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\nThe work team was working at a rate fast enough to process $1250$ items in ten hours. But after working for six hours, the tea... | 30 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFunctions $f$ and $g$ are defined so that $f(1) = 4$, $g(1) = 9$, and for each integer $n \\ge 1$, $f(n+1) = 2f(n) + 3g(n) + 2... | 1999 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWe have two positive integers both less than $1000$. The arithmetic mean and the geometric mean of these numbers are consecuti... | 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\nThe rodent control task force went into the woods one day and caught $200$ rabbits and $18$ squirrels. The next day they went ... | 5491 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn how many ways can $100$ be written as the sum of three positive integers $x, y$, and $z$ satisfying $x < y < z$ ?\n\nRememb... | 784 |
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{
"content": "Solve the following math problem step by 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 $36$. Point $E$ is on side $AB$ a distance $12$ from $B$, point $F$ is the midpoint of side $BC... | 144 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA dart board looks like three concentric circles with radii of 4, 6, and 8. Three darts are thrown at the board so that they ... | 617 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSix chairs sit in a row. Six people randomly seat themselves in the chairs. Each person randomly chooses either to set their ... | 1106 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nPenelope plays a game where she adds $25$ points to her score each time she wins a game and deducts $13$ points from her score... | 97 |
[
{
"content": "Solve the following math problem step by 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 $Answer is the answer to the problem.\n\nThree congruent ellipses are mutually tangent. Their major axes are parallel. Two of the ellipses are tangent at the end poin... | 776 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe four sets A, B, C, and D each have $400$ elements. The intersection of any two of the sets has $115$ elements. The inter... | 1094 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA city is laid out with a rectangular grid of roads with 10 streets numbered from 1 to 10 running east-west and 16 avenues num... | 544 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEach of the distinct letters in the following subtraction problem represents a different digit. Find the number represented by... | 6852 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the sum of all the integers $N > 1$ with the properties that the each prime factor of $N $ is either $2, 3,$ or $5,... | 2820 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf you roll six fair dice, let $\\mathsf{ p}$ be the probability that exactly five different numbers appear on the upper faces... | 133 |
[
{
"content": "Solve the following math problem step by 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 all the digits in the decimal representations of all the positive integers less than $1000.$\n\nRemember to pu... | 13500 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhen Tim was Jim’s age, Kim was twice as old as Jim. When Kim was Tim’s age, Jim was 30. When Jim becomes Kim’s age, Tim wil... | 221 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $S$ S be a point chosen at random from the interior of the square $ABCD$, which has side $AB$ and diagonal $AC$. Let $P$ ... | 160 |
[
{
"content": "Solve the following math problem step by 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 quadrilateral with vertices at the four points in three dimensional space $(0,0,0)$, $(2,6,1)$, $(-3,0,3)$ and... | 53 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAt Mallard High School there are three intermural sports leagues: football, basketball, and baseball. There are 427 students ... | 238 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA right circular cone pointing downward forms an angle of $60^\\circ$ at its vertex. Sphere $S$ with radius $1$ is set into th... | 364 |
[
{
"content": "Solve the following math problem step by 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 cylinder radius $12$ and a cylinder radius $36$ are held tangent to each other with a tight band. The length of the band is ... | 107 |
[
{
"content": "Solve the following math problem step by 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 isosceles triangle $ABC$ sides $AB$ and $BC$ have length $125$ while side $AC$ has length $150$. Point $D$ is the midpoint ... | 1728 |
[
{
"content": "Solve the following math problem step by 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 non-congruent scalene triangles whose sides all have integral length, and the longest side has length $11$.... | 20 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFive men and seven women stand in a line in random order. Let m and n be relatively prime positive integers so that $\\tfrac{m... | 287 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $ABCD$ be a trapezoid where $AB$ is parallel to $CD.$ Let $P$ be the intersection of diagonal $AC$ and diagonal $BD.$ If t... | 81 |
