prompt listlengths 1 1 | label stringlengths 1 23 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven distinct straight lines $OA$ and $OB$. From a point on $OA$, a perpendicular is drawn to $OB$; from the foot of this per... | 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 $Answer is the answer to the problem.\n\nThe sum of the base-\\( 10 \\) logarithms of the divisors of \\( 10^n \\) is \\( 792 \\). What is the value of \\( n \\)?\n\nR... | 11 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nChloé chooses a real number uniformly at random from the interval \\([0, 2017]\\). Independently, Laurent chooses a real numbe... | 7 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the sum of the last two digits of $7^{42} + 7^{43}$ in base $10$?\n\nRemember to put your answer on its own line after... | 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\nLet $N = 123456789101112\\dots4344$ be the $79$-digit number obtained by writing the integers from $1$ to $44$ in order, one a... | 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 baseball is dropped on top of a basketball. The basketball hits the ground, rebounds with a speed of $4.0 \\, \\text{m/s}$, ... | 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\nHow many distinct four-tuples $(a, b, c, d)$ of rational numbers are there such that:\n\n$$a \\log_{10} 2 + b \\log_{10} 3 + c... | 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\nA circle is inscribed in a triangle with side lengths $8$, $13$, and $17$. Let the segments of the side of length $8$, made by... | 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 $\\triangle ABC$ with $|BC| > |BA|$, $D$ is a point inside the triangle such that $\\angle ABD = \\angle DBC$, $\\... | 60 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the number of real numbers which satisfy the equation $x|x-1|-4|x|+3=0$.\n\nRemember to put your answer on its own line a... | 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\nThree equally spaced parallel lines intersect a circle, creating three chords of lengths $38$, $38$, and $34$. What is the dis... | 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\nPhillip and Paula both pick a rational number, and they notice that Phillip's number is greater than Paula's number by $12$. T... | 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 $Answer is the answer to the problem.\n\nWhat is the value of \n\\[ \\log_3 7 \\cdot \\log_5 9 \\cdot \\log_7 11 \\cdot \\log_9 13 \\cdots \\log_{21} 25 \\cdot \\log_{... | 6 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the largest integer that divides each of the numbers in the sequence \\(1^5 - 1, 2^5 - 2, 3^5 - 3, \\cdots, n^5 - n, \\cd... | 30 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA unit cube has vertices \\(P_1, P_2, P_3, P_4, P_1', P_2', P_3'\\), and \\(P_4'\\). Vertices \\(P_2, P_3\\), and \\(P_4\\) ar... | 9 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEach integer $1$ through $9$ is written on a separate slip of paper and all nine slips are put into a hat. Jack picks one of t... | 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\nA drawer in a darkened room contains $100$ red socks, $80$ green socks, $60$ blue socks, and $40$ black socks. A youngster sel... | 23 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA ladder is formed by removing some consecutive unit squares of a $10 \\times 10$ chessboard such that for each $k$-th row ($k... | 715 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven a quadrilateral $ABCD$ inscribed in a circle with side $AB$ extended beyond $B$ to point $E$, if $\\measuredangle BAD=92... | 68 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEvery high school in the city of Euclid sent a team of 3 students to a math contest. Each participant in the contest received ... | 23 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSquare \\(ABCD\\) has an area of \\(36\\), and \\(\\overline{AB}\\) is parallel to the x-axis. Vertices \\(A\\), \\(B\\), and ... | 10 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA three-quarter sector of a circle with a radius of \\(4\\) inches, together with its interior, can be rolled up to form the l... | 10 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nPoints $A, B, Q, D,$ and $C$ lie on the circle shown. The measures of arcs $\\widehat{BQ}$ and $\\widehat{QD}$ are $42^\\circ$... | 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\nCircles $A$, $B$, and $C$ are externally tangent to each other and internally tangent to circle $D$. Circles $B$ and $C$ are c... | 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\nIf $(0.2)^x = 2$ and $\\log 2 = 0.3010$, then if the value of $x$ to the nearest tenth is $y$, what is the value of $10y$?\n\n... | -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\nLet $n$ be the number of ways that $10$ dollars can be changed into dimes and quarters, with at least one of each coin being u... | 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 $Answer is the answer to the problem.\n\nHow many sequences of zeros and ones of length 20 have all the zeros consecutive, or all the ones consecutive, or both?\n\nRem... | 382 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $AD$ be a median of $\\triangle ABC$ such that $m(\\widehat{ADB})=45^{\\circ}$ and $m(\\widehat{ACB})=30^{\\circ}$. If the... | 105 |
