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\nGiven a rational number $r$ that, when expressed in base-$10$, is a repeating, non-terminating decimal, we define $f(r)$ to be... | 600 |
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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 $Answer is the answer to the problem.\n\nRyan is messing with Brice’s coin. He weights the coin such that it comes up on one side twice as frequently as the other, and... | 8 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $I$ be the incenter of a triangle $ABC$ with $AB = 20$, $BC = 15$, and $BI = 12$. Let $CI$ intersect the circumcircle $\\o... | 13 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f$ be a function which takes in $0, 1, 2$ and returns $0, 1, $ or $2$. The values need not be distinct: for instance we c... | 2 |
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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 $n = 2^{2015} - 1$. For any integer $1 \\le x < n$, let \\[f_n(x) = \\sum\\limits_p s_p(n-x) + s_p(x) - s_p(n),\\] where $... | 382 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $P, A, B, C$ be points on circle $O$ such that $C$ does not lie on arc $\\widehat{BAP}$, $\\overline{P A} = 21, \\overline... | 16 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $S$ be the set of ordered integer pairs $(x, y)$ such that $0 < x < y < 42$ and there exists some integer $n$ such that $x... | 252 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nMarie is painting a $4 \\times 4$ grid of identical square windows. Initially, they are all orange but she wants to paint $4$ ... | 252 |
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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 ordered pairs $(a, b)$ of positive integers with $1 \\le a, b \\le 10$ are there such that in the geometric sequence ... | 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\nEvery day, Heesu talks to Sally with some probability $p$. One day, after not talking to Sally the previous day, Heesu resolve... | 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\nGiven a positive integer $k$, let $f(k)$ be the sum of the $k$-th powers of the primitive roots of $73$. For how many positive... | 1847 |
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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 $a\\%b$ denote the remainder when $a$ is divided by $b$. Find $\\Sigma_{i=1}^{100}(100\\%i)$.\n\nRemember to put your answ... | 1701 |
[
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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 $z^{3}=2+2i$, where $i=\\sqrt{-1}$. The product of all possible values of the real part of $z$ can be written in the f... | 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 sequence of positive integers $a_1, a_2, \\dots$ has the property that $\\gcd(a_m, a_n) > 1$ if and only if $|m - n| = 1$.... | 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\nThe area of parallelogram $ABCD$ is $51\\sqrt{55}$ and $\\angle{DAC}$ is a right angle. If the side lengths of the parallelogr... | 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 $Answer is the answer to the problem.\n\nTristan is eating his favorite cereal, Tiger Crunch, which has marshmallows of two colors, black and orange. He eats the marsh... | 37 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor a sequence of $10$ coin flips, each pair of consecutive flips and count the number of “Heads-Heads”, “Heads-Tails”, “Tails... | 180 |
[
{
"content": "Solve the following math problem step by 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 regular hexagonal prism has bases $ABCDEF$, $A'B'C'D'E'F'$ and edges $AA'$, $BB'$, $CC'$, $DD'$, $EE'$, $FF'$, each of... | 323 |
[
{
"content": "Solve the following math problem step by 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 the sequences $a_n$ and $b_n$ as follows: $a_1 = 2017$ and $b_1 = 1$. \nFor $n > 1$, if there is a greatest integer $k ... | 8002 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTogether, Kenneth and Ellen pick a real number $a$. Kenneth subtracts $a$ from every thousandth root of unity (that is, the th... | 2 |
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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\nZack keeps cutting the interval $[0, 1]$ of the number line, each time cutting at a uniformly random point in the interval, un... | 34 |
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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 $Answer is the answer to the problem.\n\nLet\n\\[ \\frac{p}{q} = \\frac{2017}{2-\\frac{1}{3-\\frac{2}{2-\\frac{1}{2-\\frac{1}{3-\\frac{2}{2-\\frac{1}{2-\\frac{1}{2-\\f... | 16 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the minimum value attained by $\\sum_{m=1}^{100} \\gcd(M - m, 400)$ for $M$ an integer in the range $[1746, 2017]$.\n\n\n... | 680 |
