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\nAdam has a box with $15$ pool balls numbered from $1$ to $15$. He picks out $5$ of them and sorts them in increasing order. He... | 990 |
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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 $ABC$ be an equilateral triangle with side length $1$. This triangle is rotated by some angle about its center to form tri... | 7 |
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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\nDefine a common chord between two intersecting circles as the line segment connecting their two intersection points. Let $\\om... | 96 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute the number of positive four-digit multiples of $11$ whose sum of digits (in base ten) is divisible by $11$.\n\nRemembe... | 72 |
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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 rectangle has side lengths $6$ and $8$. There are relatively prime positive integers $m$ and $n$ such that $\\frac{m}{n}$ is... | 203 |
[
{
"content": "Solve the following math problem step by 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 natural numbers $n$ less than $2015$ are divisible by $\\lfloor\\sqrt[3]{n}\\rfloor$?\n\nRemember to put your answer ... | 266 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nMarcus and four of his relatives are at a party. Each pair of the five people are either $\\textit{friends}$ or $\\textit{enem... | 52 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA polynomial $P$ is of the form $\\pm x^6 \\pm x^5 \\pm x^4 \\pm x^3 \\pm x^2 \\pm x \\pm 1$. Given that $P(2) = 27$, what is ... | 439 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA car has a 4-digit integer price, displayed in a digital format (similar to digital numbers on a watch). While the salesman i... | 6815 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the smallest integer $k > 1$ for which $n^k - n$ is a multiple of $2010$ for every positive integer $n$.\n\nRemember to p... | 133 |
[
{
"content": "Solve the following math problem step by 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 different integers that can be selected from the set $\\{1, 2, \\ldots, 2013\\}$ such that no two i... | 1010 |
[
{
"content": "Solve the following math problem step by 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 $n$ is considered lucky if the numbers $2n + 1$, $3n + 1$, and $4n + 1$ are all composite. Determine the sm... | 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 $\\alpha$ and $\\beta$ be real numbers. Find the least possible value of:\n\\[\n(2 \\cos \\alpha + 5 \\sin \\beta - 8)^2 +... | 100 |
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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 number $1995$ is divisible by both $19$ and $95$. How many four-digit numbers are there that are divisible by the two-digi... | 155 |
[
{
"content": "Solve the following math problem step by 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 $x_1, x_2, \\ldots, x_6$ be non-negative reals such that $x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 1$ and $x_1x_3x_5 + x_2x_4x_... | 559 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn each square of an $11 \\times 11$ board, we are to write one of the numbers $-1$, $0$, or $1$ in such a way that the sum of... | 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\nIf $xy:yz:zx = 6:8:12$, and the ratio $\\frac{x^3+y^3+z^3}{xyz}$ is $\\frac{m}{n}$, where $m$ and $n$ are relatively prime pos... | 41 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the sum of all integers between $-\\sqrt{1442}$ and $\\sqrt{2020}$.\n\nRemember to put your answer on its own line after ... | 287 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine \\[ \\left\\lfloor \\prod_{n=2}^{2022} \\frac{2n+2}{2n+1} \\right\\rfloor, \\] given that the answer is relatively p... | 29 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $S(M)$ denote the sum of the digits of a positive integer $M$ written in base $10$. Let $N$ be the smallest positive integ... | 18 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nUsing $600$ cards, where $200$ of them have the number $5$, $200$ have the number $2$, and the other $200$ have the number $1$... | 150 |
[
{
"content": "Solve the following math problem step by 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 alien script has $n$ letters $b_1, b_2, \\dots, b_n$. For some $k < \\frac{n}{2}$, assume that all words formed by any of t... | 600 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf there is only \\(1\\) complex solution to the equation \\(8x^3 + 12x^2 + kx + 1 = 0\\), what is \\(k\\)?\n\nRemember to put... | 6 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\triangle ABC$ be an isosceles triangle with $AB = AC$ and $BC = 14$. This triangle is inscribed in a circle $\\omega$. ... | 105 |
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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 number of nonempty subsets $S$ of the set $\\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\\}$ such that $\\frac{\\max S + \\min ... | 234 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA set $S$ of positive integers is \\textit{sum-complete} if there are positive integers $m$ and $n$ such that an integer $a$ i... | 223 |
