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 $Answer is the answer to the problem.\n\nSeven distinct pieces of candy are to be distributed among three bags. The red bag and the blue bag must each receive at least... | 1932 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEach of $2010$ boxes in a line contains a single red marble, and for $1 \\le k \\le 2010$, the box in the $k{th}$ position als... | 45 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nMonic quadratic polynomial $P(x)$ and $Q(x)$ have the property that $P(Q(x))$ has zeros at $x=-23, -21, -17,$ and $-15$, and $... | -100 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nJim starts with a positive integer $n$ and creates a sequence of numbers. Each successive number is obtained by subtracting th... | 3 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a > 0$, and let $P(x)$ be a polynomial with integer coefficients such that \n$P(1) = P(3) = P(5) = P(7) = a$, and \n$P... | 315 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTwo quadrilaterals are considered the same if one can be obtained from the other by a rotation and a translation. How many dif... | 568 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $N$ be the second smallest positive integer that is divisible by every positive integer less than $7$. What is the sum of ... | 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 $Answer is the answer to the problem.\n\nThe players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They score... | 13 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA majority of the $30$ students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought ... | 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 $Answer is the answer to the problem.\n\nTwo real numbers are selected independently and at random from the interval \\([-20,10]\\). Find the probability that the prod... | 14 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nKeiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the end... | 4 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTwo congruent squares, $ABCD$ and $PQRS$, have side length $15$. They overlap to form the $15$ by $25$ rectangle $AQRD$ shown.... | 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\nA segment through the focus $F$ of a parabola with vertex $V$ is perpendicular to $\\overline{FV}$ and intersects the parabola... | 8 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to ... | 4 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nRhombus \\(ABCD\\) has side length \\(2\\) and \\(\\angle B = 120^\\circ\\). Region \\(R\\) consists of all points inside the ... | 8 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the given circle, the diameter $\\overline{EB}$ is parallel to $\\overline{DC}$, and $\\overline{AB}$ is parallel to $\\ove... | 130 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f(x) = 10^{10x}, g(x) = \\log_{10}\\left(\\frac{x}{10}\\right), h_1(x) = g(f(x))$, and $h_n(x) = h_1(h_{n-1}(x))$ for int... | 16089 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nQuadrilateral $ABCD$ is a trapezoid, $AD = 15$, $AB = 50$, $BC = 20$, and the altitude is $12$. What is the area of the trapez... | 750 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet \\(f_{1}(x)=\\sqrt{1-x}\\), and for integers \\(n \\geq 2\\), let \\(f_{n}(x)=f_{n-1}(\\sqrt{n^2 - x})\\). If \\(N\\) is t... | -226 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the hundreds digit of \\(2011^{2011}\\)?\n\nRemember to put your answer on its own line after \"Answer:\".",
"role... | 6 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $P(z) = z^8 + \\left(4\\sqrt{3} + 6\\right)z^4 - \\left(4\\sqrt{3} + 7\\right)$. Find the minimum perimeter among all the ... | 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\nSuppose that the euro is worth $1.3$ dollars. If Diana has $500$ dollars and Etienne has $400$ euros, by what percent is the v... | 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 $Answer is the answer to the problem.\n\nIn a bag of marbles, \\(\\frac{3}{5}\\) of the marbles are blue and the rest are red. If the number of red marbles is doubled ... | 11 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nMary divides a circle into $12$ sectors. The central angles of these sectors, measured in degrees, are all integers and they f... | 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 $Answer is the answer to the problem.\n\nAn iterative average of the numbers \\(1, 2, 3, 4\\), and \\(5\\) is computed the following way. Arrange the five numbers in s... | 25 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA triangle has an area of $30$, one side of length $10$, and the median to that side has a length of $9$. Let $\\theta$ be the... | 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\nPoint $B$ is due east of point $A$. Point $C$ is due north of point $B$. The distance between points $A$ and $C$ is $10\\sqrt{... | 63 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA square region \\(ABCD\\) is externally tangent to the circle with equation \\(x^2+y^2=1\\) at the point \\((0,1)\\) on the s... | 30 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many sequences of zeros and ones of length $20$ have all the zeros consecutive, or all the ones consecutive, or both?\n\nR... | 380 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nBernardo and Silvia play the following game. An integer between $0$ and $999$ inclusive is selected and given to Bernardo. Whe... | 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 $Answer is the answer to the problem.\n\nJesse cuts a circular disk of radius \\(12\\), along \\(2\\) radii to form \\(2\\) sectors, one with a central angle of \\(120... | 20 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAmy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore,... | 132 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA unit cube has vertices \\(P_1,P_2,P_3,P_4,P_1',P_2',P_3',\\) and \\(P_4'\\). Vertices \\(P_2\\), \\(P_3\\), and \\(P_4\\) ar... | 9 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe sum of the first $m$ positive odd integers is $212$ more than the sum of the first $n$ positive even integers. Find the su... | 255 |
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{
