prompt listlengths 1 1 | label stringlengths 1 210 |
|---|---|
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a$ and $b$ be natural numbers such that $2a-b$, $a-2b$, and $a+b$ are all distinct squares. What is t... | 21 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine the value of the parameter $m$ such that the equation $(m-2)x^2 + (m^2-4m+3)x - (6m^2-2) = 0$ ha... | 3 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all positive integers $l$ for which the equation $$a^3+b^3+ab=(lab+1)(a+b)$$ has a solution over posi... | 1 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven a regular polygon with $n$ sides, it is known that there are 1200 ways to choose three of the vertic... | 50 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nKobar and Borah are playing a game on a whiteboard with the following rules: They start with two distinct ... | 2179 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn quadrilateral $ABCD$, it is given that $AB = DB$ and $AD = BC$. If the angles are $\nabla ABD = 36^{\\c... | 102 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all positive integers $n$ such that $n$ has exactly 6 positive divisors $1 < d_{1} < d_{2} < d_{3} < ... | 1519 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the smallest natural number $n$ such that for any coloration of the numbers $1, 2, \\dots, n$ with th... | 25 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor some real number $c$, the graphs of the equation $y = |x - 20| + |x + 18|$ and the line $y = x + c$ in... | 18 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all positive integers $n$ such that $n^3 - 5n^2 + 9n - 6$ is a perfect square number.\n\nRemember to ... | 2 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all values of the parameter $a$ for which the sum of all real solutions of the equation $x^4 - 5x + a... | 0 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDraw a regular hexagon. Then, construct a square on each edge of the hexagon. Form equilateral triangles b... | 1 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $n$ be a positive integer. If $S$ is a nonempty set of positive integers, then we say $S$ is $n$-compl... | 32 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $p$ be a polynomial with integer coefficients such that $p(15) = 6$, $p(22) = 1196$, and $p(35) = 26$.... | 28 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the smallest positive integer $N$ satisfying the following three properties:\n- $N$ leaves a remainde... | 502 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA time is chosen randomly and uniformly in a 24-hour day. The probability that at that time, the non-refle... | 211 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all integers $n$ such that $1 < n < 10^6$ and $n^3 - 1$ is divisible by $10^6 n - 1$.\n\nRemember to ... | 1000 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a$ and $b$ be positive integers not divisible by $5$. A sequence of integers is constructed as follow... | 5 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the sum of all integers $n$ such that $n^4 + n^3 + n^2 + n + 1$ is a perfect square.\n\nRemember to p... | 2 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nMichael, David, Evan, Isabella, and Justin compete in the NIMO Super Bowl, a round-robin cereal-eating tou... | 1116 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $x,y$ be real numbers such that $xy=1$. Let $T$ and $t$ be the largest and smallest values of the expr... | 25 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nMereduth has many red boxes and many blue boxes. Coloon has placed five green boxes in a row on the ground... | 21 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWe are given weights ranging from 1 to 5771, i.e., 1, 2, 3, ..., 5770, 5771. These weights are to be parti... | 2886 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine all prime numbers $p$ such that $5^p + 4p^4$ is a perfect square, i.e., the square of an integer... | 5 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the smallest positive integer $n$ such that $n$ is divisible by $100$ and has exactly $100$ divisors.... | 162000 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA frog is located at the origin of the Cartesian coordinate system. Every second, the frog jumps either ho... | 4064256 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nNumbers $a$, $b$, and $c$ form an arithmetic sequence if $b - a = c - b$. Let $a$, $b$, and $c$ be positiv... | 9 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere are $64$ booths arranged around a circular table, each containing a chip. The chips and booths are n... | 32 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the number of positive integer solutions $(a,b,c,d)$ to the equation $(a^2+b^2)(c^2-d^2)=2020$. Note:... | 8 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nConsider an isosceles triangle $ABC$ with sides $BC = 30$, $CA = AB = 20$. Let $D$ be the foot of the perp... | 25 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $q$ be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes th... | 0 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere are $12$ students in a classroom; $6$ of them are Democrats and $6$ of them are Republicans. Every h... | 6 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$ABC$ is a triangle with $AB = 33$, $AC = 21$, and $BC = m$, where $m$ is an integer. There are points $D$... | 30 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all prime numbers $p$ such that the number of distinct positive factors of $p^2 + 2543$ is less than ... | 2 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nInside a circle with radius $6$ lie four smaller circles with centers $A$, $B$, $C$, and $D$. These circle... | 24 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $D$, $E$, and $F$ respectively be the feet of the altitudes from $A$, $B$, and $C$ of acute triangle $... | 14 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all positive integers $n$ such that there exists a prime number $p$ where $p^n - (p-1)^n$ is a power ... | 2 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the number of all integer-sided isosceles obtuse-angled triangles with perimeter $2008$.