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\nConvex quadrilateral $ABCD$ satisfies the following conditions: \\( \\angle{CAB} = \\angle{ADB} = 30^{\\circ} \\), \\( \\angle... | 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\nFind the number of integers $n$ such that $1 \\le n \\le 2017$ and $(n-2)(n-0)(n-1)(n-7)$ is a multiple of $1001$.\n\nRemember... | 99 |
[
{
"content": "Solve the following math problem step by 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 integer $$\\frac{2^{\\left(2^5-2\\right)/5-1}-2}{5}.$$\n\nRemember to put your answer on its own line after \"Answ... | 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\nFind $100m+n$ if $m$ and $n$ are relatively prime positive integers such that \n\\[ \\sum_{\\substack{i,j \\ge 0 \\\\ i+j \\te... | 504 |
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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\nA sequence $a_n$ of real numbers satisfies the following conditions:\n\n- $a_1 = 1$\n- $a_2 = 0$\n- $a_n = (S_{n-1} + 1) S_{n-... | 10 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet \\( \\gamma = \\alpha \\times \\beta \\) where \\[ \\alpha = \\underbrace{999 \\cdots 9}_{2010 \\text{ '9's}} \\] and \\[ ... | 18090 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many positive five-digit integers have the product of their five digits equal to $900$?\n\nRemember to put your answer on ... | 210 |
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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 $Answer is the answer to the problem.\n\nA tailor met a tortoise sitting under a tree. When the tortoise was the tailor's age, the tailor was only a quarter of his cur... | 77 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the largest positive integer $n$ for which there are no positive integers $a, b$ such that $8a + 11b = n$?\n\nRemember... | 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 $Answer is the answer to the problem.\n\n(MON 1) Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits fr... | 1944 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe function $f: \\mathbb{N} \\to \\mathbb{N}_0$ is defined such that:\n- $f(2) = 0$\n- $f(3) > 0$\n- $f(6042) = 2014$\n- For ... | 671 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a, b, c, d$ be four positive integers such that $a > b > c > d$. Given that $ab + bc + ca + d^2 \\mid (a + b)(b + c)(c + ... | 3 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTriangle $ABC$ has $AB = 3$, $BC = 4$, and $CA = 5$. Points $D$, $E$, $F$, $G$, $H$, and $I$ are the reflections of $A$ over $... | 78 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA four-digit number has the following properties:\n1. The units digit equals the tens digit increased by $1$.\n2. The hundreds... | 9634 |
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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 $Answer is the answer to the problem.\n\nWe record one number every workday: Monday, Tuesday, Wednesday, Thursday, and Friday. On the first Monday, the number we recor... | 252 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many ways are there to write $657$ as a sum of powers of two, where each power of two is used at most twice in the sum? Fo... | 41 |
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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 $Answer is the answer to the problem.\n\n$A, B, C$ are points in the plane such that $\\angle ABC = 90^\\circ$. Circles with diameters $BA$ and $BC$ meet at $D$. If $B... | 449 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA $3 \\times 3$ determinant has three entries equal to $2$, three entries equal to $5$, and three entries equal to $8$. Find t... | 405 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the 2000th positive integer that is not the difference between any two integer squares.\n\nRemember to put your answer on... | 7998 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFive bricklayers working together finish a job in $3$ hours. Working alone, each bricklayer takes at most $36$ hours to finish... | 270 |
[
{
"content": "Solve the following math problem step by 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 two parallel lines $a$ and $b$. The circles $C, C_1$ are tangent to each other and to the line $a$. The circles $C, C... | 24 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe circle centered at point $A$ with radius $19$ and the circle centered at point $B$ with radius $32$ are both internally ta... | 140 |
[
{
"content": "Solve the following math problem step by 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 a sequence using all the digits from $1$ to $9$ without repetition, such that any two consecutive digits form a number di... | 784913526 |
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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\nLet $ABCD$ be a cyclic quadrilateral with circumcircle $\\omega$ and let $AC$ and $BD$ intersect at $X$. Let the line through ... | 421 |
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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 $Answer is the answer to the problem.\n\nReimu has a wooden cube. In each step, she creates a new polyhedron from the previous one by cutting off a pyramid from each v... | 974 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the least natural number $n$, which has at least 6 different divisors $1 = d_1 < d_2 < d_3 < d_4 < d_5 < d_6 < \\ldots$, ... | 494 |
