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\nHow many positive three-digit integers have a remainder of $2$ when divided by $6$, a remainder of $5$ when divided by $9$, an... | 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\nConsider polynomials $P(x)$ of degree at most $3$, each of whose coefficients is an element of $\\{0, 1, 2, 3, 4, 5, 6, 7, 8, ... | 220 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn $\\triangle PAT,$ $\\angle P=36^{\\circ},$ $\\angle A=56^{\\circ},$ and $PA=10.$ Points $U$ and $G$ lie on sides $\\overlin... | 80 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAjay is standing at point $A$ near Pontianak, Indonesia, $0^\\circ$ latitude and $110^\\circ \\text{ E}$ longitude. Billy is s... | 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\nLet $\\lfloor x \\rfloor$ denote the greatest integer less than or equal to $x$. Find the number of real numbers $x$ that sati... | 199 |
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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 positive integer $n$ and nonzero digits $a$, $b$, and $c$, let $A_n$ be the $n$-digit integer each of whose digits is eq... | 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\nWhat is the hundreds digit of $(20! - 15!)$?\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "... | 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\nThree identical rectangles are put together to form rectangle $ABCD$, as shown in the figure below. Given that the length of t... | 150 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the sum of all real numbers $x$ for which the median of the numbers $4,6,8,17,$ and $x$ is equal to the mean of those ... | -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 $Answer is the answer to the problem.\n\nThe figure below shows a square and four equilateral triangles, with each triangle having a side lying on a side of the square... | 19 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What ... | 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\nWhat is the greatest three-digit positive integer $n$ for which the sum of the first $n$ positive integers is $\\underline{not... | 996 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA rectangular floor that is $10$ feet wide and $17$ feet long is tiled with $170$ one-foot square tiles. A bug walks from one ... | 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\nThe figure below shows $13$ circles of radius $1$ within a larger circle. All the intersections occur at points of tangency. F... | 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\nFor some positive integer $k$, the repeating base-$k$ representation of the (base-ten) fraction $\\frac{7}{51}$ is $0.\\overli... | 16 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the greatest possible sum of the digits in the base-seven representation of a positive integer less than $2019$?\n\nRe... | 22 |
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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 $Answer is the answer to the problem.\n\nRight triangle \\(ACD\\) with right angle at \\(C\\) is constructed outwards on the hypotenuse \\(\\overline{AC}\\) of isoscel... | 16 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many ways are there to paint each of the integers $2, 3, \\cdots , 9$ either red, green, or blue so that each number has a... | 432 |
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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 base-ten representation for $19!$ is $121,6T5,100,40M,832,H00$, where $T$, $M$, and $H$ denote digits that are not given. ... | 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\nThere are lily pads in a row numbered $0$ to $11$, in that order. There are predators on lily pads $3$ and $6$, and a morsel o... | 271 |
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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 child builds towers using identically shaped cubes of different colors. How many different towers with a height of $8$ cubes... | 1260 |
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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 $s_k$ denote the sum of the $\\textit{k}$th powers of the roots of the polynomial $x^3-5x^2+8x-13$. In particular, $s_0=3$... | 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 $Answer is the answer to the problem.\n\nA sphere with center \\(O\\) has radius \\(6\\). A triangle with sides of length \\(15, 15,\\) and \\(24\\) is situated in spa... | 7 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nRaashan, Sylvia, and Ted play the following game. Each starts with $＄$ $1$. A bell rings every $15$ seconds, at which time eac... | 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 $Answer is the answer to the problem.\n\nReal numbers between \\(0\\) and \\(1\\), inclusive, are chosen in the following manner. A fair coin is flipped. If it lands h... | 23 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet \\[z=\\frac{1+i}{\\sqrt{2}}.\\] Calculate the value of \\[\\left(z^{1^2}+z^{2^2}+z^{3^2}+\\dots+z^{{12}^2}\\right) \\cdot ... | 36 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nRaashan, Sylvia, and Ted play the following game. Each starts with $＄$ $1$. A bell rings every $15$ seconds, at which time eac... | 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\nCircles $\\omega$ and $\\gamma$, both centered at $O$, have radii $20$ and $17$, respectively. Equilateral triangle $ABC$, who... | 130 |