\n\nRemember ... | 86 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the tribe of Zimmer, a boy is not considered a man until he completes a rite of passage involving a ser... | 73 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f(n)$ be the number of ways to write $n$ as a sum of powers of $2$, where the order of the summation ... | 2047 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine the smallest positive integer $A$ with an odd number of digits such that both $A$ and the number... | 100902018 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine all positive integers $n$ for which the equation\n\\[\nx^n + (2+x)^n + (2-x)^n = 0\n\\]\nhas an ... | 1 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nMatilda drew $12$ quadrilaterals. The first quadrilateral is a rectangle with integer sides, where the wid... | 2023 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine all real solutions of the equation:\n\\[\n\\frac{x^{2}}{x-1} + \\sqrt{x-1} + \\frac{\\sqrt{x-1}}... | 2 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the number of values of $x$ such that the area of the isosceles triangle with sides $x$, $65$, and $6... | 4224 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine all functions $f : \\mathbb{N}_0 \\to \\mathbb{R}$ satisfying $f(x+y) + f(x-y) = f(3x)$ for all ... | 0 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $p$ be a prime number. All natural numbers from $1$ to $p$ are written in a row in ascending order. Fi... | 3 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFive airway companies operate in a country consisting of 36 cities. Between any pair of cities, exactly on... | 3780 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn each cell, a strip of length $100$ is worth a chip. You can change any $2$ neighboring chips and pay $1... | 50 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all positive integers $n$ such that all positive divisors of $n$ can be arranged in a rectangular tab... | 1 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the game of *Winners Make Zeros*, a pair of positive integers \\((m,n)\\) is written on a sheet of pape... | 999 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAssume that $f(a+b) = f(a) + f(b) + ab$, and that $f(75) - f(51) = 1230$. Find $f(100)$.\n\nRemember to pu... | 3825 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all natural numbers $n > 1$ for which the following applies: The sum of the number $n$ and its second... | 1342 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nOne hundred and one of the squares of an $n \\times n$ table are colored blue. It is known that there exis... | 101 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA convex 2011-gon is drawn on the board. Peter keeps drawing its diagonals in such a way that each newly d... | 4016 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the sum of all positive integers $n$ such that $\\frac{2n+1}{n(n-1)}$ has a terminating decimal repre... | 52 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor which positive integer $n$ is the quantity $\\frac{n}{3} + \\frac{40}{n}$ minimized?\n\nRemember to pu... | 11 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nBob starts with an empty whiteboard. He then repeatedly chooses one of the digits $1, 2, \\ldots, 9$ (unif... | 8102 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHari is obsessed with cubics. He comes up with a cubic polynomial with leading coefficient 1, rational coe... | 31 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$A$ and $B$ are on a circle with radius $20$ centered at $C$, and $\\angle ACB = 60^\\circ$. $D$ is chosen... | 20 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the smallest integer $n > 3$ such that, for each partition of $\\{3, 4, \\ldots, n\\}$ into two sets,... | 243 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the largest number of obtuse triangles that can be composed of $16$ different segments (each trian... | 35 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all prime numbers $p$ such that both $4p^2 + 1$ and $6p^2 + 1$ are also prime numbers.\n\nRemember to... | 5 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTriangle $ABC$ with $\\angle A = 90^\\circ$ has incenter $I$. A circle passing through $A$ with center $I$... | 7 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDefine $A(n)$ as the average of all positive divisors of the positive integer $n$. Find the sum of all sol... | 1374 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$f(x), g(x)$ are two quadratic trinomials and $a, b, c, d$ are some real numbers. Given the following valu... | 8 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all prime numbers $p$ such that the expression $p^2 - p + 1$ is a perfect cube.