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\nA $12 \\times 12$ board is divided into $144$ unit squares by drawing lines parallel to the sides. Two rooks placed on two uni... | 73 |
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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 $Answer is the answer to the problem.\n\nThe Stromquist Comet is visible every 61 years. If the comet was visible in 2017, what is the next leap year when the comet wi... | 2444 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f(x) = (x-1)(x-2)(x-3)^2(x-4)(x-5)(x-6)$. Compute $f''(3) - f'(3) + f(3)$.\n\nRemember to put your answer on its own line... | -24 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $S = \\{1, 2, \\dots, 9\\}$. Compute the number of functions $f : S \\rightarrow S$ such that, for all $s \\in S$, $f(f(f(... | 288 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAn equilateral pentagon $AMNPQ$ is inscribed in triangle $ABC$ such that $M \\in \\overline{AB}$, $Q \\in \\overline{AC}$, and... | 5073 |
[
{
"content": "Solve the following math problem step by 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 sum of all possible positive integers $n$, such that the product of its digits equals $n^2 - 15n - 27$.\n\nRemem... | 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\nA prime number $p$ is called a \\textbf{moderate} number if for every $2$ positive integers $k > 1$ and $m$, there exist $k$ p... | 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\nWhat is the smallest positive odd integer that has the same number of positive divisors as $360$?\n\nRemember to put your answ... | 3465 |
[
{
"content": "Solve the following math problem step by 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 such that $f(x) + f(x+1) = 2^x$ and $f(0) = 2010$. Find the last two digits of $f(2010)$.\n\nRemember to... | 51 |
[
{
"content": "Solve the following math problem step by 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 largest positive integer $n$ for which there exists a set $S$ with exactly $n$ numbers such that: \n- Each membe... | 1958 |
[
{
"content": "Solve the following math problem step by 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 triangle $ABC$, the median $AD$ (with $D$ on $BC$) and the angle bisector $BE$ (with $E$ on $AC$) are perpendicular to ea... | 47 |
[
{
"content": "Solve the following math problem step by 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 $\\operatorname{cif}(x)$ denote the sum of the digits of the number $x$ in the decimal system. Define $a_1 = 1997^{1996^{1... | 1 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA fair $6$-sided die is repeatedly rolled until a $1$, $4$, $5$, or $6$ is rolled. What is the expected value of the product o... | 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\nDefine\n\\[ S = \\tan^{-1}(2020) + \\sum_{j = 0}^{2020} \\tan^{-1}(j^2 - j + 1). \\]\nThen $S$ can be written as $\\frac{m \\p... | 4045 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFelix chooses a positive integer as the starting number and writes it on the board. He then repeats the following step:\n- If ... | 21 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe equation $x^5 - 2x^4 - 1 = 0$ has five complex roots $r_1, r_2, r_3, r_4, r_5$. Find the value of \n\\[\n\\frac{1}{r_1^8} ... | 16 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $N$ be the number of positive divisors of $2010^{2010}$ that end in the digit 2. What is the remainder when $N$ is divided... | 503 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many ways are there to color the vertices of a cube using the colors red, blue, or green such that no edge connects two ve... | 114 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a > 0$. If the inequality $22 < ax < 222$ holds for precisely $10$ positive integers $x$, find how many positive integers... | 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\nLet circle $O$ have radius $5$ with diameter $\\overline{AE}$. Point $F$ is outside circle $O$ such that lines $\\overline{FA}... | 15 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA word is an ordered, non-empty sequence of letters, such as $word$ or $wrod$. How many distinct 3-letter words can be made fr... | 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\nTwenty-five of the numbers $1, 2, \\ldots, 50$ are chosen. Twenty-five of the numbers $51, 52, \\ldots, 100$ are also chosen. ... | 2525 |
