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\nSuppose $a, b$ are positive real numbers such that $a + a^2 = 1$ and $b^2 + b^4 = 1$. Compute $a^2 + b^2$.\n\nRemember to put ... | 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\nPrinceton’s Math Club recently bought a stock for $\\$2$ and sold it for $\\$9$ thirteen days later. Given that the stock eith... | 273 |
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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 that $x_{n+2}=\\dfrac{20x_{n+1}}{14x_n}$, $x_0=25$, $x_1=11$, it follows that $\\sum_{n=0}^\\infty\\dfrac{x_{3n}}{2^n}=\... | 1685 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCamilla drove $20$ miles in the city at a constant speed and $40$ miles in the country at a constant speed that was $20$ miles... | 35 |
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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 $x$ be a real number such that $3 \\sin^4 x - 2 \\cos^6 x = -\\frac{17}{25}$. Then, $3 \\cos^4 x - 2 \\sin^6 x = \\frac{m}... | 445 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $m$ be the answer to this question. What is the value of $2m - 5$?\n\nRemember to put your answer on its own line after \"... | 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\nA student walks down a hallway with a row of closed lockers numbered from $1$ to $1024$. He opens locker No. $1$, skips the ne... | 1024 |
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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\n12. In a rectangle $ABCD$ with $AB = 8$ and $BC = 20$, let $P$ be a point on $AD$ such that $\\angle BPC = 90^\\circ$. If $r_1... | 8 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA quadrilateral $ABCD$ has a right angle at $\\angle ABC$ and satisfies $AB = 12$, $BC = 9$, $CD = 20$, and $DA = 25$. Determi... | 769 |
[
{
"content": "Solve the following math problem step by 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 $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, \\ldots$ Find $n$ such that the first $n$ terms sum up to ... | 1027 |
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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 $Answer is the answer to the problem.\n\nThe initial number of inhabitants in a city, which is greater than 150, is a perfect square. With an increase of 1000 inhabita... | 249001 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven $2018$ ones in a row: $$\\underbrace{1\\,\\,\\,1\\,\\,\\,1\\,\\,\\,1 \\,\\,\\, ... \\,\\,\\,1 \\,\\,\\,1 \\,\\,\\,1 \\,\... | 1991 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nRight triangle $ABC$ has hypotenuse $AB = 26$, and the inscribed circle of $\\triangle ABC$ has radius $5$. The largest possib... | 1814 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThree fair six-sided dice are labeled with the numbers $\\{1, 2, 3, 4, 5, 6\\},$ $\\{1, 2, 3, 4, 5, 6\\},$ and $\\{1, 2, 3, 7,... | 1136 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a, b, c, d$ be natural numbers such that $a + b + c + d = 2018$. Find the minimum value of the expression:\n$$E = (a-b)^2... | 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 $Answer is the answer to the problem.\n\nThere are several teacups in the kitchen, some with handles and others without handles. The number of ways to select two cups ... | 29 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the number of pairs of integers $x, y$ with different parities such that $\\frac{1}{x} + \\frac{1}{y} = \\frac{1}{2520}$.... | 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\nLet $v$ and $w$ be real numbers such that, for all real numbers $a$ and $b$, the inequality \\[(2^{a+b}+8)(3^a+3^b) \\leq v(12... | 62208 |
[
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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 $N$ for which there exists a unique ordered triple of non-negative integers $(a, b, c... | 280 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDefine a *growing spiral* in the plane as a sequence of points with integer coordinates $P_0=(0,0), P_1, \\dots, P_n$ such tha... | 10053 |
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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\nCalculate the value of $\\sqrt{\\frac{11^4 + 100^4 + 111^4}{2}}$ and provide the answer as an integer.\n\nRemember to put your... | 11221 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the largest integer $n$ such that $n$ is divisible by all positive integers less than $\\sqrt[3]{n}$.\n\nRemember to put ... | 420 |
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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 $a$ be a real number greater than $1$ such that $\\frac{20a}{a^2+1} = \\sqrt{2}$. Find $\\frac{14a}{a^2 - 1}$.\n\nRemember... | 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 $Answer is the answer to the problem.\n\nYou are given that \\(17! = 355687ab8096000\\) for some digits \\(a\\) and \\(b\\). Find the two-digit number \\(\\overline{ab... | 42 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDarwin takes an $11 \\times 11$ grid of lattice points and connects every pair of points that are 1 unit apart, creating a $10... | 220 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nYou are given a rectangular playing field of size $13 \\times 2$ and any number of dominoes of sizes $2 \\times 1$ and $3 \\ti... | 257 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor real numbers $x$ and $y$, define \\[ \\nabla(x,y)=x-\\frac{1}{y}. \\] If \\[ \\underbrace{\\nabla(2, \\nabla(2, \\nabla(2,... | 203817 |