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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 is a box containing $100$ balls, each of which is either orange or black. The box is equally likely to contain any numbe... | 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\nLet $X =\\{1, 2, ... , 2017\\}$. Let $k$ be a positive integer. Given any $r$ such that $1\\le r \\le k$, there exist $k$ sub... | 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\nCall an ordered triple $(a, b, c)$ of integers feral if $b -a, c - a$ and $c - b$ are all prime. \nFind the number of feral tr... | 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\n Triangle $ABC$ is so that $AB = 15$, $BC = 22$, and $AC = 20$. Let $D, E, F$ lie on $BC$, $AC$, and $AB$, respectively, so $A... | 497 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nBilly the baker makes a bunch of loaves of bread every day, and sells them in bundles of size $1, 2$, or $3$. On one particula... | 1011 |
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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 that f is a function $f : R_{\\ge 0} \\to R$ so that for all $x, y \\in R_{\\ge 0}$ (nonnegative reals) we have that $... | 10104 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\gamma_1$ and $\\gamma_2$ be circles centered at $O$ and $ P$ respectively, and externally tangent to each other at poin... | 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 $Answer is the answer to the problem.\n\nCary has six distinct coins in a jar. Occasionally, he takes out three of the coins and adds a dot to each of them. Determine ... | 79200 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the number of ordered pairs of integers $(x, y)$ such that $2167$ divides $3x^2 + 27y^2 + 2021$ with $0 \\le x, y \\le 21... | 2352 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $C$ be a circle centered at point $O$, and let $P$ be a point in the interior of $C$. Let $Q$ be a point on the circumfere... | 2500 |
[
{
"content": "Solve the following math problem step by 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 defined as below for integers $n \\ge 0$ and $a_0, a_1, ...$ such that $\\sum_{i\\ge 0}a_i$ is finite:\n$$f(n; a_0,... | 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\nLet $P$ be a $10$-degree monic polynomial with roots $r_1, r_2, . . . , r_{10} \\ne $ and let $Q$ be a $45$-degree monic polyn... | 19 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $ABCD$ be a cyclic quadrilateral with circumcenter $O$ and radius $10$. Let sides $AB$, $BC$, $CD$, and $DA$ have midpoint... | 78 |
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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 there be a tiger, William, at the origin. William leaps $ 1$ unit in a random direction, then leaps $2$ units in a random ... | 1240 |
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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 a triangle with sides $AB = 34$, $BC = 15$, $AC = 35$ and let $\\Gamma$ be the circle of smallest possible radius... | 173 |
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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\\} = x- \\lfloor x \\rfloor$ . Consider a function f from the set $\\{1, 2, . . . , 2020\\}$ to the half-open interv... | 1019595 |
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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\nThree (not necessarily distinct) points in the plane which have integer coordinates between $ 1$ and $2020$, inclusive, are ch... | 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\nLet $X, Y$ , and $Z$ be concentric circles with radii $1$, $13$, and $22$, respectively. Draw points $A, B$, and $C$ on $X$, $... | 24300 |
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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 smallest integer $a_0$ such that, for every positive integer $n$, there exists a sequence of positive integers $a_... | 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\nGiven two positive integers $a \\ne b$, let $f(a, b)$ be the smallest integer that divides exactly one of $a, b$, but not both... | 706 |
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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 You are walking along a road of constant width with sidewalks on each side. You can only walk on the sidewalks or cross the r... | 200 |
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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 ordered triples of nonzero integers $(a, b, c)$ satisfy $2abc = a + b + c + 4$?\n\nRemember to put your answer on its... | 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\nWill and Lucas are playing a game. Will claims that he has a polynomial $f$ with integer coefficients in mind, but Lucas doesn... | 289 |
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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 points $P \\in Z^2$ that satisfy the following two conditions:\n1) If $Q$ is a point on the circle of radiu... | 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\nJacob has a piece of bread shaped like a figure $8$, marked into sections and all initially connected as one piece of bread. T... | 1515 |
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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\nConsider a regular $2020$-gon circumscribed into a circle of radius $ 1$. Given three vertices of this polygon such that they ... | 5049 |
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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\nRuney is speaking his made-up language, Runese, that consists only of the “letters” zap, zep, zip, zop, and zup. Words in Rune... | 111110 |