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{
"content": "Solve the following math problem step by 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 chose an even 4-digit number $n$. She wrote down all the divisors of $n$ in increasing order from left to right: $1, 2, \... | 340 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn a table tennis tournament, every participant played every other participant exactly once. There were twice as many right-ha... | 36 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTriangles $ABC$ and $ABD$ are isosceles with $AB = AC = BD$, and $BD$ intersects $AC$ at $E$. If $BD$ is perpendicular to $AC$... | 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 three representation of $x$ is \n\\[ 12112211122211112222. \\]\nWhat is the first digit (on the left) of the base nin... | 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 $Answer is the answer to the problem.\n\nFor all positive numbers \\(x, y, z\\), evaluate the product:\n\\[\n(x+y+z)^{-1}(x^{-1}+y^{-1}+z^{-1})(xy+yz+xz)^{-1}[(xy)^{-1... | 6 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven the function $f$ defined as follows: \\[ f(0) = 0, \\] \\[ f(1) = 1, \\] and for every integer $n \\geq 1$, \\[ f(3n-1) ... | 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\nAn integer $x$, with $10 \\le x \\le 99$, is to be chosen. If all choices are equally likely, find the probability that at lea... | 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 $P(x) = x^{2022} + x^{1011} + 1$. Find the polynomial of the form ax^6 + bx^3 + c that divides $P(x)$. Please provide the ... | 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\nIn $\\triangle ABC$, $\\angle ABC = 45^\\circ$. Point $D$ is on $\\overline{BC}$ such that $2 \\cdot BD = CD$ and $\\angle DAB... | 75 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAn unfair coin lands on heads with a probability of $\\frac{1}{4}$. When tossed $n$ times, the probability of exactly two head... | 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 $Answer is the answer to the problem.\n\nPoints \\(A(6,13)\\) and \\(B(12,11)\\) lie on circle \\(\\omega\\) in the plane. Suppose that the tangent lines to \\(\\omega... | 93 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the product of the real roots of the equation $x^4+3x^3+5x^2 + 21x -14=0$. The answer is an integer. Please provide the i... | -2 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $ABC$ be a triangle such that angle $C$ is obtuse. Let $D \\in [AB]$ and $[DC] \\perp [BC]$. If $m(\\widehat{ABC}) = \\alp... | 20 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor the infinite series \\(1 - \\frac{1}{2} - \\frac{1}{4} + \\frac{1}{8} - \\frac{1}{16} - \\frac{1}{32} + \\frac{1}{64} - \\... | 9 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA semi-palindrome is a four-digit number whose first two digits and last two digits are identical. For instance, $2323$ and $5... | 8989 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the least integer $n \\geq 100$ such that $77$ divides $1+2+2^2+2^3+\\dots + 2^n$? Find the value of $n$.\n\nRemember ... | 119 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nUsain is walking for exercise by zigzagging across a $100$-meter by $30$-meter rectangular field, beginning at point $A$ and e... | 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\nConsider the sequence of numbers defined recursively by $t_1=1$ and for $n>1$ by $t_n=1+t_{\\left(\\frac{n}{2}\\right)}$ when ... | 15 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA house and a store were each sold for $\\$12000$. The house was sold at a loss of $20\\%$ of its cost, and the store at a gai... | 1000 |
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{