[
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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 $Answer is the answer to the problem.\n\nFour mathletes and two coaches sit at a circular table. How many distinct arrangements are there of these six people if the tw... | 24 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAlice and Bob play a game together as a team on a $100 \\times 100$ board with all unit squares initially white. Alice sets up... | 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\nWe have a calculator with two buttons that displays an integer $x$. Pressing the first button replaces $x$ by $\\lfloor \\frac... | 233 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $k$ be the smallest positive integer such that the binomial coefficient $\\binom{10^9}{k}$ is less than the binomial coeff... | 38 |
[
{
"content": "Solve the following math problem step by 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 $n$ that has at least $7$ positive divisors $1 = d_1 < d_2 < \\ldots < d_k = n$, $k \\geq 7... | 2024 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe sequence of digits \\(1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 \\dots\\) is obtained by writing t... | 1984 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nBill draws two circles which intersect at $X,Y$. Let $P$ be the intersection of the common tangents to the two circles, and le... | 4807 |
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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 $p(x)$ of degree $1000$ is such that $p(n) = (n+1)2^n$ for all nonnegative integers $n$ such that $n \\leq 1000$.... | 500 |
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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 $f_1(x) = x^2 - 1$, and for each positive integer $n \\geq 2$, define $f_n(x) = f_{n-1}(f_1(x))$. How many distinct real r... | 2005 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA large number of rocks are placed on a table. On each turn, one may remove some rocks from the table following these rules: o... | 18 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAt a party, every guest is a friend of exactly fourteen other guests (not including themselves). Every two friends have exactl... | 64 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn a right trapezoid $ABCD$ where $AB \\parallel CD$, the angle at vertex $B$ measures $75^\\circ$. Point $H$ is the foot of t... | 8 |
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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 $P$ be the parabola in the plane determined by the equation $y = x^2$. Suppose a circle $C$ in the plane intersects $P$ at... | 1247 |
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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\nDilhan has objects of 3 types: $A$, $B$, and $C$. He also has 6 functions: $f_{A,B}$, $f_{A,C}$, $f_{B,A}$, $f_{B,C}$, $f_{C,A... | 18 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose $x$ is a random real number between $1$ and $4$, and $y$ is a random real number between $1$ and $9$. If the expected ... | 1112 |
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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 $\\alpha, \\beta$ be the distinct positive roots of the equation $2x = \\tan x$. Evaluate the integral:\n\n$$\\int_0^1 \\s... | 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\nWe are given 5040 balls in \\(k\\) different colors, where the number of balls of each color is the same. The balls are put in... | 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\nQuadrilateral $ABCD$ is neither a kite nor a rectangle. It is known that its side lengths are integers, $AB = 6$, $BC = 7$, an... | 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\nLet $f : \\mathbb{R} \\to \\mathbb{R}$ satisfy $f(x + f(y)) = 2x + 4y + 2547$ for all reals $x, y$. Compute $f(0)$.\n\nRemembe... | 849 |
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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 functions $f : \\{1, 2, \\ldots, 16\\} \\to \\{1, 2, \\ldots, 16\\}$ have the property that $f(f(x)) - 4x$ is divisib... | 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\nDaniel had a string that formed the perimeter of a square with area $98$. Daniel cut the string into two pieces. With one piec... | 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\nFind the remainder when $26!^{26} + 27!^{27}$ is divided by $29$.\n\nRemember to put your answer on its own line after \"Answe... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn triangle $ABC$, we have $AB=36$, $BC=48$, and $CA=60$. The incircle of $\\triangle ABC$ is centered at $I$ and touches $AB$... | 720 |