[
{
"content": "Solve the following math problem step by 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 is counting up by fives, starting with the number $3$. Meanwhile, Bob is counting down by fours, starting with the numbe... | 101 |
[
{
"content": "Solve the following math problem step by 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 two of its sides have lengths of $1$ and $4$. Additionally, a diagonal of length $2$ divides t... | 11 |
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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\nDetermine the last (rightmost) three decimal digits of $n$ where:\n\\[\nn=1 \\times 3 \\times 5 \\times 7 \\times \\ldots \\ti... | 875 |
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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\nLet $N \\ge 5$ be given. Consider all sequences $(e_1,e_2,...,e_N)$ with each $e_i$ equal to $1$ or $-1$. Per move, one can ch... | 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\nFind the largest positive integer value of $n < 1000$ such that $\\phi(36n) = \\phi(25n) + \\phi(16n)$, where $\\phi(n)$ denot... | 996 |
[
{
"content": "Solve the following math problem step by 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 $\\angle ABC = 76^\\circ$ and $\\angle ACB = 72^\\circ$. Points $P$ and $Q$ lie on the... | 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\nWhat is the remainder when $128^{2023}$ is divided by $126$?\n\nRemember to put your answer on its own line after \"Answer:\".... | 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 $0 < \\theta < 2\\pi$ be a real number for which $\\cos (\\theta) + \\cos (2\\theta) + \\cos (3\\theta) + \\cdots + \\cos ... | 1926 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nToner Drum and Celery Hilton are both running for president. A total of $2015$ people cast their vote, giving $60\\%$ to Toner... | 605 |
[
{
"content": "Solve the following math problem step by 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 $a_1$, $a_2$, $a_3$, \\ldots$ is chosen so that $a_n = a_{n - 1} - a_{n - 2}$ for each $n \\ge 3$. \n\... | 986 |
[
{
"content": "Solve the following math problem step by 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. If the equation \\[x^3 + (4-i)x^2 + (2+5i)x = z\\] has two roots that form a conjugate pair, find... | 423 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many 6-tuples \\((a_1,a_2,a_3,a_4,a_5,a_6)\\) are there such that each of \\(a_1,a_2,a_3,a_4,a_5,a_6\\) is from the set \\... | 40 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nMr. Rose, Mr. Stein, and Mr. Schwartz start at the same point around a circular track and run clockwise. Mr. Stein completes e... | 45 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEach square on an $8 \\times 8$ checkers board contains either one or zero checkers. The number of checkers in each row is a m... | 30 |
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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 $Answer is the answer to the problem.\n\nFind the sum of \\( \\frac{\\sigma(n) \\cdot d(n)}{ \\phi (n)} \\) over all positive \\( n \\) that divide \\( 60 \\).\n\n**No... | 350 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf $a$ and $b$ are natural numbers such that $a + 13b$ is divisible by $11$ and $a + 11b$ is divisible by $13$, then find the ... | 28 |
[
{
"content": "Solve the following math problem step by 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 $256$ players in a tennis tournament ranked from $1$ to $256$, where $1$ is the highest rank and $256$ is the lowest... | 103 |
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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 trapezoid $ABCD$, where $AD \\parallel BC$ and $\\angle ABC + \\angle CDA = 270^{\\circ}$, compute $AB^2$ given the followi... | 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\nLet $a_1 = 11^{11}$, $a_2 = 12^{12}$, $a_3 = 13^{13}$, and \n\\[a_n = |a_{n-1} - a_{n-2}| + |a_{n-2} - a_{n-3}|, \\quad n \\ge... | 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 $Answer is the answer to the problem.\n\nA BWM tree is defined recursively:\n\n- An empty tree is a BWM tree of height 0 and size 0.\n- A nonempty BWM tree consists of... | 154 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe polynomial $R(x)$ is the remainder when $x^{2007}$ is divided by $x^2 - 5x + 6$. Express $R(0)$ in the form $ab(a^c - b^c)... | 2005 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEach number in the list $1, 2, 3, \\ldots, 10$ is either colored red or blue. Numbers are colored independently, and both colo... | 455 |
[
{
"content": "Solve the following math problem step by 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 $s_n$ be the number of solutions to the equation \\(a_1 + a_2 + a_3 + a_4 + b_1 + b_2 = n\\), where \\(a_1, a_2, a_3,\\) a... | 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\nWhat are the last two digits of $2017^{2017}$?\n\nRemember to put your answer on its own line after \"Answer:\".",
"role":... | 77 |
[
{