[
{
"content": "Solve the following math problem step by 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 possible sum $a + b + c + d + e$ where $a, b, c, d,$ and $e$ are positive integers satisfying the conditions... | 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\nThere are positive integers $b$ and $c$ such that the polynomial $2x^2 + bx + c$ has two real roots which differ by $30.$ Find... | 126 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the number of ordered pairs of integers $(m, n)$ that satisfy $20m-10n = mn$.\n\nRemember to put your answer on its own l... | 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\nIn the coordinate plane a parabola passes through the points $(7,6)$, $(7,12)$, $(18,19)$, and $(18,48)$. The axis of symmetry... | 71 |
[
{
"content": "Solve the following math problem step by 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 $\\frac{1}{1}+\\frac{1}{2}+\\frac{1}{3}+\\frac{1}{4}+\\frac{1}{5}+\\frac{1}{6}=\\frac{m}{n}$ where $m$ and $n$ are rel... | 69 |
[
{
"content": "Solve the following math problem step by 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 least positive integer $k$ so that $k + 25973$ is a palindrome (a number which reads the same forward and backwards).... | 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\nHalf the volume of a 12 foot high cone-shaped pile is grade A ore while the other half is grade B ore. The pile is worth \\$62... | 608 |
[
{
"content": "Solve the following math problem step by 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 $x$ and $y$ be real numbers satisfying \\[(x^2+x-1)(x^2-x+1)=2(y^3-2\\sqrt{5}-1)\\] and \\[(y^2+y-1)(y^2-y+1)=2(x^3+2\\sqr... | 20 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $x_1$, $x_2$, and $x_3$ be the roots of the polynomial $x^3+3x+1$. There are relatively prime positive integers $m$ and $... | 10 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFive congruent circles have centers at the vertices of a regular pentagon so that each of the circles is tangent to its two ne... | 1320 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many numbers are there that appear both in the arithmetic sequence $10,\n16, 22, 28, ... 1000$ and the arithmetic sequence... | 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 $Answer is the answer to the problem.\n\nFind the smallest prime number that does not divide \\[9+9^2+9^3+\\cdots+9^{2010}.\\]\n\nRemember to put your answer on its ow... | 17 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the area of the region in the coordinate plane satisfying the three conditions\n\n$\\star$ $x \\le\u0014 2y$\n$\\star$... | 600 |
[
{
"content": "Solve the following math problem step by 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 distinct positive integers are randomly chosen between $1$ and $2011;$ inclusive. The probability that some pair of the si... | 100 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the number of ordered quadruples $(a, b, c, d)$ where each of $a, b, c,$ and $d$ are (not necessarily distinct) elements ... | 2017 |
[
{
"content": "Solve the following math problem step by 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 congruent spheres are stacked so that each is tangent to the other three. A larger sphere, $R$, contains the four congrue... | 2449 |
[
{
"content": "Solve the following math problem step by 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 of sets $(A, B)$ have the properties:\n\n1. $ A\\subseteq \u0012\\{1, 2, 3, 4, 5, 6\\} $\n2. $ B\\subse... | 720 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $V$ be the set of vertices of a regular $25$ sided polygon with center at point $C.$ How many triangles have vertices in $... | 650 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many positive integer solutions are there to $w+x+y+z=20$ where $w+x\\ge 5$ and $y+z\\ge 5$?\n\nRemember to put your answe... | 781 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $A=\\{1, 3, 5, 7, 9\\}$ and $B=\\{2, 4, 6, 8, 10\\}$. Let $f$ be a randomly chosen function from the set $A\\cup B$ into i... | 253 |
[
{
"content": "Solve the following math problem step by 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 value of $x$ that satisfies $\\log_{3}(\\log_9x)=\\log_9(\\log_3x)$\n\nRemember to put your answer on its own line af... | 81 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nMeredith drives 5 miles to the northeast, then 15 miles to the southeast, then 25 miles to the southwest, then 35 miles to the... | 20 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA rectangle has side lengths $6$ and $8$. There are relatively prime positive integers $m$ and $n$ so that $\\tfrac{m}{n}$ is ... | 203 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFloyd looked at a standard $12$ hour analogue clock at $2\\!:\\!36$. When Floyd next looked at the clock, the angles through w... | 14 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSix children stand in a line outside their classroom. When they enter the classroom, they sit in a circle in random order. The... | 13 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the least six-digit palindrome that is a multiple of $45$. Note that a palindrome is a number that reads the same forward... | 504405 |