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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 n lilypads in a row labeled $1, 2, \\dots, n$ from left to right. Fareniss the Frog picks a lilypad at random to sta... | 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\nWe say that a polynomial $p$ is respectful if $\\forall x, y \\in Z$, $y - x$ divides $p(y) - p(x)$, and $\\forall x \\in Z$, ... | 290 |
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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\nSelect two distinct diagonals at random from a regular octagon. What is the probability that the two diagonals intersect at a ... | 26 |
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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 that $f : Z\\times Z \\to R$, satisfies the equation $f(x, y) = f(3x+y, 2x+ 2y)$ for all $x, y \\in Z$. Determine the ... | 8983 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine the number of pairs $(a, b)$, where $1 \\le a \\le b \\le 100$ are positive integers, so that $\\frac{a^3+b^3}{a^2+b... | 122 |
[
{
"content": "Solve the following math problem step by 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 roots of the polynomial $f(x) = x^8 +x^7 -x^5 -x^4 -x^3 +x+ 1 $ are all roots of unity. We say that a real number $r \\in ... | 31 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nNelson is having his friend drop his unique bouncy ball from a $12$ foot building, and Nelson will only catch the ball at the ... | 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\nCompute the remainder when $2^{3^5}+ 3^{5^2}+ 5^{2^3}$ is divided by $30$.\n\nRemember to put your answer on its own line afte... | 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\nEighteen people are standing in a (socially-distanced) line to enter a grocery store. Five people are wearing a black mask, $6... | 116 |
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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\nKris is asked to compute $\\log_{10} (x^y)$, where $y$ is a positive integer and $x$ is a positive real number. However, they ... | 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\nGiven an integer $a_0$, we define a sequence of real numbers $a_0, a_1, . . .$ using the relation $$a^2_i = 1 + ia^2_{i-1},$$\... | 4946 |
[
{
"content": "Solve the following math problem step by 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 Abby and Ben have a little brother Carl who wants candy. Abby has $7$ different pieces of candy and Ben has $15$ different pi... | 96 |
[
{
"content": "Solve the following math problem step by 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, \\gamma \\in C$ be the roots of the polynomial $x^3 - 3x2 + 3x + 7$. For any complex number $z$, let $f(... | 12 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\vartriangle ABC$ be a triangle. Let $Q$ be a point in the interior of $\\vartriangle ABC$, and let $X, Y,Z$ denote the ... | 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\nProvided that $\\{a_i\\}^{28}_{i=1}$ are the $28$ distinct roots of $29x^{28} + 28x^{27} + ... + 2x + 1 = 0$, then the absolut... | 275 |
[
{
"content": "Solve the following math problem step by 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 $\\vartriangle ABC$ have sidelengths $BC = 7$, $CA = 8$, and, $AB = 9$, and let $\\Omega$ denote the circumcircle of $\\va... | 59 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA sequence of integers $a_1, a_2, \\ldots, a_n$ is said to be [i]sub-Fibonacci[/i] if $a_1=a_2=1$ and $a_i \\le a_{i-1}+a_{i-2... | 238 |
[
{
"content": "Solve the following math problem step by 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 set of real values $a$ such that the equation $x^4-3ax^3+(2a^2+4a)x^2-5a^2x+3a^2$ has exactly two nonreal solutions is the... | 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\n6. Let $f(p)$ denote the number of ordered tuples $\\left(x_{1}, x_{2}, \\ldots, x_{p}\\right)$ of nonnegative integers satisf... | 475 |
[
{
"content": "Solve the following math problem step by 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 $\\vartriangle ABC$ be a triangle with $AB = 13$, $BC = 14$, and $CA = 15$. Let $D$, $E$, and $F$ be the midpoints of $AB$... | 80 |
[
{
"content": "Solve the following math problem step by 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 set of degree $4$ polynomials $f$ with complex number coefficients satisfying $f(1)=f(2)^2=f(3)^3$ $=$ $f(4)^4=... | 1643751 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n10. The sum $\\sum_{k=1}^{2020} k \\cos \\left(\\frac{4 k \\pi}{4041}\\right)$ can be written in the form\n\n$$\n\\frac{a \\co... | 4049 |
[
{
"content": "Solve the following math problem step by 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 $\\vartriangle ABC$ have $AB = 15$, $AC = 20$, and $BC = 21$. Suppose $\\omega$ is a circle passing through $A$ that is ta... | 361 |
[
{
"content": "Solve the following math problem step by 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 $AB = AC =\\sqrt{7}, BC=1$. Let $G$ be the centroid of $\\triangle{ABC}$. ... | 1843 |
