prompt listlengths 1 1 | label stringlengths 1 23 |
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{
"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 universe of Pi Zone, points are labeled with $2 \\times 2$ arrays of positive reals. One can teleport from point $M$ to... | 17 |
[
{
"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 square $ABCD$ with side length $16$. Points $E$ and $F$ are on $CD$ such that $CE = EF = FD$. The lines $BF$ and $A... | 96 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $P(x) = x^3 + x^2 - r^2 x - 2020$ be a polynomial with roots $r, s, t$. What is $P(1)$?\n\nRemember to put your answer on ... | -4038 |
[
{
"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 nonnegative integers $n$, let $f(n)$ be the number of digits of $n$ that are at least $5$. Let $g(n)=3^{f(n)}$. Compute \\... | 8000 |
[
{
"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 box contains $28$ red balls, $20$ green balls, $19$ yellow balls, $13$ blue balls, $11$ white balls, and $9$ black balls. Wh... | 76 |
[
{
"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 incenter $I$ and incircle $\\omega$. It is given that there exist points $X$ and $Y$ on the circu... | 59 |
[
{
"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 $N$ such that there exist 6 distinct integers $a_1, a_2, a_3, a_4, a_5, a_6 > 0$ satis... | 911 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA boat is traveling upstream at \\(5\\) mph relative to the current flowing against it at \\(1\\) mph. If a tree branch \\(10\... | 2 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the smallest positive multiple of $225$ that can be written using digits $0$ and $1$ only?\n\nRemember to put your ans... | 11111111100 |
[
{
"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 largest positive integer $n$ which cannot be written as the sum of three numbers greater than $1$ that are pairw... | 17 |
[
{
"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 $x$ be the answer to this problem. For what real number $a$ is the answer to this problem also $a-x$?\n\nRemember to put y... | 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\nLet $f(x)$ be a function mapping real numbers to real numbers. Given that $f(f(x)) = \\frac{1}{3x}$, and $f(2) = \\frac{1}{9}$... | 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\nTriangle $ABC$ has side lengths $AB = 14$, $BC = 15$, and $CA = 13$. We draw a circle with diameter $AB$ such that it intersec... | 3308 |
[
{
"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 $x, y, z$ be real numbers such that $x+y+z=-2$ and\n\\[\n(x^2+xy+y^2)(y^2+yz+z^2) + (y^2+yz+z^2)(z^2+zx+x^2) + (z^2+zx+x^2... | 14 |
[
{
"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 positive integer $n$, we define $a_{n}$ as the last digit of the sum $1 + 2 + \\cdots + n$. Compute $a_{1} + a_{2} +... | 6984 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all positive integers $n \\geq 1$ such that there exists a pair $(a, b)$ of positive integers, satisfying the following c... | 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\nHow many distinct sums can be made from adding together exactly 8 numbers that are chosen from the set $\\{ 1, 4, 7, 10 \\}$, ... | 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\nA semicircle has diameter $\\overline{AD}$ with $AD = 30$. Points $B$ and $C$ lie on $\\overline{AD}$, and points $E$ and $F$ ... | 52 |
[
{
"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 five-digit positive integers $\\overline{vwxyz}$ for which \n\\[ (10v+w) + (10w+x) + (10x+y) + (10y+z) =... | 164 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe pentagon below has three right angles. Find its area.\n\n\\begin{asy}\nsize(150);\ndefaultpen(linewidth(1));\ndraw((4,10)-... | 76 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEvaluate \\[ \\frac{1}{729} \\sum_{a=1}^{9} \\sum_{b=1}^{9} \\sum_{c=1}^{9} \\left( abc + ab + bc + ca + a + b + c \\right). \... | 215 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven that the base-$17$ integer $\\overline{8323a02421_{17}}$ (where $a$ is a base-$17$ digit) is divisible by $\\overline{16... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all positive integers $n$ that have 4 digits, all of which are perfect squares, and such that $n$ is divisible by 2, 3, 5... | 4410 |
[
{
"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 $x$ is a real number, let $\\lfloor x \\rfloor$ be the greatest integer that is less than or equal to $x$. If $n$ is a posi... | 108 |
[
{
"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 subsets of $\\{1, 3, 5, 7, 9, 11, 13, 15, 17, 19\\}$ where the sum of the elements in the subset is $49$.\n... | 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\nFind the smallest positive integer $n$ such that $\\underbrace{2^{2^{\\cdots^{2}}}}_{n} > 3^{3^{3^3}}$. \n\n(The notation $\\u... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $S$ be the set of numbers of the form $n^5 - 5n^3 + 4n$, where $n$ is an integer that is not a multiple of $3$. What is th... | 360 |
