prompt listlengths 1 1 | label stringlengths 1 210 |
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[
{
"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 positive integer $m$ is called perfect if the sum of all its positive divisors, including $1$ and $m$, e... | 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 the number of ordered triples of nonnegative integers $(a, b, c)$ that satisfy:\n\\[\n(ab + 1)(bc + 1... | 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\nOn rectangular coordinates, point $A = (1,2)$ and point $B = (3,4)$. Point $P = (a, 0)$ is on the $x$-axis... | 100 |
[
{
"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\nIf integers $m$, $n$, and $k$ satisfy the equation $m^2 + n^2 + 1 = kmn$, what values can $k$ have?\n\nRem... | 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 the greatest exponent $k$ for which $2001^k$ divides $2000^{2001^{2002}} + 2002^{2001^{2000}}$.\n\nRe... | 2001 |
[
{
"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 the function $f(x) = x^3 - 6x^2 + 17x$, if $f(a) = 16$ and $f(b) = 20$, find the value of $a + b$.\n... | 4 |
[
{
"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 $a$ is a real number such that $3a + 6$ is the greatest integer less than or equal to $a$ and $4a ... | -3 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTwo circles, both with the same radius $r$, are placed in the plane without intersecting each other. A lin... | 13 |
[
{
"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 $Answer is the answer to the problem.\n\nA table tennis club hosts a series of doubles matches following several rules:\n1. Each player belongs to ... | 6 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAssume the earth is a perfect sphere with a circumference of $60$ units. A great circle is a circle on a s... | 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\nOn a table, there are $100$ red and $k$ white buckets, all initially empty. In each move, a red and a whit... | 100 |
[
{
"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 $Answer is the answer to the problem.\n\n$n$ consecutive positive integers are arranged in a row (not necessarily in order) such that the sum of an... | 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\nTo each positive integer $n$, assign a non-negative integer $f(n)$ such that the following conditions are ... | 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\nFind all the ways of placing the integers $1, 2, 3, \\ldots, 16$ in the boxes below, such that each intege... | 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\nFind all integers $n$ such that $n^4 + 8n + 11$ is a product of two or more consecutive integers.\n\nRemem... | 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 $Answer is the answer to the problem.\n\nSeven students in Princeton Juggling Club are searching for a room to meet in. However, they must stay at ... | 12 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThree identical circles are packed into a unit square. Each of the three circles is tangent to each other ... | 2 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn an abandoned chemistry lab, Gerome found a two-pan balance scale and three 1-gram weights, three 5-gram... | 129 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the largest $k$ such that for every positive integer $n$, there exist at least $k$ numbers in the set... | 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\nFor integers $0 \\le m,n \\le 2^{2017}-1$, let $\\alpha(m,n)$ be the number of nonnegative integers $k$ fo... | 1382 |
[
{
"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 are the last two digits of $2^{3^{4^{\\cdots^{2019}}}}$?\n\nRemember to put your answer on its own li... | 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 $Answer is the answer to the problem.\n\nDefine a \\textit{subsequence} of a string \\( \\mathcal{S} \\) of letters to be a positive-length string ... | 255 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nPositive integers $a$, $b$, and $c$ are all powers of $k$ for some positive integer $k$. It is known that ... | 64 |
[
{
"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 $x$, $y$, and $z$ are positive real numbers satisfying the following system of equations:\n\n... | 69 |
[
{
"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 $Answer is the answer to the problem.\n\nFind the number of ordered pairs of integers \\((p, q)\\) satisfying the equation \\(p^2 - q^2 + p + q = 2... | 16 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWe call a pair $(a,b)$ of positive integers a \"pupusa\" if $a < 391$ and \\(\\text{lcm}(a,b) > \\text{lcm... | 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\nLet $x$ be a real number in the interval $(0, \\frac{\\pi}{2})$ such that $\\frac{1}{\\sin x \\cos x} + 2\... | 8 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f(x) = (x^4 + 2x^3 + 4x^2 + 2x + 1)^5$. Compute the prime $p$ satisfying $f(p) = 418,195,493$.\n\nRem... | 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 number of all ordered triplets of positive integers $(a, b, c)$, which satisfy the equalitie... | 70 |
[
{
"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 $Answer is the answer to the problem.\n\nA cross-pentomino is a shape that consists of a unit square and four other unit squares, each sharing a di... | 250 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the smallest multiple of $2019$ that is of the form $abcabc\\dots abc$, where $a$, $b$, and $c$ are d... | 673673673 |
[
{
"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\nChip and Dale play the following game. Chip starts by splitting $222$ nuts between two piles, so Dale can ... | 111 |
[
{
"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 positive integers $n$, let $c_n$ be the smallest positive integer for which $n^{c_n} - 1$ is divisible... | 329 |