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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 42 points $P_1, P_2, \\ldots, P_{42}$ lie on a straight line in that order, such that the distance between $P_n$ and $P_{n... | 861 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLil Wayne, the rain god, determines the weather. If Lil Wayne makes it rain on any given day, the probability that he makes it... | 9 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many positive integers less than 2015 have exactly 9 positive integer divisors?\n\nRemember to put your answer on its own ... | 13 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor how many integers $k$ does the following system of equations have a solution other than $a=b=c=0$ in the set of real numbe... | 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 $Answer is the answer to the problem.\n\nFind the remainder when \\( \\prod_{n=3}^{33} \\left( 2n^4 - 25n^3 + 33n^2 \\right) \\) is divided by \\( 2019 \\).\n\nRemembe... | 0 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose there exist constants $A$, $B$, $C$, and $D$ such that \n\\[n^4 = A\\binom{n}{4} + B\\binom{n}{3} + C\\binom{n}{2} + D... | 75 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuzie flips a fair coin 6 times. The probability that Suzie flips 3 heads in a row but not 4 heads in a row is given by \\(\\f... | 19 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all positive integers $P$ such that the sum and product of all its divisors are $2P$ and $P^2$, respectively.\n\nRemember... | 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\nLet $\\ell$ be a line with negative slope passing through the point $(20,16)$. What is the minimum possible area of a triangle... | 640 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nBy dividing $2023$ by a natural number $m$, the remainder is $23$. How many numbers $m$ are there with this property?\n\nRemem... | 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\nThere is a right triangle $\\triangle ABC$ where $\\angle A$ is the right angle. On side $AB$, there are three points $X$, $Y$... | 277 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn trapezoid $ABCD$, where $AD \\parallel BC$ and $\\angle ABC + \\angle CDA = 270^{\\circ}$, compute $AB^2$. Given that $AB \... | 260 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEvaluate the following expression:\n\n\\[\n\\frac{2005\\displaystyle \\int_0^{1002}\\frac{dx}{\\sqrt{1002^2-x^2}+\\sqrt{1003^2... | 2005 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn triangle $ABC$, $AB = 8$, $AC = 9$, and $BC = 10$. Let $M$ be the midpoint of $BC$. Circle $\\omega_1$ with area $A_1$ pass... | 16295 |
[
{
"content": "Solve the following math problem step by 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 target below is made up of concentric circles with diameters $4$, $8$, $12$, $16$, and $20$. The area of the dark region i... | 60 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor a positive integer $n$, let $p(n)$ denote the number of distinct prime numbers that divide evenly into $n$. Determine the ... | 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\nIt takes $3$ rabbits $5$ hours to dig $9$ holes. It takes $5$ beavers $36$ minutes to build $2$ dams. At this rate, how many m... | 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\nFind the smallest positive integer $n$ such that if $n = p_1^{a_1} p_2^{a_2} \\ldots p_s^{a_s}$, then \\[ \\phi(n) = n \\left(... | 59455 |
[
{
"content": "Solve the following math problem step by 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 the two smallest positive integers $b$ with the following property: there are at least ten integers $0 \\le... | 540 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n\\( \\triangle ABC \\) has side lengths \\( AB=20 \\), \\( BC=15 \\), and \\( CA=7 \\). Let the altitudes of \\( \\triangle AB... | 15 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA box of strawberries, containing $12$ strawberries total, costs $2$. A box of blueberries, containing $48$ blueberries total,... | 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 $Answer is the answer to the problem.\n\nDetermine the number of integer pairs \\((x, y)\\) such that \\((|x| - 2)^2 + (|y| - 2)^2 < 5\\).\n\nRemember to put your answ... | 48 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor any set $S$, let $P(S)$ be its power set, the set of all its subsets. Consider all sets $A$ of 2015 arbitrary finite sets.... | 872 |
[
{
"content": "Solve the following math problem step by 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 circles $C_i$ are given in the plane: $C_1$ has diameter $AB$ of length $1$; $C_2$ is concentric and has diameter $k$ ($... | 1 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the total number of integer solutions of the equation $x^5 - y^2 = 4$.\n\nRemember to put your answer on its own line aft... | 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 $Answer is the answer to the problem.\n\nHow many reorderings of \\(2, 3, 4, 5, 6\\) have the property that every pair of adjacent numbers are relatively prime?\n\nRem... | 4 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose $ABCD$ is a rectangle whose diagonals meet at $E$. The perimeter of triangle $ABE$ is $10\\pi$ and the perimeter of tr... | 47 |
[
{
"content": "Solve the following math problem step by 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 numbers in the sequence 101, 104, 109, 116, $\\dots$ are of the form $a_n = 100 + n^2$, where $n = 1, 2, 3, \\dots$. For e... | 401 |