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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 $Answer is the answer to the problem.\n\nTravis has to babysit the terrible Thompson triplets. Knowing that they love big numbers, Travis devises a counting game for t... | 5979 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDefine binary operations $\\diamondsuit$ and $\\heartsuit$ by \\[a \\, \\diamondsuit \\, b = a^{\\log_{7}(b)} \\qquad \\text{a... | 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\nHow many sequences of $0$s and $1$s of length $19$ are there that begin with a $0$, end with a $0$, contain no two consecutive... | 65 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\triangle A_0B_0C_0$ be a triangle whose angle measures are exactly $59.999^\\circ$, $60^\\circ$, and $60.001^\\circ$. F... | 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\nFour friends do yardwork for their neighbors over the weekend, earning $＄$ $15$, $＄$ $20$, $＄$ $25,$ and $＄$ $40,$ respectivel... | 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\nHow many positive even multiples of $3$ less than $2020$ are perfect squares?\n\nRemember to put your answer on its own line 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\nWhat is the median of the following list of $4040$ numbers? \n$1, 2, 3, \\ldots, 2020, 1^2, 2^2, 3^2, \\ldots, 2020^2$\nCalcu... | 19765 |
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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 single bench section at a school event can hold either $7$ adults or $11$ children. When $N$ bench sections are connected en... | 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 $Answer is the answer to the problem.\n\nIn unit square \\(ABCD,\\) the inscribed circle \\(\\omega\\) intersects \\(\\overline{CD}\\) at \\(M,\\) and \\(\\overline{AM... | 15 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nMs. Carr asks her students to read any $5$ of the $10$ books on a reading list. Harold randomly selects $5$ books from this li... | 88 |
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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 value of $\\sqrt{\\log_2{6}+\\log_3{6}}$? The original answer is in the form $\\sqrt{\\log_k{3}}+\\sqrt{\\log_m{2}... | 5 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nQuadrilateral $ABCD$ satisfies $\\angle ABC = \\angle ACD = 90^{\\circ}$, $AC=20$, and $CD=30$. Diagonals $\\overline{AC}$ and... | 360 |
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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设有三个函数，第一个是 $y=\\varphi(x)$ ，它的反函数是第二个函数，而第三个函数的图象与第二个函数的图象关于 $x+y=0$ 对称。第三个函数的表达式为 $y=ax+b\\varphi(cx)$，请给出a+b+c的值。\n\nRememb... | -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 $Answer is the answer to the problem.\n\nTwo different cubes of the same size are to be painted, with the color of each face being chosen independently and at random t... | 1171 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor how many positive integers $n \\le 1000$ is\n$\\left\\lfloor \\dfrac{998}{n} \\right\\rfloor+\\left\\lfloor \\dfrac{999}{n... | 22 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the maximum value of \\(\\frac{(2^t-3t)t}{4^t}\\) for real values of \\(t?\\) The original answer is in \\(\\frac{k}{m... | 13 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nJason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, p... | 43 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many positive integers $n$ satisfy \n$$\\dfrac{n+1000}{70} = \\lfloor \\sqrt{n} \\rfloor?$$\n\nRemember to put your answer... | 6 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $D(n)$ denote the number of ways of writing the positive integer $n$ as a product\\[n = f_1\\cdot f_2\\cdots f_k,\\]where ... | 112 |
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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\nMenkara has a $4 \\times 6$ index card. If she shortens the length of one side of this card by $1$ inch, the card would have a... | 20 |
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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 six-digit number $\\underline{2}\\,\\underline{0}\\,\\underline{2}\\,\\underline{1}\\,\\underline{0}\\,\\underline{A}$ is ... | 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\nWhen a student multiplied the number $66$ by the repeating decimal, \n$\\underline{1}.\\underline{a} \\ \\underline{b} \\ \\u... | 75 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe least positive integer with exactly $2021$ distinct positive divisors can be written in the form $m \\cdot 6^k$, where $m$... | 58 |