\n\nRemember to put yo... | 19 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere exists a polynomial $P$ of degree $5$ with the following property: if $z$ is a complex number such t... | 1 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine the least odd number $a > 5$ satisfying the following conditions: There are positive integers $m... | 261 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all positive integers $n$ such that the number $\\frac{(2n)!+1}{n!+1}$ is a positive integer.\n\nReme... | 3 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nOn a $1000 \\times 1000$-board, we place dominoes such that each domino covers exactly two squares on the ... | 250000 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all positive integers $r$ such that there exist positive prime numbers $p$ and $q$ satisfying the equ... | 7 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTriangle $ABC$ has side lengths $AB = 20$, $AC = 14$, and $BC = 22$. The median from $B$ intersects $AC$ a... | 331 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA quadratic polynomial $p(x)$ with integer coefficients satisfies $p(41) = 42$. For some integers $a, b > ... | 2842 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the minimum number $n$ such that for any coloring of the integers from $1$ to $n$ into two colors, on... | 11 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSergei chooses two different natural numbers $a$ and $b$. He writes four numbers in a notebook: $a$, $a+2$... | 2 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine the real number $a$ such that $f(a) = a$ is a relative minimum of the function $f(x) = x^4 - x^3... | 1 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $ABCD$ be a square with side length $10$, and let $P$ be a point on side $BC$. By folding the paper al... | 20 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSolve the equation:\n\\[\n \\sin 9^\\circ \\sin 21^\\circ \\sin(102^\\circ + x^\\circ) = \\sin 30^\\circ ... | 9 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere are $100$ countries participating in an olympiad. Suppose $n$ is a positive integer such that each o... | 20 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEach person in Cambridge drinks a (possibly different) $12$ ounce mixture of water and apple juice, where ... | 7 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f(x) = x^3 + ax^2 + bx + c$ have solutions that are distinct negative integers. If $a + b + c = 2014$... | 1440 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose an integer $x$, a natural number $n$, and a prime number $p$ satisfy the equation $7x^2 - 44x + 12... | 47 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThree fair six-sided dice are rolled. The expected value of the median of the numbers rolled can be writte... | 9 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA triangle has side lengths of $x$, $75$, and $100$, where $x < 75$, and altitudes of lengths $y$, $28$, a... | 56 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA square has been divided into $2022$ rectangles, with no two of them having a common interior point. What... | 2025 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the diagram, $\\angle AOB = \\angle BOC$ and $\\angle COD = \\angle DOE = \\angle EOF$. Given that $\\a... | 118 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nKermit the frog enjoys hopping around the infinite square grid in his backyard. It takes him $1$ Joule of ... | 10201 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere is a table with $n$ rows and $18$ columns. Each cell contains either a $0$ or a $1$. The table satis... | 12376 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nInitially, all edges of the $K_{2024}$ are painted with $13$ different colors. If there exist $k$ colors s... | 7 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the smallest natural number $k$ for which there exist natural numbers $m$ and $n$ such that $1324 + 2... | 3 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a, b, c, d$ be the four roots of the polynomial:\n\\[\nx^4 + 3x^3 - x^2 + x - 2.\n\\]\nGiven that:\n\... | 39 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the real number $k$ such that $a$, $b$, $c$, and $d$ are real numbers that satisfy the system of equa... | 1018 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine the value of $p$ such that the numbers $p$, $4p^2 + 1$, and $6p^2 + 1$ are all prime numbers.\n\... | 5 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTwo semicircles are tangent to a middle circle, and both semicircles and the middle circle are tangent to ... | 15 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nJack and Jill are playing a chance game. They take turns alternately rolling a fair six-sided die labeled ... | 16 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA finite set $M$ of positive integers consists of distinct perfect squares and the number $92$. The averag... | 22 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all values of $a$ for which the equation $x^3 - x + a = 0$ has three different integer solutions.\n\n... | 0 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSolve in positive integers the following equation:\n\n$$ \\left[ \\sqrt{1} \\right] + \\left[ \\sqrt{2} \\... | 6 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose that $x, y, z$ are non-zero real numbers such that:\n\\[\n\\begin{cases}\n x = 2 - \\dfrac{y}{z} ... | 3 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.