MathSE ID int64 795 4.87M | Math Question stringlengths 23 305 | Tag stringclasses 2
values |
|---|---|---|
377,705 | How many repetitions do I need to find a prime between n and 2n? | Ambiguous |
726,827 | 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 |
3,564,289 | What is the expected number of returns to the origin for a random walk? | Ambiguous |
1,232,385 | How do I prove $\mathbb{Z}[w]/(a)$ is isomorphic to $\mathbb{Z}_{N(a)}$ using the norm? | Ambiguous |
700,216 | 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 |
3,533,343 | Counter examples in measures | Ambiguous |
1,612,241 | What is the logarithmic identity for the sum of logarithms of multiple terms? | Unambiguous |
3,743,564 | Another coins weighing puzzle | Ambiguous |
4,336 | If all eigenvalues of a Hermitian matrix are positive, is the matrix positive definite? | Unambiguous |
166,553 | 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 |
532,830 | How can I get more intuition about the complex numbers (C) and their multiplication and exponentiation? | Unambiguous |
2,054,106 | What are some common generalizations or variations of the Lebesgue Dominated Convergence Theorem in measure theory? | Unambiguous |
29,758 | Prove that every injective entire function from the complex plane to itself is a linear polynomial. | Unambiguous |
20,813 | How does a number system without irrational numbers handle circles? | Ambiguous |
2,730,855 | Water is poured into a container at a given rate. How fast does the level rise at a certain height? | Ambiguous |
4,194,118 | Area between square and circumscribed circle is less than half the area of the circle. | Ambiguous |
2,614,579 | How do I evaluate the limit $\lim_{x\to1} \frac{\sqrt{2x-1} -1}{x^2-1}$ step-by-step? | Unambiguous |
1,483,533 | How do I solve a linear equation for a real variable $x$? | Unambiguous |
1,145,215 | What is the definition of a matrix inverse in linear algebra and how does it relate to the identity matrix? | Unambiguous |
218,977 | Let $\Omega$ be a rectangle. What is the smallest space to which $\delta_0$ belongs? | Ambiguous |
2,361,340 | What is the limit as $x$ approaches infinity of $\frac{1}{x^{n-1}}$ for $n > 1$? | Unambiguous |
2,770,657 | Find the Maclaurin series for $\ln(1-x)$ and state its interval of convergence using sigma notation. | Unambiguous |
2,344,095 | Find coefficient of $x^2$ in a complicated expansion | Ambiguous |
1,725,118 | What are the axioms needed to state the Fixed Point Theorem? Does it apply to ZF, Robinson arithmetic, or PA? | Ambiguous |
2,732,822 | 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 |
2,805,228 | 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 |
20,096 | Basic addition and subtraction word problems for preschool and kindergarten children. | Unambiguous |
2,824,634 | Prove that $\sin(x/2) + 2\sin(x/4) \leq 3\sqrt{3}/2$ for all $x \in [0, 2\pi]$. | Unambiguous |
292,425 | How do I solve a piecewise function if it's not defined on $(-4,-1)$? | Ambiguous |
2,059,626 | choose 1 song to be played 3 times and the rest only once | Ambiguous |
494,832 | What is the definition of a ring that is complete with respect to the I-adic topology defined by an ideal $I$? | Unambiguous |
1,613,067 | subtracting the eigenvalue matrix from the original transformation of a 2x2 hit's the other eigenvalue proof | Ambiguous |
546,701 | Find the number of positive integers with exactly 5 digits whose digits add up to 42 | Unambiguous |
3,440,597 | 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 |
2,512,671 | 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 |
62,621 | Is there at least one irrational number with the property that it cannot be defined by a finite string of information? | Ambiguous |
4,494,965 | How many ways I can make \$5 in 50c, \$1 and \$2 coins? | Ambiguous |
2,112,893 | How to calculate the 100th term of an arithmetic and a geometric series? | Unambiguous |
2,108,420 | 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 |
2,806,957 | Under what conditions does a line intersect a circle in a 2D plane? | Unambiguous |
311,580 | Coordinate-free proof of $\operatorname{Tr}(AB)=\operatorname{Tr}(BA)$? | Ambiguous |
1,726,540 | How to differentiate $x^{\frac1x}$? | Ambiguous |
1,432,379 | 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 |
2,999,909 | Does Hom(X, Y) always form an abelian group? | Ambiguous |
1,891,318 | How many binary strings of length 8 do not contain the substring '100'? | Unambiguous |
11,194 | Solving the recurrence relation that contains summation of nth term | Ambiguous |
942,204 | How to derive the general term of a sequence formed by the repeated convolution of a probability distribution with itself? | Unambiguous |
3,254,331 | Prove that the definite integral $\int_0^\infty(\arctan(1/x))^2 \mathrm{d}x$ converges to $\pi\ln 2$. | Unambiguous |
2,326,677 | What is the signifigance of a saddle point? | Ambiguous |
