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\n方程 $19 x+93 y=4 x y$ 的整数解的组数是多少？\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
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] | 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 $Answer is the answer to the problem.\n\n从圆 \\(x^{2}+y^{2}=4\\) 上的点向椭圆 \\(C: \\frac{x^{2}}{2}+y^{2}= 1\\) 引切线，两个切点间的线段称为切点弦。求所有的切点弦所在的直线围成区域的面积。原始答案的形式为\\(\\frac{k\\pi... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在四面体 \\(P A B C\\) 中，\\(A B = P A = 2 A C = 2\\)，且 \\(P A \\perp\\) 平面 \\(A B C\\)，\\(\\angle B A C = 120^{\\circ}\\)。求该四面体的外接... | 40 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 P 为单位圆上一动点, 且 $A(0,2), B(0,-1)$, 求 $|A P| \\cdot|B P|^{2}$ 的最大值。答案的形式为k\\sqrt{m}，请给出k + m的值。\n\nRemember to put your answer... | 6 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在三棱柱 $A B C-A_{1} B_{1} C_{1}$ 中, 已知 $\\angle A B C=90^{\\circ}, A B=6, B C=B B_{1}=3 \\sqrt{2}$, 动点 P 在线段 $B_{1} C$ 上，求 $A_{1... | 13 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n令 $z_{k}=\\cos \\frac{2 k-2}{5} \\pi+i \\sin \\frac{2 k-2}{5} \\pi(k=1,2,3,4,5)$ ，若 $a_{i}=\\prod_{i \\neq j}\\left(z_{i}-z_{j... | 3125 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA truck is initially moving at velocity $v$. The driver presses the brake in order to slow the truck to a stop. The brake appl... | 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 $Answer is the answer to the problem.\n\nLet \\(f(x)\\) be a real-valued function satisfying \\(af(x) + bf(-x) = px^2 + qx + r\\). Here, \\(a\\) and \\(b\\) are distin... | 2 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设复数 $z$ 满足 $|3 z-7 i|=3$ ，求 $\\left|\\frac{z^{2}-2 z+2}{z-1+\\mathrm{i}}\\right|$ 的最大值。答案的形式为\\frac{k}{m}，请计算k+m的值。\n\nRemembe... | 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 $Answer is the answer to the problem.\n\n设 \\(\\alpha, \\beta\\) 为锐角，且 \\(\\cos (\\alpha+\\beta)=\\frac{\\sin \\alpha}{\\sin \\beta}\\)，求 \\(\\tan \\alpha\\) 的最大值。答案的形... | 6 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设多项式 \\(f(x)\\) 的各项系数都是非负实数，且 \\(f(1)=f^{\\prime}(1)=f^{\\prime \\prime}(1)=f^{\\prime \\prime \\prime}(1)=1\\)。求 \\(f(x)\\) 的... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设定点 $A, B, C, D$ 是以点 $O$ 为中心的正四面体的顶点，用 $\\sigma$ 表示空间以直线 $O A$ 为轴且满足条件 $\\sigma(B)=C$ 的旋转，用 $\\tau$ 表示空间关于点 $O, C, D$ 所在平面的镜面反... | 6 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若 $A+B=\\frac{2 \\pi}{3}$, 求 $\\cos ^{2} A+\\cos ^{2} B$ 的最小值和最大值，分别记为 $x$ 和 $y$。答案的形式分别为x = \\frac{m}{n}和y = \\frac{p}{q}，请给出... | 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\n将函数 $y=\\sqrt{4+6 x-x^{2}}-2(x \\in[0,6])$ 的图象逆时针方向旋转 $\\theta(0 \\leqslant \\theta \\leqslant \\alpha)$, 得到曲线 C。若对于每一个旋转角 $\\... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n有 $n$ 个质点, 每个质点质量为 $m_{k}$, 则质心位置 $x=\\frac{\\sum m_{k} x_{k}}{\\sum m_{k}}$。对于一杆，长 3 m，放于 $x \\in[-1,2]$ 间，且线密度满足 $\\beta=2+x... | 9 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 $[x]$ 为高斯函数，求满足 $\\left[\\frac{x}{2}\\right]+\\left[\\frac{x}{3}\\right]+\\left[\\frac{x}{5}\\right]=x$ 的解的组数。\n\nRemember ... | 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 $Answer is the answer to the problem.\n\n设复数 \\(z\\) 满足 \\(|3z - 7i| = 3\\)，求 \\(\\left|\\frac{z^2 - 2z + 2}{z - 1 + i}\\right|\\) 的最大值和最小值。如果最大值为 \\(\\frac{m}{n}\\)，最... | 16 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若复数 $z$ 满足 $\\left|\\frac{1}{z}+z\\right|=1$, 则求 $|z|$ 的两个可能的值。答案的形式分别为 \\frac{\\sqrt{k}-m}{n} 和 \\frac{\\sqrt{p}+q}{r}，请计算k+m... | 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 $Answer is the answer to the problem.\n\n由 \\(y \\geqslant 2|x|-1\\) 和 \\(y \\leqslant-3|x|+5\\) 所围成的平面区域的面积为多少？请注意，答案的形式为\\(\\frac{k}{m}\\)，其中k和m是互质的正整数。请给出k+m的值。