[
{
"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 $Answer is the answer to the problem.\n\nHow many ordered pairs of integers \\((x, y)\\) are there such that\n\\[ 0 < \\left\\vert xy \\right\\vert... | 524 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many distinct permutations of the letters in the word REDDER are there that do not contain a palindrom... | 6 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe numbers $\\frac{1}{1}, \\frac{1}{2}, \\ldots, \\frac{1}{2010}$ are written on a blackboard. A student ... | 2010 |
[
{
"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 such that all but $2009$ positive integers are expressible in the for... | 133 |
[
{
"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\nAn isosceles trapezoid $ABCD$ with bases $AB$ and $CD$ has $AB=13$, $CD=17$, and height $3$. Let $E$ be th... | 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 $Answer is the answer to the problem.\n\nAsheville, Bakersfield, Charter, and Darlington are four small towns along a straight road in that order. ... | 72 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the sum of all possible values of $\\cos(2\\theta)$ if $\\cos(2\\theta) = 2\\cos(\\theta)$ for a r... | 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\nSuppose that $ABC$ is a triangle with $AB = 6$, $BC = 12$, and $\\angle B = 90^{\\circ}$. Point $D$ lies o... | 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\nThe digits of a positive integer $n$ are four consecutive integers in decreasing order when read from left... | 217 |
[
{
"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\nNine positive integers $a_1, a_2, \\ldots, a_9$ have their last two-digit parts equal to $11, 12, 13, 14, ... | 85 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $P(x)$ be a polynomial with integer coefficients, leading coefficient 1, and $P(0) = 3$. If the polyno... | 133 |
[
{
"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 $ABC$ be a triangle with $AB=4$, $AC=5$, $BC=6$, and circumcircle $\\Omega$. Points $E$ and $F$ lie on... | 1763 |
[
{
"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\nSquare $A$ is adjacent to square $B$, which is adjacent to square $C$. The three squares all have their bo... | 54 |
[
{
"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 $n \\geq 1$ be a positive integer. We say that an integer $k$ is a fan of $n$ if $0 \\leq k \\leq n-1$... | 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\nFor a set $S \\subseteq \\mathbb{N}$, define $f(S) = \\{\\lceil \\sqrt{s} \\rceil \\mid s \\in S\\}$. Find... | 5043 |
[
{
"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\nProblem 1: The set $S = \\{1, 2, 3, \\ldots, 2006\\}$ is partitioned into two disjoint subsets $A$ and $B$... | 154 |
[
{
"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\nIt is given a positive real number $a$ such that: \n$$ \\left\\{ \\frac{1}{a} \\right\\} = \\{ a^2 \\} $$\... | 233 |
[
{
"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 remainder when $2024^{2023^{2022^{2021 \\ldots^{3^{2}}}}} + 2025^{2021^{2017^{2013 \\ldots^{5^{1}... | 17 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nConsider an isosceles triangle $ABC$ with side lengths $AB = AC = 10\\sqrt{2}$ and $BC = 10\\sqrt{3}$. Con... | 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\nAt a round table, there are $10$ students. Each student thinks of a number and communicates this number to... | 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\nWhat is the smallest integer $n$, greater than one, for which the root-mean-square of the first $n$ positi... | 337 |
[
{
"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 rectangular pool table has vertices at $(0, 0)$, $(12, 0)$, $(0, 10)$, and $(12, 10)$. There are pockets... | 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\nLet $S$ be the sum of all real $x$ such that $4^x = x^4$. Find the nearest integer to $S$.\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\nLet $m$ and $n$ be positive integers. Fuming Zeng gives James a rectangle, such that $m-1$ lines are drawn... | 1289 |
[
{
"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 maximum number of permutations of the set $\\{1, 2, 3, \\ldots, 2014\\}$ such that for every two ... | 2014 |
[
{
"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 $a, b, c > 0$ are integers such that:\n\\[\nabc - bc - ac - ab + a + b + c = 2013.\n\\]\nFind the ... | 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 $Answer is the answer to the problem.\n\nFind the number of strictly increasing sequences of nonnegative integers with the following properties:\n\... | 144 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f(x) = 1 + x + x^2 + \\cdots + x^{100}$. Find $f'(1)$.\n\nRemember to put your answer on its own line... | 5050 |
[
{
"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\nAlice picks a number uniformly at random from the first $5$ even positive integers, and Palice picks a num... | 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\nLet $S = \\{ A = (a_1, \\ldots, a_s) \\mid a_i = 0 \\text{ or } 1, \\ i = 1, \\ldots, 8 \\}$. For any two ... | 4 |
[
{
"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 $Answer is the answer to the problem.\n\nOne writes 268 numbers around a circle, such that the sum of 20 consecutive numbers is always equal to 75.... | -1 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nOn a clock, there are two instants between $12$ noon and $1\\,\\mathrm{PM}$ when the hour hand and the min... | 51 |
[
{
"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 sequence of integers $x_1, x_2, \\ldots$ is called \"double-dipped\" if it satisfies the recurrence rela... | 48 |
[
{