[
{
"content": "Solve the following math problem step by 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. Let $E$ be the midpoint of $AB$ and $F$ be the midpoint of $CD$. Points $P$ and $Q$ are on segm... | 8 |
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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 $O$, $A$, $B$, and $C$ be points in space such that $\\angle AOB = 60^{\\circ}$, $\\angle BOC = 90^{\\circ}$, and $\\angle... | 109 |
[
{
"content": "Solve the following math problem step by 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 constant such that exactly three real values of $x$ satisfy $$x - |x^2 - 4x + 3| = k.$$ The sum of all possible v... | 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\nLet $x, y \\in \\mathbb{R}$ such that $x > 2, y > 3$. Find the minimum value of \\( \\frac{(x+y)^2}{\\sqrt{x^2-4} + \\sqrt{y^2... | 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 $Answer is the answer to the problem.\n\n$p, q, r$ are real numbers satisfying the following equations: \n\\[ \\frac{(p+q)(q+r)(r+p)}{pqr} = 24 \\] \n\\[ \\frac{(p-2... | 67 |
[
{
"content": "Solve the following math problem step by 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 a positive integer. When the leftmost digit of the standard base 10 representation of $n$ is shifted to the rightmo... | 126 |
[
{
"content": "Solve the following math problem step by 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 two non-negative integers $m > n$, let's say that $m$ ends in $n$ if we can obtain $n$ by erasing some digits (from left... | 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 $Answer is the answer to the problem.\n\nHow many sets of positive integers \\((a, b, c)\\) satisfy \\(a > b > c > 0\\) and \\(a + b + c = 103\\)?\n\nRemember to put y... | 833 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $ABCD$ be a parallelogram such that $AB = 35$ and $BC = 28$. Suppose that $BD \\perp BC$. Let $\\ell_1$ be the reflection ... | 113 |
[
{
"content": "Solve the following math problem step by 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 two-digit integer for which the product of its digits is $17$ more than their sum?\n\nRemember to put your... | 74 |
[
{
"content": "Solve the following math problem step by 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 unit cube $ABCDEFGH$ (with faces $ABCD$, $EFGH$ and connecting vertices labeled so that $\\overline{AE}$, $\\overline{B... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEight red boxes and eight blue boxes are randomly placed in four stacks of four boxes each. The probability that exactly one o... | 843 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA mason has bricks with dimensions $2 \\times 5 \\times 8$ and other bricks with dimensions $2 \\times 3 \\times 7$. She also ... | 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 $Answer is the answer to the problem.\n\nCompute the largest integer \\( N \\leq 2012 \\) with four distinct digits.\n\nRemember to put your answer on its own line aft... | 1987 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA competition room of HOMC has $m \\times n$ students where $m, n$ are integers larger than $2$. Their seats are arranged in $... | 18 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nConsider two geometric sequences $16$, $a_1$, $a_2$, $\\ldots$ and $56$, $b_1$, $b_2$, $\\ldots$ with the same common nonzero ... | 490 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDoes there exist a four-digit positive integer with different non-zero digits, which has the following property: if we add the... | 1234 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $B = (20, 14)$ and $C = (18, 0)$ be two points in the plane. For every line $\\ell$ passing through $B$, we color red the ... | 157 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat number is nine more than four times the answer to this question?\n\nRemember to put your answer on its own line after \"A... | -3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a_n$ denote the remainder when $(n+1)^3$ is divided by $n^3$; in particular, $a_1=0$. Compute the remainder when $a_1+a_2... | 693 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor each positive integer $n \\ge 2$, define $k(n)$ to be the largest integer $m$ such that $(n!)^m$ divides $2016!$. What is ... | 89 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA number consists of 3 different digits. The sum of the 5 other numbers formed with those digits is 2003. Find the number.\n\n... | 217 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nRthea, a distant planet, is home to creatures whose DNA consists of two distinguishable strands of bases with a fixed orientat... | 28812 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine all non-negative integers $k$ such that there exists a function $f : \\mathbb{N} \\to \\mathbb{N}$ satisfying the eq... | 0 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $m$, $n$, $a$, and $k$ be positive integers with $k > 1$ such that the equation \\(5^m + 63n + 49 = a^k\\) holds. Find the... