prompt listlengths 1 1 | label stringlengths 1 23 |
|---|---|
[
{
"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 $m$ be a positive integer, and let $T$ denote the set of all subsets of $\\{1, 2, \\dots, m\\}$. Call a subset $S$ of $T$ ... | 2048 |
[
{
"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 $y=x+c$ intersect at exactly one point. Wh... | 18 |
[
{
"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\nRachel has the number $1000$ in her hands. When she puts the number $x$ in her left pocket, the number changes to $x+1.$ When ... | 13 |
[
{
"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\nConvex hexagon $ABCDEF$ is drawn in the plane such that $ACDF$ and $ABDE$ are parallelograms with area 168. $AC$ and $BD$ inte... | 196 |
[
{
"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 $N = 2^{\\left(2^2\\right)}$ and $x$ be a real number such that $N^{\\left(N^N\\right)} = 2^{(2^x)}$. Find $x$.\n\nRemembe... | 66 |
[
{
"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 $\\mathbb{N}$ be the set of positive integers, and let $f: \\mathbb{N} \\to \\mathbb{N}$ be a function satisfying\n[list]\... | 1899 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the smallest positive integer that cannot be written as the sum of two nonnegative palindromic integers? (An integer i... | 21 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor a positive integer $N$, we color the positive divisors of $N$ (including 1 and $N$) with four colors. A coloring is called... | 192 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many distinct permutations of the letters in the word REDDER are there that do not contain a palindromic substring of leng... | 6 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven an $8\\times 8$ checkerboard with alternating white and black squares, how many ways are there to choose four black squa... | 20736 |
[
{
"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 triangle $ABC$ be such that $AB = AC = 22$ and $BC = 11$. Point $D$ is chosen in the interior of the triangle such that $A... | 36104 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe numbers $1, 2, . . . , 10$ are randomly arranged in a circle. Let $p$ be the probability that for every positive integer $... | 1390 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven a positive integer $k$, let $||k||$ denote the absolute difference between $k$ and the nearest perfect square. For examp... | 89800 |
[
{
"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 $(a_1, a_2, ..., a_8)$ be a permutation of $(1, 2, ... , 8)$. Find, with proof, the maximum possible number of elements of... | 5 |
[
{
"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 $ABC$ be a triangle with $\\angle A = 60^o$. Line $\\ell$ intersects segments $AB$ and $AC$ and splits triangle $ABC$ into... | 21 |
[
{
"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, with proof, the maximum positive integer $k$ for which it is possible to color $6k$ cells of $6 \\times 6$ grid such tha... | 4 |
[
{
"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\nPoint $P$ is located inside a square $ABCD$ of side length $10$. Let $O_1$, $O_2$, $O_3$, $O_4$ be the circumcenters of $P AB$... | 16902 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nMichel starts with the string HMMT. An operation consists of either replacing an occurrence of H with HM, replacing an occurre... | 144 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose $P(x)$ is a monic polynomial of degree $2023$ such that $P(k) = k^{2023}P(1-\\frac{1}{k})$ for every positive integer ... | 406 |
[
{
"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\nRichard starts with the string $HHMMMMTT$. A move consists of replacing an instance of $HM$ with $MH$, replacing an instance o... | 70 |
[
{
"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 $E$, $I$, $L$, $V$ are (not necessarily distinct) nonzero digits in base ten for which\n[list]\n[*] the four-digit num... | 9954 |
[
{
"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\nPhilena and Nathan are playing a game. First, Nathan secretly chooses an ordered pair $(x, y)$ of positive integers such that ... | 9 |
[
{
"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 any positive integers $a$ and $b$ with $b > 1$, let $s_b(a)$ be the sum of the digits of $a$ when it is written in base $b... | 81 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $S$ be the set of ordered pairs $(a, b)$ of positive integers such that $\\gcd(a, b) = 1$. Compute \\[ \\sum_{(a, b) \\in ... | 7400 |
[
{
"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\nElbert and Yaiza each draw $10$ cards from a $20$-card deck with cards numbered $1,2,3,\\dots,20$. Then, starting with the pla... | 324 |
[
{
