932,095 | Find $a$ so that a line is tangent, secant or external from a sphere | Ambiguous |
176,625 | Prove that there do not exist positive integers $x$ and $y$ with $x^2 - y^2 = n$ | Ambiguous |
2,035,929 | How to prove bounds of area integral? | Ambiguous |
1,024,146 | Is a group divisible if and only if it has no maximal subgroup? | Ambiguous |
2,140,493 | What are some classic counterintuitive examples or paradoxes in probability theory, such as the Monty Hall problem or the Birthday paradox? | Unambiguous |
2,657,826 | How to find values of $a$ and $b$ if the rest is $15$ when $P(x)$ is divided by $(x-2)$? | Ambiguous |
2,604,608 | How do I prove $f(x)$ is continuous when it only has constants $a$ and $b$? | Ambiguous |
795,652 | How to find nth moment? | Ambiguous |
1,391,080 | Can I express the theorem about $\sqrt{2}$ as $p\Rightarrow q$? | Ambiguous |
83,035 | How to prove $(1+1/x)^x$ is increasing when $x>0$? | Ambiguous |
36,545 | What are some examples of mathematically simple puzzles or conjectures that remained unsolved or were counterintuitive to professional mathematicians for a long time? | Unambiguous |
2,044,396 | Can this be proven for all groups? | Ambiguous |
184,593 | Is it necessary to have $\theta$ in radians to obtain $\frac{\sin{\theta}}{\theta} \to 1$ as $\theta\to 0$? | Ambiguous |
3,083,195 | How to solve a linear first-order ODE for various initial conditions? | Unambiguous |
3,219,211 | Can a first-order homogeneous ordinary differential equation have a singular solution that is not a member of the general solution family? | Unambiguous |
225,922 | 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 |
3,172,708 | I have the inverse of a sparse matrix, and only 4 entries change. I need a quick way to get the new inverse. | Ambiguous |
2,457,906 | 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 |
289,461 | Is the set of midpoints of the open intervals that make up the complement of the Cantor ternary set countable? | Unambiguous |
3,123,246 | What is the closed-form value of the definite integral $\int_0^\pi \ln(1+\sin^2 t) dt$? | Unambiguous |
19,236 | Algorithm to compute Gamma function | Ambiguous |
3,415,231 | How do I prove the universal generalization theorem in my discrete math image? | Ambiguous |
262,629 | 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 |
1,386,774 | What is the definition and convergence criteria for a power series representation of a function? | Unambiguous |
1,890,614 | Mathematical definition of infinity | Ambiguous |
3,762,261 | What is the geometric intuition behind the spectral theorem for Hermitian operators on a finite-dimensional complex Hilbert space? | Unambiguous |
314,809 | Super or subscript notation on the left hand side of a symbol? | Ambiguous |
4,814 | Is there a name for graphs with 2 connections per node? | Ambiguous |
1,824,176 | Why is the graph of the function $f(x) = \sin(\arccos(x))$ a semicircle with radius $1$ centered at the origin? | Unambiguous |
262,371 | How do you solve the area of a trapezoid using diagonals | Ambiguous |
3,524,864 | Solve the first-order linear partial differential equation $-x^2u_x+u_y=1$ with the initial condition $u(e^y,y)=y$. | Unambiguous |
1,291,623 | Direct Proof on Divisibilty | Ambiguous |
2,445,959 | Find all natural numbers $n$ such that $n - 2$ divides $n^2 - 2$. | Unambiguous |
1,176,875 | Find all integral solutions to $a+b+c=abc$. | Ambiguous |
67,407 | Prove that every group of order 15 is abelian. | Unambiguous |
457,969 | What is the definition and property of the minimum of a function on a closed interval? | Unambiguous |
125,846 | L'Hopital's rule, why is this incorrect? | Ambiguous |
2,344,095 | How do I find the coefficient of $x^2$ in the binomial expansion of $(1 + 2x + 3x^2)^10$? | Unambiguous |
71,874 | 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 |
2,338,603 | What is the formula and definition for the binomial coefficient ${n\choose k}$? | Unambiguous |
1,327,420 | Solve the system of equations $x + xy + y = 11$ and $yx^2 + xy^2 = 30$ for real numbers $x$ and $y$. | Unambiguous |
1,831,188 | How to compute the 2006th power of a given square matrix $M$? | Unambiguous |
1,067,717 | 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 |
3,976,203 | 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 |
456,595 | Prove that the infinite series $\sum_{n=1}^\infty \frac{1}{n^2}$ converges and that its sum is less than 2. | Unambiguous |
3,005,021 | 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 |
1,951,021 | Why in calculus the angles are measured in radians? | Ambiguous |
1,831,188 | Calculating the matrix $M^{2006}$ | Ambiguous |
3,832,187 | Distributing 3 objects in 5 boxes | Ambiguous |
2,903,468 | Prove that the definite integral $\int_0^1\frac{\ln(x)}{\sqrt{1-x^2}}dx = -\frac{\pi}{2}\ln(2)$. | Unambiguous |