[
{
"content": "Solve the following math problem step by 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 George Washington Bridge is $2016$ meters long. Sally is standing on the bridge, $1010$ meters from its left end. Each ste... | 1016060 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAlfred and Bonnie play a game where they take turns tossing a fair coin. The winner of the game is the first person to obtain ... | 93 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nBob, a spherical person, is floating around peacefully when Dave the giant orange fish launches him straight up at 23 m/s with... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nBroady The Boar is playing a board game consisting of a circle with $2021$ points on it, labeled $0$, $1$, $2$, \\ldots, $2020... | 2021 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the greatest positive integer $n$ such that $3^n$ divides $70! + 71! + 72!$.\n\nRemember to put your answer on its own li... | 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 $Answer is the answer to the problem.\n\nWu starts with exactly one coin. Each year, Wu flips every coin he has at once. For each heads he flips, Wu receives a coin, a... | 71622400 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThree non-zero real numbers $a$, $b$, and $c$ satisfy the conditions $a + b + c = 0$ and $a^4 + b^4 + c^4 = 128$. Determine al... | -8 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a$, $b$, $c$ be positive real numbers for which:\n\\[\n \\frac{5}{a} = b+c, \\quad\n \\frac{10}{b} = c+a, \\quad \\text... | 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\nAn acute triangle $ABC$ has side lengths $a$, $b$, $c$ such that $a$, $b$, $c$ form an arithmetic sequence. Given that the are... | 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\nThe three-digit number $999$ has a special property: It is divisible by $27$, and its digit sum is also divisible by $27$. The... | 75 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many ordered triples of integers $(x, y, z)$ are there such that $x^2 + y^2 + z^2 = 34$?\n\nRemember to put your answer on... | 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\nFor a bijective function $g : \\mathbb{R} \\to \\mathbb{R}$, we say that a function $f : \\mathbb{R} \\to \\mathbb{R}$ is its ... | 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\nFind how many integer values $3 \\le n \\le 99$ satisfy that the polynomial $x^2 + x + 1$ divides $x^{2^n} + x + 1$.\n\nRememb... | 49 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA chicken breeder went to check the price per chick he charged the previous year. He found an invoice, half erased, which read... | 511 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all positive integers $n$ such that $n$ is a perfect number and $\\varphi(n)$ is a power of $2$.\n\nRemember to put your ... | 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\nLet $ABCD$ be a square with side length $2$. Let $M$ and $N$ be the midpoints of $\\overline{BC}$ and $\\overline{CD}$ respect... | 3225 |
[
{
"content": "Solve the following math problem step by 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 natural numbers that cannot be written as the sum of three squares. Legendre's three-square theorem stat... | 111 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf \\((2^x - 4^x) + (2^{-x} - 4^{-x}) = 3\\), find the numerical value of the expression \n\\[\n(8^x + 3\\cdot 2^x) + (8^{-x} ... | -1 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nBeto plays the following game with his computer: initially, the computer randomly picks $30$ integers from $1$ to $2015$, and ... | 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\nXavier takes a permutation of the numbers $1$ through $2011$ at random, where each permutation has an equal probability of bei... | 1006 |
[
{
"content": "Solve the following math problem step by 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 of the first $1000$ positive integers can be written as the sum of finitely many distinct numbers from the sequence $... | 105 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn preparation for the family's upcoming vacation, Tony puts together five bags of jelly beans, one bag for each day of the tr... | 45 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the largest $n$ for which there exists a sequence $(a_0, a_1, \\ldots, a_n)$ of non-zero digits such that, for each $k$, ... | 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\nFind the sum of the first $55$ terms of the sequence $\\binom{0}{0}, \\binom{1}{0}, \\binom{1}{1}, \\binom{2}{0}, \\binom{2}{1... | 1023 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe alphabet in its natural order $\\text{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$ is $T_0$. We apply a permutation to $T_0$ to get $T_1$,... | 252 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the number of integers $n$ such that $1 \\leq n \\leq 2017$ and $(n-2)(n-0)(n-1)(n-7)$ is a multiple of $1001$.\n\nRememb... | 99 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nOn the blackboard, there are $25$ points arranged in a grid as shown in the figure below. Gastón needs to choose $4$ points th... | 50 |