[
{
"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 rectangle with $AB=10$ and $BC=26$. Let $\\omega_1$ be the circle with diameter $\\overline{AB}$ and $\\omega_... | 26 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nKris is asked to compute $\\log_{10} (x^y)$, where $y$ is a positive integer and $x$ is a positive real number. However, they ... | 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\nLet $s_1, s_2, s_3$ be the three roots of the polynomial $x^3 + x^2 + \\frac{9}{2}x + 9$.\n\nFind the value of $a + b + c$ if ... | 16 |
[
{
"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 $n$, an $n$-branch $B$ is an ordered tuple $(S_1, S_2, \\dots, S_m)$ of nonempty sets (where $m$ is any... | 76 |
[
{
"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 that $20^{22} + 1$ has exactly four prime divisors $p_1 < p_2 < p_3 < p_4$, determine $p_1 + p_2$.\n\nRemember to put yo... | 490 |
[
{
"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 ants crawl along the sides of the $49$ squares of a $7 \\times 7$ board. Each ant passes through all $64$ vertices exactly... | 16 |
[
{
"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\nClaire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of $2017$. She then discovers ... | 143 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nNeo has an infinite supply of red pills and blue pills. When he takes a red pill, his weight will double, and when he takes a ... | 13 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\alpha$ and $\\beta$ be positive integers such that $\\frac{43}{197} < \\frac{\\alpha}{\\beta} < \\frac{17}{77}$. Find t... | 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\nA two-digit positive integer is $\\textit{primeable}$ if one of its digits can be deleted to produce a prime number. A two-dig... | 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 $f(n)$ denote the largest odd factor of $n$, including possibly $n$ itself. Determine the value of \\[ \\frac{f(1)}{1} + \... | 1365 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nConsider the following three lines in the Cartesian plane:\n\n\\[\n\\begin{cases}\n\\ell_1: & 2x - y = 7\\\\\n\\ell_2: & 5x + ... | 260 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many ways are there to fill in a $2 \\times 2$ square grid with the numbers $1, 2, 3,$ and $4$ such that the numbers in an... | 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 $Answer is the answer to the problem.\n\nA multiple choice test consists of 100 questions. If a student answers a question correctly, he will get 4 marks; if he answer... | 495 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTriangle $ABC$ satisfies $AB=104$, $BC=112$, and $CA=120$. Let $\\omega$ and $\\omega_A$ denote the incircle and $A$-excircle ... | 49 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEvaluate \\( \\frac{1}{\\int _0^{\\frac{\\pi}{2}} \\cos ^{2006}x \\cdot \\sin 2008 x\\, dx} \\).\n\nRemember to put your answe... | 2007 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the smallest positive integer $k$ such that there is exactly one prime number of the form $kx + 60$ for the integers $0 \... | 17 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn a tournament, each player played exactly one game against each of the other players. In each game, the winner was awarded $... | 25 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet \\[ f(x) = \\frac{1}{1 - \\frac{1}{1-x}} \\]. Compute \\( f^{2016}(2016) \\), where \\( f \\) is composed upon itself 2016... | 2016 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe integers from 1 to $2008^2$ are written on each square of a $2008 \\times 2008$ board. For every row and column, the diffe... | 16184704896 |
[
{
"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 digits $1$, $4$, $9$, and $2$ are each used exactly once to form a $4$-digit number $N$. What is the sum of all possible v... | 106656 |
[
{
"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 $1000 \\leq n = \\text{ABCD}_{10} \\leq 9999$ be a positive integer whose digits $\\text{ABCD}$ satisfy the divisibility c... | 1729 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many ways are there to color the edges of a hexagon orange and black, assuming that two hexagons are indistinguishable if ... | 14 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $z$ be a complex number satisfying $(z+\\frac{1}{z})(z+\\frac{1}{z}+1)=1$. Evaluate $(3z^{100}+\\frac{2}{z^{100}}+1)(z^{10... | 36 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nYou and your friend play a game on a $7 \\times 7$ grid of buckets. Your friend chooses $5$ \"lucky\" buckets by marking an \"... | 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 $a_n$ be the last digit of the sum of the digits of the number formed by repeating the block $2005$ exactly $n$ times. Cal... | 9025 |