110,579 | In how many ways can a batsman score a century in all $6$'s and $4$'s? | Ambiguous |
2,246,596 | What is the max function exactly and how do I examine its contour maps? | Ambiguous |
510,991 | If every continuous function on $X$ is uniformly continuous, is $X$ compact? | Ambiguous |
614,812 | 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 |
1,290,291 | Identification of a quadrilateral as a trapezoid, rectangle, or square | Ambiguous |
3,483,693 | Find the smallest integer $n > 1$ such that $n^2 \equiv n \pmod{1000}$. | Unambiguous |
1,153,474 | Is the function $f(n) = 2^{\log_3(n) + 1}$ in $\Theta(n)$ for asymptotic running time analysis? | Unambiguous |
2,206,714 | Evaluate the indefinite integral $\int \sin(2x) \frac{e^{\cos^2 x}}{e^{\sin^2 x}} dx$. | Unambiguous |
2,470,166 | What are the conditions for the eigenvectors of this matrix to be to be real and orthogonal? | Ambiguous |
2,307,642 | Proof of the spectral theorem for normal operators on a finite-dimensional complex inner product space | Unambiguous |
1,054,529 | the purpose of induction | Ambiguous |
2,669,751 | How do you count all unique groups that can be made with items | Ambiguous |
2,049,971 | 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 |
1,731,227 | Find he local maximum and minimum value and saddle points of the function? | Ambiguous |
4,343,109 | For a graph $(G, V)$, $\# \text{edges}$ is given by the rank of $H_1(G, V)$. | Ambiguous |
1,271,168 | How do I square a logarithm? | Ambiguous |
292,425 | 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 |
2,897,989 | Let $D$ be defined by $D(a)=f_a$ for $f\in L^2(\mathbb{R})$. Show that $D$ is continuous. | Ambiguous |
241,316 | Does the infinite series $\sum_{k=1}^{\infty} \frac{\sqrt{k}}{(k+1)^2}$ converge? | Unambiguous |
1,961,686 | 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 |
1,438,319 | Prove that $\lim_\limits{x\to 0}{\frac{e^x-1}{x}}=1$ without derivatives | Ambiguous |
33,296 | How do you prove that a differentiable function is monotonic given information about the sign of its derivative? | Unambiguous |
4,317,139 | Bianchi classification of solvable Lie groups and cocompact subgroups | Ambiguous |
820,492 | Are there algorithms to simulate the continuous deformation of a cup into a donut? | Ambiguous |
521,814 | I have this picture and I know a height. How do I find its other proportions using perspective projection? | Ambiguous |
709,004 | Evaluate $\lim_{x\to\infty}\left(\frac{x^2+2x-1}{2x^3-3x-2}\right)^{1/x}$. | Unambiguous |
866,327 | 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 |
2,296,759 | What is the relationship between a symmetric matrix and the inner product it defines on a vector space? | Unambiguous |
3,538,836 | How is this inequality proved for the squeeze theorem if it doesn't hold for all $x$? | Ambiguous |
1,900,241 | What is the expected value of $X$, given its generating function? | Ambiguous |
4,595,703 | 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 |
2,537,775 | Evaluate $\prod_{r=1}^{10}(1+\tan r^\circ)\cdot\prod_{r=46}^{55}(1+\cot r^\circ)$. | Unambiguous |
2,249,320 | Find a closed-form expression for $g(x) = \cos x + \cos 3x + \cos 5x + \cdots + \cos((2n-1)x)$. | Unambiguous |
972,189 | 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 |
298,648 | What is the standard notation for the concatenation of two finite sequences in mathematics and computer science? | Unambiguous |
2,820,510 | How are quaternions used to represent 3D rotations in geometry? | Unambiguous |
1,812,294 | How to find $P(B\cup C)$ when $A$ and $B$ are independent and the union of $A,B,C$ is $1$? | Ambiguous |
1,111,222 | Is a function that's unbounded near a boundary point still integrable if we redefine its value there? | Ambiguous |
986,013 | What should I do with the fact that my Laurent series converges in $z=1+i$? | Ambiguous |
108,308 | Does Tom Apostol's proof of the Fundamental Theorem of Calculus involve a division by zero? | Unambiguous |
2,697,247 | What are the standard integral representations of the Bessel function of the first kind $J_n(x)$? | Unambiguous |
2,296,602 | Simplify the propositional logic expression $\neg ((P \to (Q \to R)) \to (Q \to (P \to R)))$ to its simplest form. | Unambiguous |
3,944,628 | 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 |
548,732 | 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 |
1,045,011 | How to find the limit $\lim_{n\to \infty}\frac{n}{2^n-1}$? | Ambiguous |
1,258,955 | Convergence of the integral $\int_0^{\pi/2}\ln(\cos(x))dx$ | Ambiguous |
2,427,685 | Calculate the limit $\lim_{n \to \infty} \sum_{k=1}^{n} \frac{(-1)^{k-1}}{k^2 \binom{n}{k}}$. | Unambiguous |
124,986 | 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 |
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