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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\nMr. Zhou places all the integers from $1$ to $225$ into a $15$ by $15$ grid. He places $1$ in the middle square (eighth row an... | 367 |
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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 $Answer is the answer to the problem.\n\nA two-digit positive integer is said to be cuddly if it is equal to the sum of its nonzero tens digit and the square of its un... | 1 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $M$ be the least common multiple of all the integers $10$ through $30,$ inclusive. Let $N$ be the least common multiple of... | 74 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTwo distinct numbers are selected from the set $\\{1,2,3,4,\\dots,36,37\\}$ so that the sum of the remaining $35$ numbers is t... | 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 $Answer is the answer to the problem.\n\n$A$ school has $100$ students and $5$ teachers. In the first period, each student is taking one class, and each teacher is tea... | -135 |
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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 base-nine representation of the number $N$ is $27{,}006{,}000{,}052_{\\text{nine}}.$ Find the remainder when $N$ is divide... | 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 $Answer is the answer to the problem.\n\nGrandma has just finished baking a large rectangular pan of brownies. She is planning to make rectangular pieces of equal size... | 60 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTriangle \\(ABC\\) has \\(AB=13, BC=14\\) and \\(AC=15\\). Let \\(P\\) be the point on \\(\\overline{AC}\\) such that \\(PC=10... | 14 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEmily sees a ship traveling at a constant speed along a straight section of a river. She walks parallel to the riverbank at a ... | 70 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nConsider two concentric circles of radius $17$ and $19.$ The larger circle has a chord, half of which lies inside the smaller ... | 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\nUna rolls $6$ standard $6$-sided dice simultaneously and calculates the product of the $6$ numbers obtained. What is the proba... | 123 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA square with side length \\(3\\) is inscribed in an isosceles triangle with one side of the square along the base of the tria... | 85 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $c = \\frac{2\\pi}{11}.$ What is the value of\n$\\frac{\\sin 3c \\cdot \\sin 6c \\cdot \\sin 9c \\cdot \\sin 12c \\cdot \\... | 1 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAll the roots of the polynomial \\(z^6-10z^5+Az^4+Bz^3+Cz^2+Dz+16\\) are positive integers, possibly repeated. The original an... | -88 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $ABCD$ be a rectangle and let $\\overline{DM}$ be a segment perpendicular to the plane of $ABCD$. Suppose that $\\overline... | 29 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the following list of numbers, the integer $n$ appears $n$ times in the list for $1 \\leq n \\leq 200$.\n$1, 2, 2, 3, 3, 3,... | 142 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the following list of numbers, the integer $n$ appears $n$ times in the list for $1 \\leq n \\leq 200$. \n$1, 2, 2, 3, 3, ... | 142 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose $a$, $b$, $c$ are positive integers such that \\[a+b+c=23\\] and \\[\\gcd(a,b)+\\gcd(b,c)+\\gcd(c,a)=9.\\] What is the... | 438 |
[
{
"content": "Solve the following math problem step by 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 architect is building a structure that will place vertical pillars at the vertices of regular hexagon $ABCDEF$, which is ly... | 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\nDistinct lines $\\ell$ and $m$ lie in the $xy$-plane. They intersect at the origin. Point $P(-1, 4)$ is reflected about line $... | -1 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEach of the $20$ balls is tossed independently and at random into one of the $5$ bins. Let $p$ be the probability that some bi... | 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\nThe area of the region bounded by the graph of $x^2+y^2 = 3|x-y| + 3|x+y|$ is $m+n\\pi$, where $m$ and $n$ are integers. What ... | 54 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose that on a parabola with vertex \\(V\\) and a focus \\(F\\) there exists a point \\(A\\) such that \\(AF=20\\) and \\(A... | 43 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA square piece of paper has side length $1$ and vertices $A,B,C,$ and $D$ in that order. As shown in the figure, the paper is ... | 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\nThe five solutions to the equation\\[(z-1)(z^2+2z+4)(z^2+4z+6)=0\\] may be written in the form $x_k+y_ki$ for $1\\le k\\le 5,$... | 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\nLet $ABCD$ be an isosceles trapezoid with $\\overline{BC}\\parallel \\overline{AD}$ and $AB=CD$. Points $X$ and $Y$ lie on dia... | 38 |