"content": "Solve the following math problem step by 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 set of nine distinct integers. Six of the elements are $2$, $3$, $4$, $6$, $9$, and $14$. What is the number of p... | 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 $Answer is the answer to the problem.\n\nA circle of radius \\(2\\) is cut into four congruent arcs. The four arcs are joined to form the star figure shown. Find the r... | 5 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nJoey and his five brothers are ages $3$, $5$, $7$, $9$, $11$, and $13$. One afternoon two of his brothers whose ages sum to $1... | 11 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSix points are equally spaced around a circle of radius \\(1\\). Three of these points are the vertices of a triangle that is ... | 5 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLine \\(l_1\\) has equation \\(3x - 2y = 1\\) and goes through \\(A = (-1, -2)\\). Line \\(l_2\\) has equation \\(y = 1\\) and... | 7 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$A$ flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are rose... | 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\nLet $S$ be the set of positive integers $n$ for which $\\frac{1}{n}$ has the repeating decimal representation $0.\\overline{ab... | 143 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTed's grandfather used his treadmill on $3$ days this week. He went $2$ miles each day. On Monday he jogged at a speed of $5$ ... | 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\nIn $\\triangle ABC$, $AB=AC=28$ and $BC=20$. Points $D,E,$ and $F$ are on sides $\\overline{AB}$, $\\overline{BC}$, and $\\ove... | 56 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many three-digit numbers are not divisible by $5$, have digits that sum to less than $20$, and have the first digit equal ... | 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\nIsabella uses one-foot cubical blocks to build a rectangular fort that is $12$ feet long, $10$ feet wide, and $5$ feet high. T... | 280 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn base $10$, the number $2013$ ends in the digit $3$. In base $9$, on the other hand, the same number is written as $(2676)_{... | 13 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe real numbers \\(c, b, a\\) form an arithmetic sequence with \\(a \\geq b \\geq c \\geq 0\\). The quadratic \\(ax^2+bx+c\\)... | 5 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe number $2013$ is expressed in the form \n$2013 = \\frac {a_1!a_2!...a_m!}{b_1!b_2!...b_n!}$, \nwhere $a_1 \\ge a_2 \... | 2 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSamantha lives $2$ blocks west and $1$ block south of the southwest corner of City Park. Her school is $2$ blocks east and $2$... | 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 $Answer is the answer to the problem.\n\nSix spheres of radius \\(1\\) are positioned so that their centers are at the vertices of a regular hexagon of side length \\(... | 5 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe regular octagon $ABCDEFGH$ has its center at $J$. Each of the vertices and the center are to be associated with one of the... | 1152 |
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{
"content": "Solve the following math problem step by 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 nonnegative integer number that reads the same forwards and backwards when written in base $10$ with no lead... | 41 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nToothpicks are used to make a grid that is $60$ toothpicks long and $32$ toothpicks wide. How many toothpicks are used altoget... | 3932 |
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{
"content": "Solve the following math problem step by 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 $ABC$, $AB=13$, $BC=14$, and $CA=15$. Distinct points $D$, $E$, and $F$ lie on segments $\\overline{BC}$, $\\overl... | 21 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSquare $ABCD$ has side length $s$, and a circle centered at $E$ has radius $r$. Both $r$ and $s$ are rational numbers. The cir... | 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 $Answer is the answer to the problem.\n\nIn \\(\\triangle ABC\\) shown in the figure, \\(AB=7\\), \\(BC=8\\), \\(CA=9\\), and \\(\\overline{AH}\\) is an altitude. Poin... | 28 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe sides of a right triangle are $a$ and $b$, and the hypotenuse is $c$. A perpendicular from the vertex divides $c$ into seg... | 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 $Answer is the answer to the problem.\n\nThe area of a circle inscribed in a regular hexagon is \\(100\\pi\\). What is the area of the hexagon? The original answer is ... | 203 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nMary is about to pay for five items at the grocery store. The prices of the items are $7.99, $4.99, $2.99, $1.99, and $0.99. M... | 5 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe graphs of $y = -|x - a| + b$ and $y = |x - c| + d$ intersect at points $(2,5)$ and $(8,3)$. Find the value of $a + c$.\n\n... | 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\nHow many unique $3$-letter sequences with no spaces can be made using the letters in \"AUGUSTIN LOUIS CAUCHY\", which contains... | 1486 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhen \\(p = \\sum_{k=1}^{6} k \\ln{k}\\), the number \\(e^p\\) is an integer. What is the largest power of \\(2\\) that is a f... | 16 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe vertices of triangle \\(PQR\\) have coordinates as follows: \\(P(0,a)\\), \\(Q(b,0)\\), \\(R(c,d)\\), where \\(a\\), \\(b\... | 5 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the adjoining figure, \\( \\triangle CDE \\) is an equilateral triangle and \\( \\square ABCD \\) and \\( \\square DEFG \\)... | 120 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf the angles $a, b, c, d,$ and $e$ are known, what is the expression for the angle $u$ in terms of $a$, $b$, $c$, $d$, and $e... | 3 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe Tigers beat the Sharks $2$ out of the first $3$ times they played. They then played $N$ more times, and the Sharks ended u... | 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\nHow many ordered pairs $(a, b)$ such that $a$ is a real positive number and $b$ is an integer between $2$ and $200$, inclusive... | 597 |
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{