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute the number of ways to erase 24 letters from the string ``OMOMO\\cdots OMO'' (with length 27), such that the three rema... | 455 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA positive integer $n$ is said to be base-able if there exist positive integers $a$ and $b$, with $b > 1$, such that $n = a^b$... | 90 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAlexis noticed her brother Joshua measuring something on the floor. He was designing a sandcastle and needed to measure the ba... | 20 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nBob's Rice ID number consists of six digits, each ranging from $1$ to $9$. Each digit can be repeated any number of times. The... | 324 |
[
{
"content": "Solve the following math problem step by 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 grid of points $X = \\{(m,n) \\mid 0 \\leq m,n \\leq 4 \\}$. We say a pair of points $\\{(a,b),(c,d)\\}$ in $X$ i... | 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\nConsider a regular $2n + 1$-gon $P$ in the plane, where $n$ is a positive integer. We say that a point $S$ on one of the sides... | 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\nIn how many ways can Irena arrange the integers $1$ through $7$ in a line such that for any three consecutive numbers, the lar... | 232 |
[
{
"content": "Solve the following math problem step by 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 $a_1$, $a_2$, $a_3$, $a_4$, $a_5$ be real numbers satisfying the following system of equations:\n\\[\n\\begin{align*}\n 2... | 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 $Answer is the answer to the problem.\n\nTalithia throws a party on the fifth Saturday of every month that has five Saturdays. That is, if a month has five Saturdays, ... | 4 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind a number with $3$ digits, knowing that:\n1. The sum of its digits is $9$.\n2. The product of its digits is $24$.\n3. The ... | 342 |
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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 trinomial $f(x)$ satisfies the equation $(f(x))^3 - f(x) = 0$, which has three real roots. Determine the y-coordinate of t... | 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\nRichard likes to solve problems from the IMO Shortlist. In 2013, Richard solves $5$ problems each Saturday and $7$ problems ea... | 1100 |
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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 $x$, $y$, $z$ be real numbers that satisfy the following equations:\n\n\\[\n(x-y)^2 + (y-z)^2 + (z-x)^2 = 8, \\\\\nx^3 + y... | 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 $Answer is the answer to the problem.\n\nThe **doubling sum** function is defined by:\n\\[ D(a,n) = a + 2a + 4a + 8a + \\ldots \\] \nThis sum contains \\( n \\) terms.... | 9765 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the smallest positive integer $k$ such that $k!$ ends in at least 43 zeroes.\n\nRemember to put your answer on its own li... | 175 |
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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\nFind the $2019$th strictly positive integer $n$ such that $\\binom{2n}{n}$ is not divisible by $5$.\n\nRemember to put your an... | 37805 |
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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 all unordered pairs $\\{A,B \\}$ of subsets of an $8$-element set, such that $A \\cap B \\neq \\emptyset$ a... | 23606 |
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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 any positive integer $a$, let $\\tau(a)$ be the number of positive divisors of $a$. Find, with proof, the largest possible... | 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\nFind the smallest natural number $n$ for which there exists a natural number $x$ such that\n\\[(x+1)^3 + (x+2)^3 + (x+3)^3 + (... | 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 $Answer is the answer to the problem.\n\nDetermine the leftmost three digits of the number \\[1^1 + 2^2 + 3^3 + \\ldots + 999^{999} + 1000^{1000}.\\]\n\nRemember to pu... | 100 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\\mathcal{B}... | 4 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe positive integers $v, w, x, y$, and $z$ satisfy the equation \\[v + \\frac{1}{w + \\frac{1}{x + \\frac{1}{y + \\frac{1}{z}... | 12354 |
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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 smallest positive integer that is $3$ more than a multiple of $5$, and twice a multiple of $6$.\n\nRemember to put... | 48 |
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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 10 real numbers $a_1, a_2, \\ldots, a_{10}$, 45 sums of two of these numbers are formed as $a_i + a_j$ for $1 \\leq i < ... | 9 |
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