"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 cell of a $3 \\times 3$ grid is labeled with a digit in the set $\\{1, 2, 3, 4, 5\\}$ Then, the maximum entry in each row... | 2664 |
[
{
"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 $a_1, a_2, \\ldots, a_{100}$ be positive integers, satisfying $$\\frac{a_1^2+a_2^2+\\ldots+a_{100}^2} {a_1+a_2+\\ldots+a_{... | 550 |
[
{
"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\nSally the snail sits on the $3 \\times 24$ lattice of points $(i, j)$ for all $1 \\le i \\le 3$ and $1 \\le j \\le 24$. S... | 4096 |
[
{
"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(n)=(n-1^3)(n-2^3)\\ldots (n-40^3)$ for positive integers $n$. Let $d$ be the largest positive integer such that $d \\mi... | 48 |
[
{
"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 [i]lame king[/i] is a chess piece that can move from a cell to any cell that shares at least one vertex with it, except for ... | 43 |
[
{
"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\nCompute the number of triples $(f,g,h)$ of permutations on $\\{1,2,3,4,5\\}$ such that \\begin{align*} \n & f(g(h(x))) = h(g(f... | 146 |
[
{
"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 polynomial $f(x)$ with integer coefficients is called $\\textit{splitty}$ if and only if for every prime $p$, there exists p... | 693 |
[
{
"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\nNine distinct positive integers summing to $74$ are put into a $3 \\times 3$ grid. Simultaneously, the number in each cell is ... | 18 |
[
{
"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\nCompute the sum of all positive integers $50 \\leq n \\leq 100$ such that $2n+3 \\nmid 2^{n!}-1$.\n\nRemember to put your answ... | 222 |
[
{
"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 $\\Omega$ and $\\omega$ be circles with radii $123$ and $61$, respectively, such that the center of $\\Omega$ lies on $\\o... | 42 |
[
{
"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 $ABC$ be a triangle with $\\angle BAC = 90^o$. Let $D$, $E$, and $F$ be the feet of altitude, angle bisector, and median f... | 20 |
[
{
"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 integer which cannot be expressed as $2008x + 2009y + 2010z$ for some positive integers $x, y$, and $z$?\n... | 2016031 |
[
{
"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 four-dimensional space, the $24$-cell of sidelength $\\sqrt{2}$ is the convex hull of (smallest convex set containing) the ... | 8 |
[
{
"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\nIf $f(x) = x^{x^{x^x}}$ , find the last two digits of $f(17) + f(18) + f(19) + f(20)$.\n\nRemember to put your answer on its o... | 32 |
[
{
"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 the values of $x$ for which $\\binom{x}{0}-\\binom{x}{1}+\\binom{x}{2}-...+\\binom{x}{2008}=0$\n\nRemember to ... | 2017036 |
[
{
"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 ordered pairs $(a, b)$ of positive integers that are solutions of the following equation: \\[a^2 + b^2 = ab... | 1 |
[
{
"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\nLines $l$ and $m$ are perpendicular. Line $l$ partitions a convex polygon into two parts of equal area, and partitions the pro... | 2141 |
[
{
"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 $\\triangle ABC$ and a point $M$ in its interior so that $\\angle MAB = 10^\\circ$, $\\angle MBA = 20^\\circ$, $\\ang... | 60 |
[
{
"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\nThe real numbers $x$, $y$, $z$, and $t$ satisfy the following equation:\n\\[2x^2 + 4xy + 3y^2 - 2xz -2 yz + z^2 + 1 = t + \\sq... | 125 |
[
{
"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 a rectangular plot of land, a man walks in a very peculiar fashion. Labeling the corners $ABCD$, he starts at $A$ and walks... | 793 |
[
{
"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 regular hexagon $ABCDEF$, $AC$, $CE$ are two diagonals. Points $M$, $N$ are on $AC$, $CE$ respectively and satisfy $AC: AM ... | 300 |
[
{
"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\nAll the diagonals of a regular decagon are drawn. A regular decagon satisfies the property that if three diagonals concur, th... | 161 |
[
{
"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\nThe expression $\\sin2^\\circ\\sin4^\\circ\\sin6^\\circ\\cdots\\sin90^\\circ$ is equal to $p\\sqrt{5}/2^{50}$, where $p$ is an... | 192 |
[
{
"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 $AB = 4$, $AC = 5$, and $BC = 6$. An angle bisector is drawn from angle $A$, and meets $BC$ at $M$. What ... | 100 |
[
{
"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 rectangular coordinates, point $A = (1,2)$, $B = (3,4)$. $P = (a, 0)$ is on $x$-axis. Given that $P$ is chosen such that $A... | 100 |