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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 $x$, let \n$$P(x)=1+\\cos(x)+i\\sin(x)-\\cos(2x)-i\\sin(2x)+\\cos(3x)+i\\sin(3x)$$\nwhere $i = \\sqrt{-1}$. ... | 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\nAzar and Carl play a game of tic-tac-toe. Azar places an $X$ in one of the boxes in a $3$-by-$3$ array of boxes, then Carl pla... | 148 |
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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 $Answer is the answer to the problem.\n\nRight triangle \\(ABC\\) has side lengths \\(BC=6\\), \\(AC=8\\), and \\(AB=10\\). A circle centered at \\(O\\) is tangent to ... | 47 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA quadratic polynomial with real coefficients and leading coefficient \\(1\\) is called \\(\\emph{disrespectful}\\) if the equ... | 21 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the average number of pairs of consecutive integers in a randomly selected subset of $5$ distinct integers chosen from... | 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 $S$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between $1$ and $30,$ inc... | 85 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $d(n)$ denote the number of positive integers that divide $n$, including $1$ and $n$. For example, $d(1)=1,d(2)=2,$ and $d... | 9 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $m\\ge 5$ be an odd integer, and let $D(m)$ denote the number of quadruples $(a_1, a_2, a_3, a_4)$ of distinct integers wi... | 11 |
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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 $n$ a positive integer, let $R(n)$ be the sum of the remainders when $n$ is divided by $2$, $3$, $4$, $5$, $6$, $7$, $8$, ... | 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\nIn rhombus $ABCD$, point $P$ lies on segment $\\overline{AD}$ so that $\\overline{BP}$ $\\perp$ $\\overline{AD}$, $AP = 3$, an... | 20 |
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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 $Answer is the answer to the problem.\n\nHow many three-digit positive integers have an odd number of even digits? Provide your answer as an integer.\n\nRemember to pu... | 450 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhen three positive integers $a$, $b$, and $c$ are multiplied together, their product is $100$. Suppose $a < b < c$. In how ma... | 4 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA donkey suffers an attack of hiccups and the first hiccup happens at $4:00$ one afternoon. Suppose that the donkey hiccups re... | 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\nFor how many values of the constant $k$ will the polynomial $x^{2}+kx+36$ have two distinct integer roots? Provide the number ... | 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\nThe taxicab distance between points $(x_1, y_1)$ and $(x_2, y_2)$ in the coordinate plane is given by \\[|x_1 - x_2| + |y_1 - ... | 841 |
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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 the following $100$ sets of $10$ elements each:\n\\begin{align*} &\\{1,2,3,\\ldots,10\\}, \\\\ &\\{11,12,13,\\ldots,2... | 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\nHenry the donkey has a very long piece of pasta. He takes a number of bites of pasta, each time eating $3$ inches of pasta fro... | 44 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\triangle ABC$ be a scalene triangle. Point $P$ lies on $\\overline{BC}$ so that $\\overline{AP}$ bisects $\\angle BAC.$... | 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\nThe graph of $y=x^2+2x-15$ intersects the $x$-axis at points $A$ and $C$ and the $y$-axis at point $B$. Find $\\tan(\\angle AB... | 11 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nQuadrilateral $ABCD$ with side lengths $AB=7, BC=24, CD=20, DA=15$ is inscribed in a circle. The area interior to the circle b... | 1565 |
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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 roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangula... | 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 midpoints of the four sides of a rectangle are $(-3,0), (2,0), (5,4),$ and $(0,4).$ What is the area of the rectangle?\n\n... | 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\nMr. Ramos gave a test to his class of $20$ students. The dot plot below shows the distribution of test scores. \nLater Mr. Ra... | 4 |
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