"content": "Solve the following math problem step by 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 $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are integers. Suppose that $f(1) = 0$, $50 < f(7) < 60$, $70 < f(8) < 80$,... | 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\nPoints $A$ and $B$ lie on a circle centered at $O$, and $\\angle AOB = 60^\\circ$. A second circle is internally tangent to th... | 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 $Answer is the answer to the problem.\n\nIn \\(\\triangle ABC\\), we have \\(AB = 1\\) and \\(AC = 2\\). Side \\(BC\\) and the median from \\(A\\) to \\(BC\\) have the... | 2 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTom, Dick, and Harry started out on a $100$-mile journey. Tom and Harry traveled by automobile at a speed of $25$ mph, while D... | 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 $Answer is the answer to the problem.\n\nIf \\(b > 1\\), \\(\\sin x > 0\\), \\(\\cos x > 0\\), and \\(\\log_b \\sin x = a\\), find the value of \\(\\log_b \\cos x\\). ... | 5 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nOn an algebra quiz, $10\\%$ of the students scored $70$ points, $35\\%$ scored $80$ points, $30\\%$ scored $90$ points, and th... | 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 $Answer is the answer to the problem.\n\n在\\(\\triangle ABC\\)中，已知边长\\(|AB|=3\\)，且从点\\(C\\)作出的高的长度为\\(2\\)。求另外两条高的长度乘积的最大值。答案的形式为\\(\\frac{k}{m}\\)，请给出k+m的值。\n\nRememb... | 169 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many positive integers not exceeding 2001 are multiples of 3 or 4 but not 5? Provide your answer as an integer.\n\nRemembe... | 801 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe student locker numbers at Olympic High are numbered consecutively beginning with locker number $1$. The plastic digits use... | 2001 |
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{
"content": "Solve the following math problem step by 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 solid spherical balls:\n\n1. **First Ball**:\n - Centered at $(0, 0, \\frac{21}{2})$\n - Radius $6$\n\n2. **S... | 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 $Answer is the answer to the problem.\n\nWe define a positive integer as \"special\" if it is divisible by all of its non-zero digits. Determine the maximum number of ... | 12 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the largest integer not exceeding $\\lfloor (n+1)a \\rfloor - \\lfloor na \\rfloor$, where $n$ is a natural number and $a... | 1 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA circular disk is divided by $2n$ equally spaced radii ($n > 0$) and one secant line. What is the maximum number of non-overl... | 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 $Answer is the answer to the problem.\n\nA traffic light runs repeatedly through the following cycle: green for \\(30\\) seconds, then yellow for \\(3\\) seconds, and ... | 8 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe numbers $\\log(a^3b^7)$, $\\log(a^5b^{12})$, and $\\log(a^8b^{15})$ are the first three terms of an arithmetic sequence, a... | 112 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe number of distinct pairs \\((x,y)\\) of real numbers satisfying both of the following equations: \n\\[ \\begin{align*} x &... | 4 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nJuan and Mary play a two-person game where the winner gains 2 points and the loser loses 1 point. If Juan won exactly 3 games ... | 7 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nRegular octagon \\(ABCDEFGH\\) has area \\(n\\). Let \\(m\\) be the area of quadrilateral \\(ACEG\\). The answer is in the for... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $r$ be the speed in miles per hour at which a wheel, 11 feet in circumference, travels. If the time for a complete rotatio... | 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 $Answer is the answer to the problem.\n\nIn Cartesian coordinates, two areas \\(M\\) and \\(N\\) are defined as follows: \n- \\(M: y \\geq 0, y \\leq x, y \\leq 2-x\\... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe graph of \\( y = x^2 + 2x - 15 \\) intersects the \\( x \\)-axis at points \\( A \\) and \\( C \\) and the \\( y \\)-axis ... | 11 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $P$ be a polynomial such that $(x-4)P(2x) = 4(x-1)P(x)$, for every real $x$. If $P(0) \\neq 0$, what is the degree of $P$?... | 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\nCalculate the difference between the number of hours in $10$ years and the number of seconds in $1$ day.\n\nRemember to put yo... | 1200 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet \\(S\\) be the set of ordered triples \\((x,y,z)\\) of real numbers for which\n\\[ \\log_{10} (x + y) = z \\text{ and } \\... | 31 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the figure, polygons $A$, $E$, and $F$ are isosceles right triangles; $B$, $C$, and $D$ are squares with sides of length $1... | 11 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA square and a circle have equal perimeters. Find the ratio of the area of the circle to the area of the square. The original ... | 7 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA small block with an initial speed $v_0$ smoothly moves onto a sloped big block of mass $M$. After the small block reaches a ... | 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 $Answer is the answer to the problem.