[
{
"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\nFrist Campus Center is located $1$ mile north and $1$ mile west of Fine Hall. The area within $5$ miles of Fine Hall that is l... | 30 |
[
{
"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 directed graph with $2013$ vertices, there is exactly one edge between any two vertices, and for every vertex, there exis... | 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\nFor any positive integer $n$, let $r_n$ denote the greatest odd divisor of $n$. Compute the sum:\n\\[ T = r_{100} + r_{101} + ... | 10025 |
[
{
"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 $x$, $y$, $z$ be positive integers satisfying $x < y < z$ and $x + xy + xyz = 37$. Find the greatest possible value of $x ... | 20 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe diagram below shows a parallelogram $ABCD$ with $AB = 36$ and $AD = 60$. Diagonal $BD$ is perpendicular to side $AB$. Poin... | 288 |
[
{
"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 real number $\\alpha > 0$, consider the infinite real sequence defined by $x_1 = 1$ and \n\\[ \\alpha x_n = x_1 + x_2 + ... | 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\nIn the $xy$-coordinate plane, the $x$-axis and the line $y=x$ are mirrors. If you shoot a laser beam from the point $(126, 21)... | 111 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $n = 2^8 \\cdot 3^9 \\cdot 5^{10} \\cdot 7^{11}$. For $k$ a positive integer, let $f(k)$ be the number of integers $0 \\le... | 24 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many ordered pairs of positive integers $(m, n)$ satisfy the system:\n\n\\[\n\\begin{align*}\n\\gcd (m^3, n^2) & = 2^2 \\c... | 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\nDefine a sequence $a_i$ as follows: $a_1 = 181$ and for $i \\ge 2$, \\( a_i = a_{i-1}^2 - 1 \\) if $a_{i-1}$ is odd and \\( a_... | 105 |
[
{
"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 degree $n$ monic polynomial with integer coefficients such that $2013$ divides $P(r)$ for exactly $1000$ v... | 50 |
[
{
"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 lattice points are exactly twice as close to $(0,0)$ as they are to $(15,0)$? (A lattice point is a point $(a,b)$ suc... | 12 |
[
{
"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 $100$ lightbulbs $B_1, B_2, \\ldots, B_{100}$ spaced evenly around a circle in this order. Additionally, there are $... | 50 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSita and Geeta are two sisters. If Sita's age is written after Geeta's age, a four-digit perfect square number is obtained. If... | 55 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the number arrangement:\n\\[\n\\begin{array}{ccccc}\n\\texttt{1}&&&&\\\\\n\\texttt{2}&\\texttt{3}&&&\\\\\n\\texttt{4}&\\tex... | 2073 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor every integer $n \\ge 1$, the function $f_n : \\{ 0, 1, \\cdots, n \\} \\to \\mathbb{R}$ is defined recursively by $f_n(0)... | 26 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA semicircle with a radius of $5$ and a quarter circle with a radius of $8$ are tangent to each other and are situated inside ... | 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\nFind the largest positive integer $n$ such that $\\sigma(n) = 28$, where $\\sigma(n)$ is the sum of the divisors of $n$, inclu... | 12 |
[
{
"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 quadrilateral is given where the lengths of two sides are $1$ and $4$. Additionally, a diagonal of length $2$ divides the qu... | 11 |
[
{
"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 the $10$-digit number $M = 9876543210$. We obtain a new $10$-digit number from $M$ according to the following rule: w... | 88 |
[
{
"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 integer $n \\ge 4$ and a sequence of real numbers $x_1, x_2, x_3, \\ldots, x_n$. An operation consists of eliminat... | 683 |
[
{
"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 $ABC$ have side lengths $AB = 13$, $BC = 14$, $AC = 15$. Let $I$ be the incenter of $\\triangle ABC$. The circle ... | 17 |
[
{