[
{
"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)=3x^3-5x^2+2x-6$. If the roots of $f$ are given by $\\alpha$, $\\beta$, and $\\gamma$, find\n\\[\n\\left(\\frac{1}{\\... | 68 |
[
{
"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 $\\sigma$ be a random permutation of $\\{0, 1, \\ldots, 6\\}$. Let $L(\\sigma)$ be the length of the longest initial monot... | 5399 |
[
{
"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(n)$ denote the number of divisors of $n$ (including itself). You are given that \\[\\sum_{n=1}^{\\infty} \\frac{1}{n^2}... | 36 |
[
{
"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 $\\ell_1$ and $\\ell_2$ be two parallel lines, a distance of 15 apart. Points $A$ and $B$ lie on $\\ell_1$ while points $C... | 23 |
[
{
"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^3-7x^2+16x-10$. As $x$ ranges over all integers, find the sum of distinct prime values taken on by $f(x)$.\n\nRe... | 2 |
[
{
"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 such that $a + bz = x^3 + y^4$ has no solutions for any integers $x, y, z$, with $b$ as s... | 91 |
[
{
"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 polynomial $p$ can be written as \n\\begin{align*}\np(x) = x^6+3x^5-3x^4+ax^3+bx^2+cx+d.\n\\end{align*}\nGiven that all root... | 64 |
[
{
"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\nAt the start of the PUMaC opening ceremony in McCosh auditorium, the speaker counts $90$ people in the audience. Every minute ... | 6 |
[
{
"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 every integer $n$ from $0$ to $6$, we have $3$ identical weights with weight $2^n$. How many ways are there to form a tota... | 60 |
[
{
"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 sum of all primes $p$ such that $7^p - 6^p + 2$ is divisible by 43? \n\n\nRemember to put your answer on its own l... | 3 |
[
{
"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\nFour circles are situated in the plane so that each is tangent to the other three. If three of the radii are $5$, $5$, and $8$... | 43 |
[
{
"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 how many ordered triplets of three positive integers is it true that their product is four more than twice their sum?\n\nR... | 15 |
[
{
"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 circles, with radii of $1, 1$, and $2$, are externally tangent to each other. The minimum possible area of a quadrilater... | 22 |
[
{
"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\nConsidering all numbers of the form $n = \\lfloor \\frac{k^3}{2012} \\rfloor$, where $\\lfloor x \\rfloor$ denotes the greates... | 1995 |
[
{
"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 circles centered at $O$ and $P$ have radii of length $5$ and $6$ respectively. Circle $O$ passes through point $P$. Let th... | 457 |
[
{
"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\nWhen some number $a^2$ is written in base $b$, the result is $144_b$. $a$ and $b$ also happen to be integer side lengths of a ... | 169 |
[
{
"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 white pentagonal pyramids, with side lengths all the same, are glued to each other at their regular pentagon bases. Some o... | 120 |
[
{
"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\nCompute the smallest positive integer $a$ for which $$\\sqrt{a +\\sqrt{a +...}} - \\frac{1}{a +\\frac{1}{a+...}}> 7$$\n\nRemem... | 43 |
[
{
"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 be a polynomial of degree $3$ with integer coefficients such that $f(0) = 3$ and $f(1) = 11$.\nIf f has exactly $2$ inte... | 0 |
[
{
"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\nCall a positive integer $x$ a leader if there exists a positive integer $n$ such that the decimal representation of $x^n$ star... | 2008 |
[
{
"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\nIf the probability that the sum of three distinct integers between $16$ and $30$ (inclusive) is even can be written as $\\frac... | 97 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n How many (possibly empty) sets of lattice points $\\{P_1, P_2, ... , P_M\\}$, where each point $P_i =(x_i, y_i)$ for $x_i\n, ... | 3432 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $M$ be the smallest positive multiple of $2012$ that has $2012$ divisors. \nSuppose $M$ can be written as $\\Pi_{k=1}^{n}... | 1012 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\vartriangle ABC$ be a triangle with $\\angle BAC = 45^o, \\angle BCA = 30^o$, and $AB = 1$. Point $D$ lies on segment $... | 2 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute $\\Sigma_{n=1}^{\\infty}\\frac{n + 1}{n^2(n + 2)^2}$ .\nYour answer in simplest form can be written as $a/b$, where $a... | 21 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet A be a regular $12$-sided polygon. A new $12$-gon B is constructed by connecting the midpoints of the sides of A. The rati... | 9 |