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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\nWill has a magic coin that can remember previous flips. If the coin has already turned up heads $m$ times and tails $n$ times,... | 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\nDetermine the value of the series $-1 + 2 + 3 + 4 - 5 - 6 - 7 - 8 - 9 + \\ldots + 10000$, where the signs change after each pe... | 1000000 |
[
{
"content": "Solve the following math problem step by 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 $n$ positive integers greater than $1$ and less than $10000$ such that neither of them is prime but any two of them ar... | 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\nFive cards labeled $1, 3, 5, 7, 9$ are laid in a row in that order, forming the five-digit number $13579$ when read from left ... | 50308 |
[
{
"content": "Solve the following math problem step by 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 perfect square whose last digit is not $0$, such that when its last two digits are removed, the resulting nu... | 121 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nJoe is given a permutation $p = (a_1, a_2, a_3, a_4, a_5)$ of $(1, 2, 3, 4, 5)$. A swap is an ordered pair $(i, j)$ with $1 \\... | 125 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind $n$ such that $\\frac{1}{2!9!} + \\frac{1}{3!8!} + \\frac{1}{4!7!} + \\frac{1}{5!6!} = \\frac{n}{10!}$.\n\nRemember to pu... | 92 |
[
{
"content": "Solve the following math problem step by 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 integer values of $k$, with $1 \\leq k \\leq 70$, are such that $x^{k}-1 \\equiv 0 \\pmod{71}$ has at least $\\sqrt{k... | 21 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn triangle $ABC$, we have $AB = 7$, $AC = 9$, and $BC = 8$. The angle bisector of $\\angle BAC$ intersects side $BC$ at point... | 21 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAn *up-right path* from $(a, b) \\in \\mathbb{R}^2$ to $(c, d) \\in \\mathbb{R}^2$ is a finite sequence $(x_1, y_1), \\dots, (... | 1750 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor a positive integer $n$, let $\\langle n \\rangle$ denote the perfect square integer closest to $n$. For example, $\\langle... | 56 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose the function $\\psi$ satisfies $\\psi(1)=\\sqrt{2+\\sqrt{2+\\sqrt{2}}}$ and $\\psi(3x)+3\\psi(x)=\\psi(x)^3$ for all r... | 5792 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTwo right triangles are placed next to each other to form a quadrilateral as shown. What is the perimeter of the quadrilateral... | 42 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the interior of the square $ABCD$, the point $P$ is located such that $\\angle DCP = \\angle CAP = 25^{\\circ}$. Find all p... | 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\nNote that there are exactly three ways to write the integer $4$ as a sum of positive odd integers where the order of the summa... | 71 |
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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 sets $A$ and $B$ are subsets of the positive integers. The sum of any two distinct elements of $A$ is an element of $B$. T... | 5 |
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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\nJeff writes down the two-digit base-$10$ prime $\\overline{ab_{10}}$. He realizes that if he misinterprets the number as the b... | 61 |
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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 $Answer is the answer to the problem.\n\nA Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.\n\nTake five good haikus. Scramble... | 3004 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the last three digits of the number $\\frac{2019!}{2^{1009}}$.\n\nRemember to put your answer on its own line after \"Ans... | 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\nFind the last five digits of the sum:\n$$1^{100} + 2^{100} + 3^{100} + \\ldots + 999999^{100}$$\n\nRemember to put your answer... | 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\nWhat is the sum of the squares of the roots of the equation $x^2 - 7 \\lfloor x \\rfloor + 5 = 0$?\n\nRemember to put your ans... | 92 |
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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\nEphram is growing $3$ different variants of radishes in a row of $13$ radishes total, but he forgot where he planted each radi... | 17 |
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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 $L, E, T, M, \\text{ and } O$ be digits that satisfy the equation:\n\\[\nLEET + LMT = TOOL\n\\]\nGiven that $O = 0$, digit... | 1880 |
[
{