"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 $Answer is the answer to the problem.\n\nEli, Joy, Paul, and Sam want to form a company; the company will have 16 shares to split among the 4 peopl... | 315 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the number of rearrangements of the letters in the word \"MATHMEET\" that begin and end with the same... | 540 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTwo circles have radii $15$ and $95$. If the two external tangents to the circles intersect at $60$ degree... | 160 |
[
{
"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 a positive integer $n$, let $S(n)$ be the sum of its decimal digits. Determine the smallest positive i... | 14499 |
[
{
"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 positive integer $n$ such that $n^2$ is the perfect square closest to the sum $8 + 16 + 24 + \\cd... | 2011 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f(n)$ be a sequence of integers defined by $f(1) = 1$, $f(2) = 1$, and $f(n) = f(n-1) + (-1)^n f(n-2)... | 89 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\triangle ABC$ be a triangle with $BC = 4$, $CA= 5$, $AB= 6$, and let $O$ be the circumcenter of $\\... | 23 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nJerry's favorite number is $97$. He knows all kinds of interesting facts about $97$:\n\n- $97$ is the larg... | 96 |
[
{
"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 number of ordered quadruples $(a,b,c,d)$ of integers, where $a,b,c,d$ are integers between $... | 66 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe summation \\( \\sum_{k=1}^{360} \\frac{1}{k \\sqrt{k+1} + (k+1)\\sqrt{k}} \\) is the ratio of two rela... | 37 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $A$ be the set of positive integers that have no prime factors other than $2$, $3$, or $5$. The infini... | 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 $Answer is the answer to the problem.\n\nThe trapezoid below has bases with lengths 7 and 17, and an area of 120. Find the difference in the areas ... | 50 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWe say that an ordered pair $(a,b)$ of positive integers with $a > b$ is square-ish if both $a + b$ and $a... | 16 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose $a$ is a real number such that $\\sin(\\pi \\cdot \\cos a) = \\cos(\\pi \\cdot \\sin a)$. Evaluate... | 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\nLet $\\underline{xyz}$ represent the three-digit number with hundreds digit $x$, tens digit $y$, and units... | 120 |
[
{
"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 largest positive integer $N$ such that the number of integers in the set $\\{1, 2, \\dots, N\\}$ ... | 65 |
[
{
"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$$n = a^2 + b^2 + c^2 + d^2,$$\nwhere $a < b < c < d$ are the s... | 130 |
[
{
"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 spider is walking on the boundary of an equilateral triangle $\\triangle{ABC}$, with vertices labeled in... | 625 |
[
{
"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\nCompute the sum of the digits of $101^6$.\n\nRemember to put your answer on its own line after \"Answer:\"... | 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 sum of all values of $a$ such that there are positive integers $a$ and $b$ satisfying $(a - b)\\s... | 209 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCircle $\\Gamma$ is centered at $(0, 0)$ in the plane with radius $2022\\sqrt{3}$. Circle $\\Omega$ is cen... | 1349 |
[
{
"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 take $100$ consecutive natural numbers $a_1, a_2, \\ldots, a_{100}$. Determine the last two digits of t... | 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\nConsider the polynomial $f(x) = ax^2 + bx + c$, with degree less than or equal to 2.\n\nWhen $f$ varies su... | 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 sum of all the real values of $x$ satisfying the equation:\n\\[(x+\\frac{1}{x}-17)^2 = x + \\frac... | 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 $Answer is the answer to the problem.\n\nFive girls and five boys randomly sit in ten seats that are equally spaced around a circle. The probabilit... | 118 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many positive integers $N$ in the segment $\\left[10, 10^{20} \\right]$ are such that if all their dig... | 171 |
[
{
"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 $Answer is the answer to the problem.\n\nThe product of a million whole numbers is equal to one million. What can be the greatest possible value of... | 1999999 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe leading coefficient $a$ in the quadratic polynomial $P(x) = ax^2 + bx + c$ is greater than $100$. Dete... | 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 $Answer is the answer to the problem.\n\nA grasshopper starts at the origin in the coordinate plane and makes a sequence of hops. Each hop has a le... | 578 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\ell$ be a line and let points $A$, $B$, $C$ lie on $\\ell$ so that $AB = 7$ and $BC = 5$. Let $m$ b... | 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 clueless ant makes the following route: starting at point $A$, it goes $1$ cm north, then $2$ cm east, t... | 29 |
[
{
"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\nThe trapezoid $ABCD$ has bases with lengths $7$ and $17$ and an area of $120$. Find the difference of the ... | 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 $Answer is the answer to the problem.\n\nAlice and Bob are independently trying to figure out a secret password to Cathy’s bitcoin wallet. Both of ... | 5940 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSquares $ABCD$ and $AEFG$, each with side length $12$, overlap such that $\\triangle AED$ is an equilatera... | 51 |
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