[
{
"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 $f_n(x) = n + x^2$. Evaluate the product $gcd\\{f_{2001}(2002), f_{2001}(2003)\\} \\times gcd\\{f_{2011}(2012), f_{2011}(2... | 125 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the smallest natural number $n$ greater than $2012$ such that the polynomial $f(x) =(x^6 + x^4)^n - x^{4n} - x^6$ is d... | 2015 |
[
{
"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 an be a sequence such that $a_0 = 0$ and:\n$a_{3n+1} = a_{3n} + 1 = a_n + 1$\n$a_{3n+2} = a_{3n} + 2 = a_n + 2$\nfor all n... | 379 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDuring chemistry labs, we oftentimes fold a disk-shaped filter paper twice, and then open up a flap of the quartercircle to fo... | 60 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose $a,b,c>0$ are integers such that \\[abc-bc-ac-ab+a+b+c=2013.\\] Find the number of possibilities for the ordered trip... | 18 |
[
{
"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(Following question 1) Now instead consider an infinite strip of squares, labeled with the integers $0, 1, 2, \\ldots$ in that... | 2002 |
[
{
"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 $d$ be the greatest common divisor of $2^{30^{10}}-2$ and $2^{30^{45}}-2$. Find the remainder when $d$ is divided by $201... | 2012 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the largest positive integer that cannot be expressed as a sum of non-negative integer multiple of $13$, $17$, and $23... | 84 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose the function $\\psi$ satisfies $\\psi(1)=\\sqrt{2+\\sqrt{2+\\sqrt2}}$ and $\\psi(3x)+3\\psi(x)=\\psi(x)^3$ for all rea... | 5792 |
[
{
"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\nCircle $w$ with center $O$ meets circle $\\Gamma$ at $X,Y,$ and $O$ is on $\\Gamma$. Point $Z\\in\\Gamma$ lies outside $w$ su... | 82 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nMereduth has many red boxes and many blue boxes. Coloon has placed five green boxes in a row on the ground, and Mereduth wants... | 3113 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nYou are eating at a fancy restaurant with a person you wish to impress. For some reason, you think that eating at least one sp... | 961 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAn integer sequence $a_1,a_2,\\ldots,a_n$ has $a_1=0$, $a_n\\leq 10$ and $a_{i+1}-a_i\\geq 2$ for $1\\leq i<n$. How many poss... | 89 |
[
{
"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\nIf $x,y$ are real, then the $\\textit{absolute value}$ of the complex number $z=x+yi$ is \\[|z|=\\sqrt{x^2+y^2}.\\] Find the ... | 43 |
[
{
"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 chords of a sphere, each having length $5,6,7$, intersect at a single point inside the sphere and are pairwise perpendic... | 15 |
[
{
"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 a random permutation of the set $\\{1, 2, . . . , 2015\\}$. In other words, for each $1 \\le i \\le 2015$, $i$ is sen... | 651222 |
[
{
"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\nJonathan has a magical coin machine which takes coins in amounts of $7, 8$, and $9$. If he puts in $7$ coins, he gets $3$ coin... | 4 |
[
{
"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 the sequence $a_i$ as follows: $a_1 = 1, a_2 = 2015$, and $a_n = \\frac{na_{n-1}^2}{a_{n-1}+na_{n-2}}$ for $n > 2$. Wha... | 7 |
[
{
"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 largest $r$ such that $4$ balls each of radius $r$ can be packed into a regular tetrahedron with side length $1$. In ... | 15 |
[
{
"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 positive integer $n$ less than $10, 000$ such that in base $4$, $n$ and $3n$ have the same number of digit... | 4369 |
[
{
"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 define the function $f(x,y)=x^3+(y-4)x^2+(y^2-4y+4)x+(y^3-4y^2+4y)$. Then choose any distinct $a, b, c \\in \\mathbb{R}$ su... | 1 |
[
{
"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 $\\overline{AB} = 5, \\overline{BC} = 4, \\overline{CA} = 6$. Points $D$ and $E$ are on sides $AB$ and $AC$... | 255 |
[
{
"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 define the ridiculous numbers recursively as follows:\n[list=a]\n[*]1 is a ridiculous number.\n[*]If $a$ is a ridiculous nu... | 9 |
[
{
"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\nPrinceton’s Math Club recently bought a stock for $\\$2$ and sold it for $\\$9$ thirteen days later. Given that the stock eith... | 273 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.