[
{
"content": "Solve the following math problem step by 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 point chosen on the interior of side $\\overline{BC}$ of triangle $\\triangle ABC$ with side lengths $\\overline{... | 2365 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $x$ be the minimal root of the equation $x^2 - 4x + 2 = 0$. Find the first two digits after the decimal point of the numbe... | 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\nThere are $10$ horses, named Horse 1, Horse 2, $\\ldots$, Horse 10. They get their names from how many minutes it takes them t... | 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\nDetermine the sum of the two largest prime factors of the integer $89! + 90!$.\n\nRemember to put your answer on its own line ... | 172 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nOn Thursday, January 1, 2015, Anna buys one book and one shelf. For the next two years, she buys one book every day and one sh... | 89 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nReimu has a wooden cube. In each step, she creates a new polyhedron from the previous one by cutting off a pyramid from each v... | 974 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor every positive integer $k$, let $\\mathbf{T}_k = (k(k+1), 0)$, and define $\\mathcal{H}_k$ as the homothety centered at $\... | 256 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nJerry and Hannah Kubik live in Jupiter Falls with their five children. Jerry works as a Renewable Energy Engineer for the Sout... | 56 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nMr. Jones teaches algebra. He has a whiteboard with a pre-drawn coordinate grid that runs from $-10$ to $10$ in both the $x$ a... | 478 |
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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 $Answer is the answer to the problem.\n\nLet\n\\[ N = \\sum_{a_1 = 0}^2 \\sum_{a_2 = 0}^{a_1} \\sum_{a_3 = 0}^{a_2} \\dots \\sum_{a_{2011} = 0}^{a_{2010}} \\left [ \\p... | 95 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a_0$, $a_1$, $a_2$, $\\dots$ be an infinite sequence of real numbers such that $a_0 = \\frac{4}{5}$ and \n\\[ \na_{n} =... | 167 |
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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 $Answer is the answer to the problem.\n\nFind the smallest three-digit divisor of the number \\(1\\underbrace{00\\ldots 0}_{100\\text{ zeros}}1\\underbrace{00\\ldots 0... | 111 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor how many integers $x \\in [0, 2007]$ is $\\frac{6x^3+53x^2+61x+7}{2x^2+17x+15}$ reducible?\n\nRemember to put your answer ... | 286 |
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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 candy store has $100$ pieces of candy to give away. When you get to the store, there are five people in front of you, number... | 3003 |
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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 $\\{a_{n}\\}_{n\\geq 0}$ is defined by:\n\n- $a_{0} = 20$\n- $a_{1} = 100$\n- $a_{n+2} = 4a_{n+1} + 5a_{n} + 20$ ... | 108 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $P$ be a point inside the equilateral triangle $ABC$ such that $6\\angle PBC = 3\\angle PAC = 2\\angle PCA$. Find the meas... | 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 $Answer is the answer to the problem.\n\nA girl and a guy are going to arrive at a train station. If they arrive within 10 minutes of each other, they will instantly f... | 10 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\mathcal{A}$ 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 $Answer is the answer to the problem.\n\nWhat is the remainder when \\( \\sum_{k=0}^{100} 10^k \\) is divided by \\( 9 \\)?\n\nRemember to put your answer on its own l... | 2 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nPoints $A$, $B$, $C$, and $D$ lie on a circle such that chords $\\overline{AC}$ and $\\overline{BD}$ intersect at a point $E$ ... | 115 |
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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 triangle $ABC$ ($\\angle BCA = 90^{\\circ}$), let $D$ be the intersection of $AB$ with a circumference having a diameter ... | 92 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA robot is standing on the bottom left vertex $(0,0)$ of a $5 \\times 5$ grid and wants to go to $(5,5)$, only moving to the r... | 1650 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA right triangle $ABC$ is inscribed in the circular base of a cone. If two of the side lengths of $ABC$ are $3$ and $4$, and t... | 12 |
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