"content": "Solve the following math problem step by 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 $n = \\left\\lfloor \\frac{k^3}{2012} \\right\\rfloor$, where $\\lfloor x \\rfloor$ denote... | 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\nA subset $B$ of the set of the first $100$ positive integers has the property that no two elements of $B$ sum to $125$. What i... | 62 |
[
{
"content": "Solve the following math problem step by 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 $k$ for which there exists a convex polyhedron $\\mathcal{P}$ with 2022 edges, which satisfi... | 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\nCall a positive integer $n \\geq 2$ \"junk\" if there exist two distinct $n$-digit binary strings $a_1a_2\\cdots a_n$ and $b_1... | 672 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe three roots of the cubic $30x^3 - 50x^2 + 22x - 1$ are distinct real numbers between $0$ and $1$. For every nonnegative in... | 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\nLet $ABCD$ be a parallelogram with an area of 160. The diagonals $AC$ and $BD$ intersect at point $E$. Point $P$ is on the lin... | 62 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the sum of all positive integers $n$ with the following properties:\n\n- $n$ is not divisible by any primes larger than $... | 416 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a$ and $b$ be complex numbers such that $(a+1)(b+1)=2$ and $(a^2+1)(b^2+1)=32.$ Compute the sum of all possible values of... | 1160 |
[
{
"content": "Solve the following math problem step by 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 prime that divides $n^2 + 5n + 23$ for some integer $n$.\n\nRemember to put your answer on its own ... | 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\nFarmer John has a flexible fence of length $L$ and two straight walls that intersect at a corner perpendicular to each other. ... | 4000 |
[
{
"content": "Solve the following math problem step by 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 such that $\\gcd(m,n)=1$ and $$\\sum_{k=0}^{2020} (-1)^k {2020 \\choose k} \\cos(2020\\c... | 209601 |
[
{
"content": "Solve the following math problem step by 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 an equilateral triangle with side length $10$. A square $PQRS$ is inscribed in it, with $P$ on $AB$, $Q$ and $R$ ... | 10 |
[
{
"content": "Solve the following math problem step by 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 there are $160$ pigeons and $n$ holes. The $1$st pigeon flies to the $1$st hole, the $2$nd pigeon flies to the $4$th h... | 326 |
[
{
"content": "Solve the following math problem step by 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 value of an integer $B$ such that for every 9 distinct natural numbers with the sum of $2023$, there must exi... | 910 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a$, $b$, $c$ be positive reals for which\n\\[\n (a+b)(a+c) = bc + 2 \\\\\n (b+c)(b+a) = ca + 5 \\\\\n (c+a)(c+b) = ab ... | 4532 |
[
{
"content": "Solve the following math problem step by 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 inscribed in circle $\\Gamma$, and let $P$ be a point on the minor arc $AB$ of $\\Gamma$. Suppose th... | 21055 |
[
{
"content": "Solve the following math problem step by 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, m$ be positive integers of different parity, and $n > m$. Find all integers $x$ such that $\\frac{x^{2^n} - 1}{x^{2^m}... | 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\nFind the number of positive integers less than 101 that cannot be written as the difference of two squares of integers.\n\nRem... | 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\nCompute \\( \\gcd \\left( \\frac{135^{90}-45^{90}}{90^2} , 90^2 \\right) \\)\n\nRemember to put your answer on its own line af... | 4050 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nS is the set of all $(a, b, c, d, e, f)$ where $a$, $b$, $c$, $d$, $e$, $f$ are integers such that $a^2 + b^2 + c^2 + d^2 + e^... | 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\nLet $ABC$ be an isosceles triangle such that $AB = AC$ and $\\angle A = 20^\\circ$. Let $M$ be the foot of the altitude from $... | 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\nDetermine the minimum value of a prime number $p > 3$ for which there is no natural number $n > 0$ such that $2^n + 3^n \\equi... | 19 |
[
{
"content": "Solve the following math problem step by 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 eight points $A, B, \\ldots, G, H$ lie on five circles as shown. Each of these letters is represented by one of the eight ... | 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\nLet $A$ and $B$ be two non-empty subsets of $X = \\{1, 2, \\ldots, 8 \\}$ with $A \\cup B = X$ and $A \\cap B = \\emptyset$. L... | 402 |
[
{
