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 $Answer is the answer to the problem.\n\nThe vertices of a regular nonagon are colored such that:\n1. Adjacent vertices are different colors.\n2. If three vertices for... | 54 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHalf the volume of a 12-foot high cone-shaped pile is grade A ore, while the other half is grade B ore. The pile is worth $62.... | 608 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEach of the first $150$ positive integers is painted on a different marble, and the $150$ marbles are placed in a bag. If $n$ ... | 101 |
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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\nSuppose we choose two numbers $x, y \\in [0,1]$ uniformly at random. If the probability that the circle with center $(x, y)$ a... | 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\nWe will call one of the cells of a rectangle $11 \\times 13$ \"peculiar\", if after removing it the remaining figure can be cu... | 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\nWhat is the largest integer that must divide $n^5 - 5n^3 + 4n$ for all integers $n$?\n\nRemember to put your answer on its own... | 120 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many integers $n$ with $10 \\leq n \\leq 500$ have the property that the hundreds digit of $17n$ and $17n + 17$ are differ... | 84 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\star$ be an operation defined in the set of nonnegative integers with the following properties: for any nonnegative int... | 579 |
[
{
"content": "Solve the following math problem step by 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 real number $x$ such that $$\\left(\\frac{x}{x-1}\\right)^2 + \\left(\\frac{x}{x+1}\\right)^2 = \\frac{325}{1... | 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\n\\( \\triangle PQR \\) is isosceles and right-angled at \\( R \\). Point \\( A \\) is inside \\( \\triangle PQR \\), such that... | 127 |
[
{
"content": "Solve the following math problem step by 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 exist primes $p$ and $q$ such that \\[ pq = 1208925819614629174706176 \\times 2^{4404} - 4503599560261633 \\times 134217... | 358 |
[
{
"content": "Solve the following math problem step by 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 fair six-sided dice are rolled. The probability that the positive difference between the two rolls is at least $4$ can be ... | 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\nHow many nonempty subsets of \\( \\{1, 2, \\dots, 15\\} \\) are there such that the sum of the squares of each subset is a mul... | 7583 |
[
{
"content": "Solve the following math problem step by 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 Monday, a school library was attended by 5 students; on Tuesday, by 6; on Wednesday, by 4; on Thursday, by 8; and on Friday... | 15 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nPam lists the four smallest positive prime numbers in increasing order. When she divides the positive integer $N$ by the first... | 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\nFind the smallest positive integer $k$ such that there exist positive integers $M, O > 1$ satisfying:\n\\[\n (O \\cdot M \\cdo... | 2823 |
[
{
"content": "Solve the following math problem step by 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$ are integers and $a+b$ is a root of $x^2 + ax + b = 0$. What is the maximum possible value of $b^2$?\n\nRemembe... | 81 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor how many natural numbers $n$ between $1$ and $2014$ (both inclusive) is $\\frac{8n}{9999-n}$ an integer?\n\nRemember to pu... | 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 construction rope is tied to two trees. It is straight and taut. It is then vibrated at a constant velocity $v_1$. The tensi... | 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 \\( \\text{Revolution}(x) = x^3 + Ux^2 + Sx + A \\), where \\( U \\), \\( S \\), and \\( A \\) are all integers and \\( U ... | 392 |
[
{
"content": "Solve the following math problem step by 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 two positive integers, both less than $1000$. The arithmetic mean and the geometric mean of these numbers are consecut... | 120 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute the sum of all positive integers $n < 200$ such that $\\text{gcd}(n, k) \\neq 1$ for every $k \\in \\{2 \\cdot 11 \\cd... | 539 |
[
{
"content": "Solve the following math problem step by 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 composite positive integers \\(m\\) such that, whenever the product of two positive integers \\(a\\) and \\(b\\) is \... | 15 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTen test papers are to be prepared for the National Olympiad. Each paper contains 4 problems, and no two papers have more than... | 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\nA four-digit number $n$ is said to be \"literally 1434\" if, when every digit is replaced by its remainder when divided by $5$... | 67384 |