"content": "Solve the following math problem step by 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 the first $31$ integers that can be written as a sum of distinct powers of $3$?\n\nRemember to put your ans... | 1936 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many trailing zeros does the value \\(300 \\cdot 305 \\cdot 310 \\cdots 1090 \\cdot 1095 \\cdot 1100\\) end with?\n\nRemem... | 161 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute the number of ordered pairs of integers \\((a, b)\\), with \\(2 \\le a, b \\le 2021\\), that satisfy the equation\n\\[... | 43 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute the $100^{\\text{th}}$ smallest positive integer $n$ that satisfies the following three congruences:\n\n\\[\n\\begin{a... | 6491 |
[
{
"content": "Solve the following math problem step by 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 $\\text{s}_b(k)$ denote the sum of the digits of $k$ in base $b$. Compute \\[ \\text{s}_{101}(33) + \\text{s}_{101}(66) + ... | 6683 |
[
{
"content": "Solve the following math problem step by 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 positive integers expressible in the form $\\frac{x^2+y}{xy+1}$, for at least 2 pairs $(x,y)$ of positive intege... | 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\nGiven a finite set $S \\subset \\mathbb{R}^3$, define $f(S)$ to be the minimum integer $k$ such that there exist $k$ planes th... | 2189 |
[
{
"content": "Solve the following math problem step by 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 given figure, the chord $CD$ is perpendicular to the diameter $AB$ and intersects it at point $H$. The length of $AB$ i... | 65 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the sum of all integers $n$ that fulfill the equation $$2^n(6-n)=8n.$$\n\nRemember to put your answer on its own line aft... | 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 $R$ be the rectangle on the Cartesian plane with vertices $(0,0)$, $(5,0)$, $(5,7)$, and $(0,7)$. Find the number of squar... | 85 |
[
{
"content": "Solve the following math problem step by 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 with $n$ players, where $n < 10$, each player competes once against every other player. Players score 1 point ... | 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\nIn triangle $ABC$, where $AB = 13$, $BC = 14$, and $CA = 15$, a circle with radius $r$ passes through point $A$ and is tangent... | 7508 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe map below shows an east/west road connecting the towns of Acorn, Centerville, and Midland, and a north/south road from Cen... | 65 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $BC = 6$, $BX = 3$, $CX = 5$, and let $F$ be the midpoint of $\\overline{BC}$. Let $\\overline{AX} \\perp \\overline{BC}$ ... | 1288 |
[
{
"content": "Solve the following math problem step by 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 prime numbers such that $a + b = 10000$. Find the sum of the smallest possible value of $a$ and the largest... | 10000 |
[
{
"content": "Solve the following math problem step by 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 positive integers satisfying $a + b + c + d = 502$ and $a^2 - b^2 + c^2 - d^2 = 502$. Calculate how man... | 124 |
[
{
"content": "Solve the following math problem step by 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 $x_1, x_2, y_1, y_2$ be real numbers satisfying the equations:\n1. $x_1^2 + 5x_2^2 = 10$\n2. $x_2y_1 - x_1y_2 = 5$\n3. $x_... | 23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA set $S \\subseteq \\mathbb{N}$ satisfies the following conditions:\n\n1. If $x, y \\in S$ (not necessarily distinct), then $... | 2068 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWe say that a four-digit number $\\overline{abcd}$ is \\textit{slippery} if the number $a^4 + b^3 + c^2 + d$ is equal to the t... | 60 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat are the last $2$ digits of the number $2018^{2018}$ when written in base $7$?\n\nRemember to put your answer on its own l... | 44 |
[
{
"content": "Solve the following math problem step by 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 $ABCD$, the diagonals intersect at $E$. The area of $\\triangle ABE$ is 72 and the area of $\\triangle CDE$ is 50... | 242 |
[
{
"content": "Solve the following math problem step by 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 how many ways can you write $12$ as an ordered sum of integers where the smallest of those integers is equal to $2$? For ex... | 70 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nPoints $A$, $B$, and $C$ lie on a circle $\\Omega$ such that $A$ and $C$ are diametrically opposite each other. A line $\\ell$... | 460 |
[
{
"content": "Solve the following math problem step by 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 remainder when $29^{30} + 31^{28} + 28! \\cdot 30!$ is divided by $29 \\cdot 31$.\n\nRemember to put your answer o... | 668 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nKayla draws three triangles on a sheet of paper. What is the maximum possible number of regions, including the exterior region... | 20 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute the largest prime factor of $357! + 358! + 359! + 360!$.\n\nRemember to put your answer on its own line after \"Answer... | 379 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all rational terms of the sequence defined by the formula: $$ a_n = \\sqrt{\\frac{9n-2}{n+1}}, \\quad n \\in \\mathbb{N} ... | 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\nDitty can bench $80$ pounds today. Every week, the amount he benches increases by the largest prime factor of the weight he be... | 69 |