"content": "Solve the following math problem step by 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 less than $2015$. Suppose that the remainder when $2015$ is divided by $m$ is $n$. Compute the l... | 1007 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n12 knights are sitting at a round table. Each knight is an enemy with the two knights adjacent to them, but not with any of th... | 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\nThe regular octagon in the figure is inscribed in a circle with a radius of $1$. Let $P$ be an arbitrary point on this circle.... | 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\nLet $ABCD$ be a square face of a cube with edge length $2$. A plane $P$ that contains point $A$ and the midpoint of line segme... | 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\nIf we divide the number $19250$ by a certain number, the remainder is $11$. Similarly, if we divide the number $20302$ by the ... | 53 |
[
{
"content": "Solve the following math problem step by 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 there are 36 objects placed at equal distances along a circle. Determine the number of ways to choose 3 objects such t... | 5412 |
[
{
"content": "Solve the following math problem step by 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 city, there are $2019$ metro stations. Some pairs of stations are connected by tunnels, and from any station, you can rea... | 1008 |
[
{
"content": "Solve the following math problem step by 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 $k$, let $f_1(k)$ be the square of the sum of the digits of $k$. Define $f_{n+1} = f_1 \\circ f_n$. Eva... | 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\nWhat is the minimum possible perimeter of a right triangle with integer side lengths whose perimeter is equal to its area?\n\n... | 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\nCompute\n$$\n2022^{\\left(2022^{\\cdot ^ {\\cdot ^{\\cdot ^{\\left(2022^{2022}\\right)}}}}\\right)} \\pmod{111}\n$$\nwhere the... | 75 |
[
{
"content": "Solve the following math problem step by 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 $a$ and $b$ are positive real numbers such that:\n\\[\n3\\log_{101}\\left(\\frac{1,030,301-a-b}{3ab}\\right) = 3 ... | 0 |
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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 positive integers $a$ and $N$, let $r(a, N) \\in \\{0, 1, \\dots, N - 1\\}$ denote the remainder of $a$ when divided by $N... | 499500 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWe have a positive integer $n$, whose sum of digits is 100. If the sum of digits of $44n$ is 800, then what is the sum of digi... | 300 |
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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 number of real numbers $x$ that satisfy the equation $(3^x)^{x+2} + (4^x)^{x+2} - (6^x)^{x+2} = 1$.\n\nRemember to pu... | 4 |
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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\nLine $\\ell$ passes through point $A$ and into the interior of the equilateral triangle $ABC$. Points $D$ and $E$ are the orth... | 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\nLet $f(x)$ be the polynomial $\\prod_{k=1}^{50} \\bigl( x - (2k-1) \\bigr)$. Let $c$ be the coefficient of $x^{48}$ in $f(x)$.... | 60 |
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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\nTriangle $\\triangle ABC$ has $AB = 8$, $AC = 10$, and $AD = \\sqrt{33}$, where $D$ is the midpoint of $BC$. Perpendiculars ar... | 33 |
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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 $Answer is the answer to the problem.\n\n$A, B, C$, and $D$ are points on a circle, forming a convex quadrilateral $ABCD$. Given that $AB = 13$, $BC = 13$, $CD = 37$, ... | 504 |
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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 rodent control task force went into the woods one day and caught $200$ rabbits and $18$ squirrels. The next day, they caug... | 5491 |
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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 $Answer is the answer to the problem.\n\nFind the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each ... | 1944 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $n=p_1^{e_1}p_2^{e_2}\\dots p_k^{e_k}=\\prod_{i=1}^k p_i^{e_i}$, where $p_1<p_2<\\dots<p_k$ are primes and $e_1, e_2, \\do... | 35 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA subset $B$ of \\( \\{1, 2, \\dots, 2017\\} \\) is said to have property \\( T \\) if any three elements of \\( B \\) are the... | 1009 |
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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\nAn ant is at a vertex of a cube. Every $10$ minutes, it moves to an adjacent vertex along an edge. If $N$ is the number of one... | 74 |
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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 $Answer is the answer to the problem.\n\nAlice, Bob, Chad, and Denise decide to meet for a virtual group project between 1 and 3 PM, but they don't decide on a specifi... | 48 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the smallest positive integer $n$ such that for any $n$ distinct integers $a_1, a_2, \\dots, a_n$, the product of all dif... | 182 |
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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\nCompute the sum of all positive integers $n$ such that the median of the $n$ smallest prime numbers is $n$.\n\nRemember to put... | 25 |
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