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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 knights placed on distinct squares of an $8 \\times 8$ chessboard, where each square is a unit square, are said to attack ... | 1848 |
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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\nMrs. Toad has a class of $2017$ students, with unhappiness levels $1, 2, \\ldots, 2017$ respectively. Today in class, there is... | 1121 |
[
{
"content": "Solve the following math problem step by 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 positive integers $x_1, x_2, \\ldots, x_7$ satisfy $x_6 = 144$ and $x_{n+3} = x_{n+2}(x_{n+1} + x_n)$ for $n = 1, 2, 3, 4$... | 3456 |
[
{
"content": "Solve the following math problem step by 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 > 1$ such that each prime factor of $N$ is either $2$, $3$, or $5$, and $N$ is not divisible b... | 2820 |
[
{
"content": "Solve the following math problem step by 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$ and $q$ be prime numbers such that $(p-1)^{q-1}-1$ is a positive integer that divides $(2q)^{2p}-1$. Compute the sum o... | 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\nThree points $X$, $Y$, and $Z$ are on a straight line such that $XY = 10$ and $XZ = 3$. What is the product of all possible va... | 91 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere are $15$ (not necessarily distinct) integers chosen uniformly at random from the range from $0$ to $999$, inclusive. Yan... | 200 |
[
{
"content": "Solve the following math problem step by 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 committee of 5 is to be chosen from a group of 9 people. How many ways can it be chosen, if Bill and Karl must serve togethe... | 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 number of integers $n$ with $n \\ge 2$ such that the remainder when $2013$ is divided by $n$ is equal to the remainde... | 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\nHow many ways are there to arrange the numbers $1$, $2$, $3$, $4$, $5$, $6$ on the vertices of a regular hexagon such that exa... | 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\nThe sum \\( \\sum_{n=0}^{2016\\cdot2017^2}2018^n \\) can be represented uniquely in the form \\( \\sum_{i=0}^{\\infty}a_i\\cdo... | 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\nThe sum $\\frac{1}{1} + \\frac{1}{2} + \\frac{1}{3} + \\frac{1}{4} + \\frac{1}{5} + \\frac{1}{6} = \\frac{m}{n}$ where $m$ and... | 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\nA parliament consists of $n$ senators. These senators are organized into 10 parties and 10 committees. Each senator belongs to... | 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\nLet $p(x) = x^4 - 5773x^3 - 46464x^2 - 5773x + 46$. Determine the sum of $\\arctan$ of its real roots.\n\nRemember to put your... | 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 smallest integer $n$ such that any subset of $\\{1, 2, \\ldots, 100\\}$ with $n$ elements contains at least two eleme... | 55 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor all integers $n \\geq 2$, let $f(n)$ denote the largest positive integer $m$ such that $\\sqrt[m]{n}$ is an integer. Evalu... | 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\nConsider an infinite sequence $x_1, x_2, \\dots$ of positive integers such that, for every integer $n \\geq 1$: \n\n- If $x_n$... | 1183 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCat and Claire are having a conversation about Cat’s favorite number. Cat says, \"My favorite number is a two-digit perfect sq... | 36 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the number of ways to order the integers $1, 2, 3, 4, 5, 6, 7$ from left to right so that each integer has all its diviso... | 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\nLet $\\triangle ABC$ be a triangle such that the area $[ABC] = 10$ and $\\tan(\\angle ABC) = 5$. If the smallest possible valu... | 42 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the greatest common divisor of all numbers of the form $(2^{a^2} \\cdot 19^{b^2} \\cdot 53^{c^2} + 8)^{16} - 1$ where $a,... | 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\nA mouse has a wheel of cheese which is cut into $2018$ slices. The mouse also has a $2019$-sided die, with faces labeled $0, 1... | 2019 |
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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 remainder of $7^{1010} + 8^{2017}$ when divided by $57$.\n\nRemember to put your answer on its own line after \"Answe... | 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\nWhich number do we need to subtract from the numerator and add to the denominator of $\\frac{\\overline{28a3}}{7276}$ such tha... | 571 |
[
{