2,732,822 | What do you call a system where $a+b=1$ and $ab=-1$? | Ambiguous |
451,799 | 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 |
1,258,718 | The "Smooth Factor" in a number sequence | Ambiguous |
846,706 | 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 |
3,103,052 | 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 |
1,491,490 | What is the group generated by $M_\pi$? | Ambiguous |
720,688 | What is the average power of this signal? | Ambiguous |
2,932,046 | Is this formula correct: $\frac{|(p-a) \times d|}{|d|}$? | Ambiguous |
2,675,393 | How to find the derivative of a power series? | Unambiguous |
1,075,740 | What happens to the angles of an isosceles triangle if one vertex is at infinity? | Ambiguous |
808,565 | What is the standard mathematical notation for the dimension of a vector space $V$? | Unambiguous |
1,194,644 | 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 |
973,713 | Problem in the second-derivative symbol. | Ambiguous |
3,328,149 | 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 |
3,481,798 | 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 |
3,472,866 | Are the sample mean and sample variance considered random variables? | Unambiguous |
345,024 | What are the applications of Catalan numbers in computer programming and algorithm design? | Unambiguous |
553,280 | What is the number of banded $(0,1)$-matrices with exactly $k$ ones and prescribed row/column sums? | Ambiguous |
1,531,029 | If an abelian group $G$ has property $N$, then $G$ is finitely generated | Ambiguous |
2,901,457 | Determine for which values of $k$ and $a$ the series $\sum_{n=1}^\infty \frac{\ln^k(n)}{n^a}$ converges. | Unambiguous |
3,255,482 | Evaluate $\lim_{x \to 0} \frac{1}{x} \int_{x}^{2x} e^{-t^{2}}\,dt$ | Unambiguous |
3,002,761 | How do I find the maximum value of a variable $x$ when it is subject to a given equality constraint? | Unambiguous |
298,648 | Is there a common symbol for concatenating two (finite) sequences? | Ambiguous |
1,720,548 | 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 |
4,768 | Prove that every permutation of a finite set is either even or odd, but not both. | Unambiguous |
314,408 | What are the conditional probabilities for my simple OR-gate where $P(A)=0$ and $P(B)=0$? | Ambiguous |
4,221,432 | What are the general integer solutions to the Diophantine equation $a^2+b^2=c^2-1$ for positive integers $a, b, c$? | Unambiguous |
744,139 | 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 |
2,368,762 | 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 |
1,673,403 | A function is never zero, takes both signs, and is continuous. How is that possible? | Ambiguous |
378,970 | For a right triangle with a hypotenuse of 12 cm, what measures of the acute angles would produce the maximum area? | Unambiguous |
736,346 | What are the geometric methods to fold a rectangular piece of A4 paper into three equal parallel strips? | Unambiguous |
3,673,715 | I need help with the heat equation in a 3D ball. | Ambiguous |
1,494,248 | Evaluate the indefinite integral $\int \frac{\sin(4x)}{\sin(x)} \, dx$. | Unambiguous |
185,267 | How do I prove that a subset of a set with Lebesgue outer measure zero also has outer measure zero? | Unambiguous |
2,492,396 | An equation is first-order nonlinear. How do I transform it into standard linear form? | Ambiguous |
289,779 | Determine whether the sequence $a_n = 3^n - 2^{n+100}$ for $n \in \mathbb{N}$ converges or diverges. | Unambiguous |
330,556 | 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 |
3,487,811 | $\lim_{x \to 0} {1-x\cot x\over x^2}$ | Ambiguous |
366,529 | 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 |
387,437 | How can you prove that a number is an algebraic integer? | Ambiguous |
2,030,790 | 3 variable diophantine equation word problem: coins | Ambiguous |
464,426 | Evaluate the limit $\lim_{x \to 1} \frac{x^{1/5}-1}{x^{1/3}-1}$ for real-valued $x$. | Unambiguous |
3,601,099 | For a symmetric matrix and a unit vector, why does a certain quartic inequality involving inner products hold? | Ambiguous |
1,455,761 | What is the relationship between the determinant of a square matrix and its inverse? | Unambiguous |
2,887,105 | Express $\frac{1}{1-i-z}$ as a power series in $z$ valid for $|-z-i|<1$. | Unambiguous |
2,614,579 | How to solve $\lim_\limits{x\to1} \frac{\sqrt{2x-1} -1}{x^2-1}$? | Ambiguous |
531,128 | What are the subspaces of a finite-dimensional vector space $V$ that have dimension $\dim V - 1$, and what are they called? | Unambiguous |
735,461 | 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 |
1,861,189 | $PC$ bisects $\angle APB$. Prove $XY \parallel AB$. | Ambiguous |
2,369,950 | Is the statement $|x|=\pm x$ for $x>0$ a correct definition of absolute value, and why is it confusing? | Unambiguous |
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