"content": "Solve the following math problem step by 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 integer $n$ such that $$n \\leq 1 + \\frac{1}{\\sqrt{2}} + \\frac{1}{\\sqrt{3}} + \\cdots + \\frac{1}{\\s... | 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\nPablo buys popsicles for his friends. The store sells single popsicles for $1$ each, 3-popsicle boxes for $2$, and 5-popsicle ... | 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\nHow many $5$-digit positive numbers which contain only odd numbers are there such that there is at least one pair of consecuti... | 1845 |
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{
"content": "Solve the following math problem step by 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 symbol $R_k$ stands for an integer whose base-ten representation is a sequence of $k$ ones. For example, $R_3=111$, $R_5=1... | 15 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nJuan rolls a fair regular octahedral die marked with the numbers $1$ through $8$. Then Amal rolls a fair six-sided die. Find t... | 3 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f(x) = x^{2}(1-x)^{2}$. What is the value of the sum\n\\begin{align*}\nf\\left(\\frac{1}{2019}\\right)-f\\left(\\frac{2}{... | 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\nBeth adds the consecutive positive integers $a$, $b$, $c$, $d$, and $e$, and finds that the sum is a perfect square. She then ... | 1127 |
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{
"content": "Solve the following math problem step by 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 $PC=10$. There are exactl... | 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\nRepresent the hypotenuse of a right triangle by $c$ and the area by $A$. What is the altitude on the hypotenuse? The original ... | 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 $Answer is the answer to the problem.\n\nA square with an area of \\(2\\) is inscribed within a larger square with an area of \\(3\\), forming four congruent triangles... | 5 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many positive integers less than $1000$ are there such that they cannot be written as the sum of $2$ or more successive po... | 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\nIf the points $(1, y_1)$ and $(-1, y_2)$ lie on the graph of $y = ax^2 + bx + c$, and $y_1 - y_2 = -6$, the original answer is... | -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\nBertha has $6$ daughters and no sons. Some of her daughters have $6$ daughters, and the rest have none. Bertha has a total of ... | 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 $Answer is the answer to the problem.\n\nLet \\(ABCD\\) be an isosceles trapezoid with parallel bases \\(\\overline{AB}\\) and \\(\\overline{CD}\\), where \\(AB > CD\\... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe amount $2.5$ is split into two nonnegative real numbers uniformly at random, for instance, into $2.143$ and $0.357$, or in... | 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 $K$ be the point of intersection of $AB$ and the line touching the circumcircle of $\\triangle ABC$ at $C$ where $m(\\wide... | 8 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA man with mass \\(m\\) jumps off a high bridge with a bungee cord attached to his ankles. The man falls through a maximum dis... | 9 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLiquid X does not mix with water. Unless obstructed, it spreads out on the surface of water to form a circular film $0.1$ cm t... | 2160 |
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{
"content": "Solve the following math problem step by 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 $U = 2 \\times 2004^{2005}$, $V = 2004^{2005}$, $W = 2003 \\times 2004^{2004}$, $X = 2 \\times 2004^{2004}$, $Y = 2004^{20... | 43 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA line passes through $A(1,1)$ and $B(100,1000)$. Find the number of other points with integer coordinates that are on the lin... | 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\nIn our number system, the base is ten. If the base were changed to four, you would count as follows: $1, 2, 3, 10, 11, 12, 13,... | 110 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAlex has $75$ red tokens and $75$ blue tokens. There is a booth where Alex can give two red tokens and receive in return a sil... | 103 |
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{
"content": "Solve the following math problem step by 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 rhombus with a side length of 5, the length of one of its diagonals is not larger than $6$, and the length of the other ... | 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\nA plane intersects a right circular cylinder of radius $1$ forming an ellipse. If the major axis of the ellipse is $50\\%$ lon... | 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\nFor the integer numbers $i, j, k$ satisfying the condition $i^2 + j^2 + k^2 = 2011$, find the largest value of $i + j + k$.\n\... | 77 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA sector with an acute central angle \\(\\theta\\) is cut from a circle of radius 6. What is the radius of the circle circumsc... | 6 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping pa... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAssuming \\(a\\neq3\\), \\(b\\neq4\\), and \\(c\\neq5\\), find the integer value of the following expression:\n\\[\\frac{a-3}{... | -1 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere are four more girls than boys in Ms. Raub's class of $28$ students. What is the ratio of the number of girls to the numb... | 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\nA fancy bed and breakfast inn has $5$ rooms, each with a distinctive color-coded decor. One day $5$ friends arrive to spend th... | 2220 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn rectangle \\(ABCD\\), \\(DC = 2 \\cdot CB\\) and points \\(E\\) and \\(F\\) lie on \\(\\overline{AB}\\) so that \\(\\overli... | 9 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA five-digit palindrome is a positive integer with respective digits $abcba$, where $a$ is non-zero. Let $S$ be the sum of all... | 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\nThe circumference of the circle with center $O$ is divided into $12$ equal arcs, marked the letters $A$ through $L$ as seen be... | 90 |
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