"content": "Solve the following math problem step by 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_1, \\dots , r_5$ be the roots of the polynomial $x^5+5x^4-79x^3+64x^2+60x+144$. What is $r_1^2+\\dots+r_5^2$?\n\nRememb... | 183 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nConsider the sequence of numbers defined by $a_1 = 7$, $a_2 = 7^7$, and in general, $a_n = 7^{a_{n-1}}$ for $n \\ge 2$. Determ... | 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\nLet $\\triangle ABC$ satisfy $AB = 17$, $AC = \\frac{70}{3}$, and $BC = 19$. Let $I$ be the incenter of $\\triangle ABC$ and $... | 22 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe eleven members of a cricket team are numbered $1, 2, \\ldots, 11$. In how many ways can the entire cricket team sit on the... | 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\nRichard and Shreyas are arm wrestling against each other. They will play $10$ rounds, and in each round, there is exactly one ... | 932 |
[
{
"content": "Solve the following math problem step by 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 lines, parallel to the base of a triangle, divide the other two sides into 10 equal segments and the area into 10 distinc... | 400 |
[
{
"content": "Solve the following math problem step by 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 integers $b$ for which the polynomial $x^2 + bx + 2008b$ can be factored over the integers. Compute ... | 168672 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCaltech's 900 students are evenly spaced along the circumference of a circle. How many equilateral triangles can be formed wit... | 808500 |
[
{
"content": "Solve the following math problem step by 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 square $ABCD$ with side length $12$ and square $EFGH$ with side length $6$, both having the same cente... | 27 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEvariste has drawn twelve triangles such that two consecutive triangles share exactly one edge. \n\nSophie colors every triang... | 4098 |
[
{
"content": "Solve the following math problem step by 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 two last digits of $2012^{2012}$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "us... | 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\nAlison is eating $2401$ grains of rice for lunch. She eats the rice in a peculiar manner: \n\n1. If she has only one grain of ... | 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\nWhen Meena turned 16 years old, her parents gave her a cake with \\( n \\) candles, where \\( n \\) has exactly 16 different p... | 120 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nYou have 5000 distinct finite sets. Their intersection is empty. However, the intersection of any two sets is nonempty. What i... | 14 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose a biased coin gives head with probability $\\frac{2}{3}$. The coin is tossed repeatedly. If it shows heads, then playe... | 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\nGeorge and two of his friends go to a famous jiaozi restaurant, which serves only two kinds of jiaozi: pork jiaozi and vegetab... | 4096 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$A, B, C$, and $D$ are all on a circle, and $ABCD$ is a convex quadrilateral. If $AB = 13$, $BC = 13$, $CD = 37$, and $AD = 47... | 504 |
[
{
"content": "Solve the following math problem step by 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 $k$ be a positive integer. A sequence $a_0, a_1, \\ldots, a_n$, where $n > 0$, of positive integers satisfies the followin... | 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\nLet $f_n(x) = n + x^2$. Evaluate the product \\( \\text{gcd}\\{f_{2001}(2002), f_{2001}(2003)\\} \\times \\text{gcd}\\{f_{2011... | 125 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe area of a circle centered at the origin, which is inscribed in the parabola $y = x^2 - 25$, can be expressed as $\\frac{a}... | 103 |
[
{
"content": "Solve the following math problem step by 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 meeting is held at a round table. It is known that:\n- 7 women have a woman on their right side.\n- 12 women have a man on t... | 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\nAmy places positive integers in each of these cells so that each row and column contains each of $1, 2, 3, 4, 5$ exactly once.... | 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\nConsider the base 27 number \\[ n = ABCDEFGHIJKLMNOPQRSTUVWXYZ \\] where each letter has the value of its position in the alph... | 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\nA point $P$ is chosen uniformly at random in the interior of triangle $ABC$ with side lengths $AB = 5$, $BC = 12$, $CA = 13$. ... | 61 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nABCD is a cyclic quadrilateral with circumcenter $O$ and circumradius 7. The line segments $AB$ and $CD$ intersect at point $E... | 33 |
[
{