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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 $\\vartriangle ABC$ have $AB = 14$, $BC = 30$, $AC = 40$ and $\\vartriangle AB'C'$ with $AB' = 7\\sqrt6$, $B'C' = 15\\sqrt... | 55 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe integers from $1$ to $25,$ inclusive, are randomly placed into a $5$ by $5$ grid such that in each row, the numbers are in... | 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 quadratic polynomial $f(x)$ is called sparse if its degree is exactly 2 , if it has integer coefficients, and if there exist... | 228 |
[
{
"content": "Solve the following math problem step by 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 $p(x) = \\sum_{j=1}^{2n-1} a_j x^j$ with real coefficients is called [i]mountainous[/i] if $n \\ge 2$ and there e... | 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\nAlice, Bob, and Carol each independently roll a fair six-sided die and obtain the numbers $a, b, c$, respectively. They then c... | 551 |
[
{
"content": "Solve the following math problem step by 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 \\[\\frac{1}{1+2} + \\frac{1}{1+2+3} + \\ldots + \\frac{1}{1+2 + \\ldots + 20} = \\frac{m}{n}\\]\nwhere $m$ and $n$ are po... | 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\nA circle radius $320$ is tangent to the inside of a circle radius $1000$. The smaller circle is tangent to a diameter of the l... | 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 $r$ be a real number such that $\\sqrt[3]{r} - \\frac{1}{\\sqrt[3]{r}}=2$. Find $r^3 - \\frac{1}{r^3}$.\n\nRemember to put... | 2786 |
[
{
"content": "Solve the following math problem step by 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 $x_1$ and $x_2$ be the roots of the equation $x^2 + 3x + 1 = 0$. Compute \\[\\left(\\frac{x_1}{x_2 + 1}\\right)^2 + \\left... | 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\nLet $f$ be a real-valued function of real and positive argument such that $f(x) + 3xf(\\tfrac1x) = 2(x + 1)$ for all real numb... | 1002 |
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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 $Answer is the answer to the problem.\n\nHow many real numbers are roots of the polynomial \\[x^9 - 37x^8 - 2x^7 + 74x^6 + x^4 - 37x^3 - 2x^2 + 74x?\\]\n\nRemember to ... | 5 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nOne rainy afternoon you write the number $1$ once, the number $2$ twice, the number $3$ three times, and so forth until you ha... | 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\nHow many non-overlapping $2$ by $2$ squares will fit into a circle with radius $8$?\n\nRemember to put your answer on its own ... | 40 |
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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 positive integers less that $200$ are relatively prime to either $15$ or $24$?\n\nRemember to put your answer on its ... | 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 $Answer is the answer to the problem.\n\nIn the addition problem \n\\[ \\setlength{\\tabcolsep}{1mm}\\begin{tabular}{cccccc}& W & H & I & T & E\\\\ + & W & A & T & E &... | 169069 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind $x$ so that $2^{2^{3^{2^{2}}}} = 4^{4^{x}}$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"rol... | 40 |
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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 $x$ such that \\[\\frac{\\frac{5}{x-50}+ \\frac{7}{x+25}}{\\frac{2}{x-50}- \\frac{3}{x+25}} = 17.\\]\n\nRemember to put y... | 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 the number of nonnegative integers $n$ for which $(n^2 - 3n + 1)^2 + 1$ is a prime number\n\nRemember to put your answer ... | 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\nWe glue together $990$ one inch cubes into a $9$ by $10$ by $11$ inch rectangular solid. Then we paint the outside of the soli... | 382 |
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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 work team was working at a rate fast enough to process $1250$ items in ten hours. But after working for six hours, the tea... | 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\nThe number $1$ is special. The number $2$ is special because it is relatively prime to $1$. The number $3$ is not special beca... | 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\nJerry is mowing a rectangular lawn which is $77$ feet north to south by $83$ feet east to west. His lawn mower cuts a path $18... | 102 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe cost of producing each item is inversely proportional to the square root of the number of items produced. The cost of prod... | 490 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA palindrome is a number that reads the same forwards and backwards such as $3773$ or $42924$. What is the smallest $9$ digit ... | 105717501 |
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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 quadruples $(a,b,c,d)$ of integers which satisfy both \\begin{align*}\\frac{1}{a} + \\frac{1}{b} + \\frac{1... | 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\nA tailor met a tortoise sitting under a tree. When the tortoise was the tailor’s age, the tailor was only a quarter of his cur... | 77 |
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