"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 $\\angle BAC = 117^\\circ$. The angle bisector of $\\angle ABC$ intersects side $AC$ at $D$. Supp... | 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\nIn an equilateral triangle $ABC$, the midpoint of $\\overline{BC}$ is $M$. If the circumcircle of triangle $MAB$ has an area o... | 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\nFind how many multiples of 360 are of the form $\\overline{ab2017cd}$, where $a$, $b$, $c$, $d$ are digits, with $a > 0$.\n\nR... | 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\nLet $\\overline{abc}$ be a three-digit number with nonzero digits such that $a^2 + b^2 = c^2$. What is the largest possible pr... | 29 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many triples \\((a,b,c)\\) of positive integers strictly less than 51 are there such that \\(a+b+c\\) is divisible by \\(a... | 221 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many positive integers $k$ are there such that $\\frac{k}{2013}(a+b) = \\text{lcm}(a,b)$ has a solution in positive intege... | 1006 |
[
{
"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 $n$, let $a_n$ be the integer consisting of $n$ digits of $9$ followed by the digits $488$. For example... | 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\nDetermine the least possible value of $f(1998)$, where $f:\\mathbb{N} \\to \\mathbb{N}$ is a function such that for all $m,n \... | 120 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn a $4 \\times 4$ table, we place $k$ blocks such that:\n\n1. Each block covers exactly 2 cells.\n2. Each cell is covered by ... | 12 |
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{
"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 decimal digits of a natural number $A$ form an increasing sequence (from left to right). Find the sum of the digits of $9A... | 9 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nMaryssa, Stephen, and Cynthia played a game. Each of them independently privately chose one of Rock, Paper, and Scissors at ra... | 916 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn triangle $ABC$, the angle $C$ is obtuse, $m(\\angle A) = 2m(\\angle B)$, and the side lengths are integers. Find the smalle... | 77 |
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{
"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 people are gathered in a meeting. Some pairs of people shake hands. An ordered triple of people $(A,B,C)$ is called a *tr... | 10 |
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{
"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\nMilo rolls five fair dice, which have 4, 6, 8, 12, and 20 sides respectively, and each one is labeled $1$ to $n$ for the appro... | 248 |
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{
"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 given sequence $\\{S_1,S_2,...,S_k\\}$, for terms $n\\ge3$, the relationship is defined as:\n$$ S_n=\\sum_{i=1}^{n-1} i\\... | -2 |
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{
"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 lattice points $(v, w, x, y, z)$ does a 5-sphere centered on the origin, with radius 3, contain on its surface or in ... | 1343 |
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"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 the sequence $2, 3, 5, 6, 7, 8, 10, \\ldots$ of all positive integers that are not perfect squares. Determine the $20... | 2056 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor all positive integers $k$, define $f(k) = k^2 + k + 1$. Compute the largest positive integer $n$ such that:\n\\[\n2015 \\c... | 44 |
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{
"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 positive integer $n \\le 2004$ such that $3^{3n+3} - 27$ is divisible by $169$.\n\nRemember to put your answe... | 2002 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFive members of the Lexington Math Team are sitting around a table. Each flips a fair coin. Given that the probability that th... | 43 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe Matini company released a special album with the flags of the $12$ countries that compete in the CONCACAM Mathematics Cup.... | 3 |
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{
"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 $n$, let $f(n)$ be the sum of the positive integers that divide at least one of the nonzero base $10$ d... | 15 |
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{
"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 diagram shows a regular pentagon $ABCDE$ and a square $ABFG$. Find the degree measure of $\\angle FAD$.\n\nRemember to put... | 18 |
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{
"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 $n$ measured in degrees, let $T(n) = \\cos^2(30^\\circ -n) - \\cos(30^\\circ -n)\\cos(30^\\circ +n) + \\cos^2(30^\\circ +n... | 1395 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the least positive integer $n$ such that the polynomial $P(X) = \\sqrt{3} \\cdot X^{n+1} - X^n - 1$ has at least one root... | 10 |
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