[
{
"content": "Solve the following math problem step by 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 diagram two sides of a square are tangent to a circle with diameter $8$. One corner of the square lies on the... | 134 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the least three digit number that is equal to the sum of its digits plus twice the product of its digits.\n\nRemember to ... | 397 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe diagram below shows the first three \ffigures of a sequence of figures. The fi\frst \ffigure shows an equilateral triangle... | 1088 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAfter Jennifer walked $r$ percent of the way from her home to the store, she turned around and walked home, got on her bicycle... | 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\nFind $n$ so that $4^{4^{4^2}}=2^{8^n}$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"... | 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\nStart with a three-digit positive integer $A$. Obtain $B$ by interchanging the two leftmost digits of $A$. Obtain $C$ by doubl... | 344 |
[
{
"content": "Solve the following math problem step by 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 sequences of six positive integers $a_1,a_2,a_3,a_4,a_5,a_6$ with the properties that $a_1=1$, and if for some $j... | 203 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $x$, $y$, $z$ be positive real numbers satisfying the simultaneous equations\n\\begin{align*}x(y^2+yz+z^2)&=3y+10z\\\\y(z^... | 31 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind $n$ such that\\[\\frac{1!\\cdot2!\\cdot3!\\cdots10!}{(1!)^2(3!)^2(5!)^2(7!)^2(9!)^2}=15\\cdot2^n.\\]\n\nRemember to put y... | 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 $Answer is the answer to the problem.\n\nThe Bell Zoo has the same number of rhinoceroses as the Carlton Zoo has lions. The Bell Zoo has three more elephants than the ... | 57 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f$ be a real-valued function such that $4f(x)+xf\\left(\\tfrac1x\\right)=x+1$ for every positive real number $x$. Find $f... | 403 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEvaluate\n$\\frac{\\log_{10}20^2 \\cdot \\log_{20}30^2 \\cdot \\log_{30}40^2 \\cdot \\cdot \\cdot \\log_{990}1000^2}{\\log_{10... | 768 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the sum of all positive integers n with the property that the digits of n add up to 2015−n.\n\nRemember to put your answe... | 4004 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nJanet played a song on her clarinet in 3 minutes and 20 seconds. Janet’s goal is to play the song at a 25%\nfaster rate. Find ... | 160 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCindy and Neil wanted to paint the side of a staircase in the six-square pattern shown below so that each\nof the six squares ... | 1166 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA courtyard has the shape of a parallelogram ABCD. At the corners of the courtyard there stand poles AA', BB', CC', and DD', e... | 14 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the greatest positive integer $n$ so that $3^n$ divides $70! + 71! + 72!.$\n\nRemember to put your answer on its own line... | 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\nThe repeating decimal $2.0151515\\ldots$ can be expressed as $\\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive ... | 199 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many subsets of {1,2,3,4,5,6,7,8,9,10,11,12} have the property that no two of its elements differ by more than 5? For examp... | 256 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nOn the inside of a square with side length 60, construct four congruent isosceles triangles each with base\n60 and height 50, ... | 360 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere are digits a and b so that the 15-digit number 7a7ba7ab7ba7b77 is divisible by 99.\nFind 10a + b.\n\nRemember to put you... | 53 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe map below shows an east/west road connecting the towns of Acorn, Centerville, and Midland, and a\nnorth/south road from Ce... | 65 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the sum of all perfect squares that divide 2016.\n\nRemember to put your answer on its own line after \"Answer:\".",