[
{
"content": "Solve the following math problem step by 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 $j$ such that for every polynomial $p(x)$ with integer coefficients and for every integer $... | 8 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA pulley system with two blocks is released from rest. The block on the table, with mass $m = 1.0 \\, \\text{kg}$, slides and ... | 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\nDetermine the least real number $k$ such that the inequality\n\\[\n\\left(\\frac{2a}{a-b}\\right)^2 + \\left(\\frac{2b}{b-c}\\... | 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 $Answer is the answer to the problem.\n\nIn a plane, 2014 lines are distributed into 3 groups. In each group, all the lines are parallel to each other. What is the max... | 302561952 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\lfloor x \\rfloor$ denote the greatest integer less than or equal to $x$. Find the value of the sum:\n\\[\n\\left\\lflo... | 4045 |
[
{
"content": "Solve the following math problem step by 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 $\\tau(x)$ be the number of positive divisors of $x$ (including $1$ and $x$). Find $\\tau\\left( \\tau\\left( \\dots \\tau... | 2 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn order to save money on gas and use up less fuel, Hannah has a special battery installed in the family van. Before the insta... | 108 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEvaluate the sum: \n\\[ \\sqrt{1 - \\frac{1}{2} \\cdot \\sqrt{1 \\cdot 3}} + \\sqrt{2 - \\frac{1}{2} \\cdot \\sqrt{3 \\cdot 5}... | 4 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA triangle with side lengths $16$, $18$, and $21$ has a circle with radius $6$ centered at each vertex. Find $n$ so that the t... | 90 |
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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 smallest positive $\\alpha$ (in degrees) for which all the numbers $\\cos{\\alpha},\\cos{2\\alpha},\\ldots,\\cos{2^n\... | 120 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe diagram below shows a rectangle with side lengths $36$ and $48$. Each of the sides is trisected, and edges are added betwe... | 136 |
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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 we divide the number $1^{1990} + 2^{1990} + 3^{1990} + \\ldots + 1990^{1990}$ by $10$, what is the remainder?\n\nRemember t... | 5 |
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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 $Answer is the answer to the problem.\n\nStarting on April 15, 2008, you can go one day backward and one day forward to get the dates 14 and 16. If you go 15 days back... | 14 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAn equilateral triangle $ABC$ has a side length of $7$. Point $P$ is located inside triangle $ABC$ such that $PB = 3$ and $PC ... | 13 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the Game of Life, each square in an infinite grid of squares is either shaded or blank. Every day, if a square shares an ed... | 7 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nJoshua rolls two dice and records the product of the numbers face up. The probability that this product is composite can be ex... | 65 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f$ be a one-to-one function from the set of natural numbers to itself such that $f(mn) = f(m)f(n)$ for all natural number... | 24 |
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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\nTwo distinct squares of the $8 \\times 8$ chessboard $C$ are said to be adjacent if they have a vertex or side in common. Also... | 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\nLet $V$ be a 10-dimensional real vector space and $U_1, U_2$ two linear subspaces such that $U_1 \\subseteq U_2$, $\\dim U_1 =... | 67 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTed loves creating haikus, which are Japanese three-line poems with syllable patterns of 5, 7, and 5 respectively. However, hi... | 28 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA right circular cone has a height equal to three times its base radius and has a volume of 1. The cone is inscribed inside a ... | 581 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAn ant leaves the anthill for its morning exercise. It walks $4$ feet east and then makes a $160^\\circ$ turn to the right and... | 36 |
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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\nMarko lives at the origin of the Cartesian plane. Every second, Marko moves $1$ unit up with probability $\\frac{2}{9}$, $1$ u... | 1811716 |
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