\n\nRectangle \\(ABCD\\) has \\(AB=4\\) and \\(BC=3\\). Segment \\(EF\\) is constructed through \\(B\\) so that \\(EF\\) is perpen... | 137 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCircle $A$ has radius $100$. Circle $B$ has an integer radius $r<100$ and remains internally tangent to circle $A$ as it rolls... | 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 particular $12$-hour digital clock displays the hour and minute of a day. Unfortunately, whenever it is supposed to display ... | 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 convex polyhedron $Q$ has vertices $V_1,V_2,\\cdots,V_n$, and $100$ edges. The polyhedron is cut by planes $P_1,P_2,\\cdots,... | 300 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nParallelogram $ABCD$ has area $1,000,000$. Vertex $A$ is at $(0,0)$ and all other vertices are in the first quadrant. Vertices... | 784 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nRachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every \\(90\\) seconds, and Robert... | 19 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe keystone arch is an ancient architectural feature. It is composed of congruent isosceles trapezoids fitted together along ... | 100 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEach face of a cube is given a single narrow stripe painted from the center of one edge to the center of the opposite edge. Th... | 19 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA month with $31$ days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the firs... | 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\nLucky Larry's teacher asked him to substitute numbers for $a$, $b$, $c$, $d$, and $e$ in the expression $a-(b-(c-(d+e)))$ and ... | 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 $Answer is the answer to the problem.\n\nA month with 31 days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first ... | 3 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHalfway through a $100$-shot archery tournament, Chelsea leads by $50$ points. For each shot, a bullseye scores $10$ points, w... | 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 $Answer is the answer to the problem.\n\nA palindrome, such as 83438, is a number that remains the same when its digits are reversed. The numbers \\( x \\) and \\( x+3... | 24 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet \\( n \\) be the smallest positive integer such that \\( n \\) is divisible by 20, \\( n^2 \\) is a perfect cube, and \\( ... | 7 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe solution of the equation \\(7^{x+7} = 8^x\\) can be expressed in the form \\(x = \\log_b 7^7\\). The original answer is in... | 15 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA palindrome between $1000$ and $10,000$ is chosen at random. Find the probability that it is divisible by $7$. The original 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\nOn a $50$-question multiple choice math contest, students receive $4$ points for a correct answer, $0$ points for an answer le... | 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\nA solid cube has side length $3$ inches. A $2$-inch by $2$-inch square hole is cut into the center of each face. The edges of ... | 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\nEquiangular hexagon $ABCDEF$ has side lengths $AB=CD=EF=1$ and $BC=DE=FA=r$. The area of $\\triangle ACE$ is $70\\%$ of the ar... | 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 $16$-step path is to go from $(-4,-4)$ to $(4,4)$ with each step increasing either the $x$-coordinate or the $y$-coordinate ... | 1698 |
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"content": "Solve the following math problem step by 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 $Answer is the answer to the problem.\n\nA high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points ... | 34 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nArithmetic sequences $\\left(a_n\\right)$ and $\\left(b_n\\right)$ have integer terms with $a_1=b_1=1<a_2 \\le b_2$ and $a_n b... | 8 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe graph of $y=x^6-10x^5+29x^4-4x^3+ax^2$ lies above the line $y=bx+c$ except at three values of $x$, where the graph and the... | 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\nLet $a > 0$, and let $P(x)$ be a polynomial with integer coefficients such that \n \n$P(1) = P(3) = P(5) = P(7) = a,$ and ... | 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\nLet \\(ABCD\\) be a cyclic quadrilateral. The side lengths of \\(ABCD\\) are distinct integers less than \\(15\\) such that \\... | 427 |
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