... | 210 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a_0$, $a_1$, ..., $a_6$ be real numbers such that $a_0 + a_1 + ... + a_6 = 1$ and\n$$a_0 + a_2 + a_3 + a_4 + a_5 + a_6 =\... | 5041 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nOmar has four fair standard six-sided dice. Omar invented a game where he rolls his four dice over and over again until the nu... | 819 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the number of functions $f$ that map the set $\\{1,2, 3,4\\}$ into itself such that the range of the function $f(x)$ is t... | 148 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a_1 = 1 +\\sqrt2$ and for each $n \\ge 1$ de\fne $a_{n+1} = 2 -\\frac{1}{a_n}$. Find the greatest integer less than or eq... | 283 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe set of positive real numbers $x$ that satisfy $2 | \fx^2 - 9\f | \\le 9 | x | $ is an interval $[m, M]$. Find $10m + M$.\n... | 21 |
[
{
"content": "Solve the following math problem step by 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 least positive integer $m$ such that $lcm(15,m) = lcm(42,m)$. Here $lcm(a, b)$ is the least common multiple of $a$ an... | 70 |
[
{
"content": "Solve the following math problem step by 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 $3$-dimensional coordinate space \fnd the distance from the point $(36, 36, 36)$ to the plane that passes\nthrough the ... | 200 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA right circular cone has a height equal to three times its base radius and has volume 1. The cone is inscribed inside a spher... | 581 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor real numbers $a, b$, and $c$ the polynomial $p(x) = 3x^7 - 291x^6 + ax^5 + bx^4 + cx^2 + 134x - 2$ has $7$ real roots whos... | 67 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a$ be a solution to the equation $\\sqrt{x^2 + 2} = \\sqrt[3]{x^3 + 45}$. Evaluate the ratio of $\\frac{2017}{a^2}$ to $a... | 6 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $ABCDE$ be a pentagon with area $2017$ such that four of its sides $AB, BC, CD$, and $EA$ have integer length. Suppose tha... | 178 |
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"content": "Solve the following math problem step by 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 integers $a_1, a_2, a_3,...,a_{240}$ such that $x(x + 1)(x + 2)(x + 3) ... (x + 239) =\\sum_{n=1}^{240}a_nx^n$. Find... | 159 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nBradley is driving at a constant speed. When he passes his school, he notices that in $20$ minutes he will be exactly $\\frac1... | 245 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the positive integer $k$ such that the roots of $x^3 - 15x^2 + kx -1105$ are three distinct collinear points in the compl... | 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\nIf you roll four standard, fair six-sided dice, the top faces of the dice can show just one value (for example, $3333$), two v... | 887 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nOne right pyramid has a base that is a regular hexagon with side length $1$, and the height of the pyramid is $8$. Two other r... | 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 $Answer is the answer to the problem.\n\nAileen plays badminton where she and her opponent stand on opposite sides of a net and attempt to bat a birdie back and forth ... | 73 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn $\\vartriangle ABC$ points $D, E$, and $F$ lie on side $\\overline{BC}$ such that $\\overline{AD}$ is an angle bisector of ... | 94 |
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"content": "Solve the following math problem step by 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 $6$. Circles $X, Y$ , and $Z$ are congruent circles with centers inside the square suc... | 195 |
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"content": "Solve the following math problem step by 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 jeweler can get an alloy that is $40\\%$ gold for $200$ dollars per ounce, an alloy that is $60\\%$ gold for $300$ dollar pe... | 240 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose $x$ and $y$ are nonzero real numbers simultaneously satisfying the equations\n$x + \\frac{2018}{y}= 1000$ and $ \\frac... | 1991 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose that $a$ and $b$ are positive real numbers such that $3\\log_{101}\\left(\\frac{1,030,301-a-b}{3ab}\\right) = 3 - 2 \\... | 0 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the three-digit positive integer $n$ for which $\\binom n3 \\binom n4 \\binom n5 \\binom n6 $ is a perfect square.\n\nRe... | 489 |
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"content": "Solve the following math problem step by 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 $a$ and $b$ be real numbers such that $\\frac{1}{a^2} +\\frac{3}{b^2} = 2018a$ and $\\frac{3}{a^2} +\\frac{1}{b^2} = 290b$... | 13 |
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