"content": "Solve the following math problem step by 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_0 = (3,1)$ and define $P_{n+1} = (x_n, y_n)$ for $n \\ge 0$ by \\[ x_{n+1} = - \\frac{3x_n - y_n}{2}, \\quad y_{n+1} = ... | 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\nSuppose that $ABCDE$ is a convex pentagon with $\\angle A = 90^\\circ$, $\\angle B = 105^\\circ$, $\\angle C = 90^\\circ$ and ... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $d$ be the greatest common divisor of $2^{30^{10}} - 2$ and $2^{30^{45}} - 2$. Find the remainder when $d$ is divided by $... | 2012 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\{a,b,c,d,e,f,g,h,i\\}$ be a permutation of $\\{1,2,3,4,5,6,7,8,9\\}$ such that $\\gcd(c,d)=\\gcd(f,g)=1$ and\n\\[(10a+b... | 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\nSuppose that $P(x, y, z)$ is a homogeneous degree 4 polynomial in three variables such that $P(a, b, c) = P(b, c, a)$ and $P(a... | 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\nConsider the arithmetic sequence of integers with first term $-7$ and second term $17$. What is the sum of the squares of the ... | 2019 |
[
{
"content": "Solve the following math problem step by 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 odd positive integer that is the middle number of five consecutive integers, all of which are composite.\n\nRem... | 93 |
[
{
"content": "Solve the following math problem step by 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: \\mathbb{N} \\to \\mathbb{N}$ be a function satisfying the following conditions:\n\n- $f(ab) = f(a)f(b)$ whenever the ... | 2002 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the remainder when $333^{333}$ is divided by $33$.\n\nRemember to put your answer on its own line after \"Answer:\".",
... | 27 |
[
{
"content": "Solve the following math problem step by 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 point \\((a,b)\\) in the plane is called sparkling if it also lies on the line \\(ax+by=1\\). Find the maximum possible dist... | 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\nAlice, Bob, Charlie, Diana, Emma, and Fred sit in a circle, in that order, and each roll a six-sided die. Each person looks at... | 1440 |
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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 rectangle $ABCD$, $BC = 5$, $EC = \\frac{1}{3} CD$, and $F$ is the point where $AE$ and $BD$ intersect. The triangle $\... | 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\nWe color the numbers $1, 2, 3, \\ldots, 20$ with two colors: white and black, ensuring that both colors are used. Determine th... | 62 |
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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 diagram below, $ABCD$ is a square and both $\\triangle CFD$ and $\\triangle CBE$ are equilateral. Find the degree measu... | 15 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven a $4 \\times 4$ squares table, how many ways can we fill the table with $\\{0,1\\}$ such that any two neighboring square... | 1234 |
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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$ be the number of functions $f:\\{1,2,3,4,5,6,7,8,9,10\\} \\rightarrow \\{1,2,3,4,5\\}$ that have the property that for... | 31 |
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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\nThere are $38$ people in the California Baseball League (CBL). The CBL cannot start playing games until people are split into ... | 16 |
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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 prime numbers $p$ with $p \\ge 5$ such that $p$ divides $(p + 3)^{p-3} + (p + 5)^{p-5}$.\n\nRemember to... | 322 |
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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 product \n$$\\left(\\frac{1+1}{1^2+1}+\\frac{1}{4}\\right)\\left(\\frac{2+1}{2^2+1}+\\frac{1}{4}\\right)\\left(\\frac{3+1}... | 1 |
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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\nUsing only once each of the digits $1, 2, 3, 4, 5, 6, 7$, and $8$, write the square and the cube of a positive integer. Determ... | 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\nStarting with a positive integer $M$ written on the board, Alice plays the following game: in each move, if $x$ is the number ... | 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\nOn a straight line $\\ell$ there are four points, $A$, $B$, $C$, and $D$ in that order, such that $AB = BC = CD$. A point $E$ ... | 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\nCompute the largest prime factor of $111^2 + 11^3 + 1^1$.\n\nRemember to put your answer on its own line after \"Answer:\".",
... | 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\nTriangle $ABC$ has $\\angle A = 90^\\circ$, $\\angle C = 30^\\circ$, and $AC = 12$. Let the circumcircle of this triangle be $... | 23 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe positive reals $x$, $y$, $z$ satisfy the relations\n\\begin{align*}\nx^2 + xy + y^2 &= 1 \\\\\ny^2 + yz + z^2 &= 2 \\\\\nz... | 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\nA $3 \\times 3 \\times 3$ cube composed of $27$ unit cubes rests on a horizontal plane. Determine the number of ways to select... | 60 |
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