\n\n... | 41 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知等腰梯形 ABCD 中，AD\\parallel BC，BC=2AD=4，\\angle ABC=60^\\circ，\\overrightarrow{CE}=\\frac{1}{3}\\overrightarrow{CD}，求 \\overrig... | -10 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在正四面体 ABCD 中，棱 AD 的中点和面 BCD 的中心的连线为 MN，棱 CD 的中点和面 ABC 的中心的连线为 PQ。求 MN 与 PQ 所成角的余弦值。答案应为\\(\\frac{k}{m}\\)的形式，请给出k+m的值。\n\nReme... | 19 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知函数 \\( f(x) = \\sqrt{2x^4-18x^2+12x+68}+x^2-x+1 \\)，求 \\( f(x) \\) 的最小值。\n\nRemember to put your answer on its own line afte... | 9 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 \\(A\\), \\(B\\), \\(C\\) 是平面上三个不同点，且 \\(BC = a\\), \\(CA = b\\), \\(AB = c\\)，求 \\(\\frac{c}{a+b}+\\frac{b}{c}\\) 的最小值。答案应... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n双曲线 \\(x^2-y^2=2022\\) 上格点（横纵坐标均为整数的点）的个数为多少？\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": ... | 0 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the $111^{\\text{st}}$ smallest positive integer which does not have $3$ and $4$ in its base-$5$ representation?\n\nRe... | 755 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose $a$ and $b$ are digits which are not both nine and not both zero. If the repeating decimal $0.\\overline{AB}$ is expre... | 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\nGiven $0 \\le x_0 <1$, let \\[ x_n= \\begin{cases} 2x_{n-1} & \\text{if}\\ 2x_{n-1} <1 \\\\ 2x_{n-1}-1 & \\text{if}\\ 2x_{n-1}... | 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\nIn the adjoining figure, $ABCD$ is a square and $CMN$ is an equilateral triangle. If the area of $ABCD$ is one square inch, fi... | 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 $Answer is the answer to the problem.\n\nA bug (of negligible size) starts at the origin on the coordinate plane. First, it moves one unit right to \\((1,0)\\). Then i... | 16 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f(x) = x^2 + 6x + 1$, and let $R$ denote the set of points $(x,y)$ in the coordinate plane such that\n\\[ f(x) + f(y) \\l... | 25 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTwo circles of radius 1 are to be constructed as follows. The center of circle \\(A\\) is chosen uniformly and at random from ... | 14 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA convex polyhedron $Q$ has vertices $V_1, V_2, \\ldots, V_n$, and 100 edges. The polyhedron is cut by planes $P_1, P_2, \\ldo... | 300 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn triangle $ABC$, lines $CE$ and $AD$ are drawn such that \\( \\frac{CD}{DB} = \\frac{3}{1} \\) and \\( \\frac{AE}{EB} = \\fr... | 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 ordered pairs of integers $(x,y)$ are there such that $2011y^2 = 2010x + 3$?\n\nRemember to put your answer on its ow... | 0 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn base $R_1$, the expanded fraction $F_1$ becomes $0.373737\\ldots$, and the expanded fraction $F_2$ becomes $0.737373\\ldots... | 19 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFermat determines that since his final exam counts as two tests, he only needs to score a $28$ on it for his test average to b... | 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\nIn triangle $ABC$, $AB = AC$. If there is a point $P$ strictly between $A$ and $B$ such that $AP = PC = CB$, then if the answe... | 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\nFind the last 5 digits of the number $M = 5^{2010}$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"... | 65625 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDave's Amazing Hotel has $3$ floors. If you press the up button on the elevator from the $3$rd floor, you are immediately tran... | 803 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAn object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one u... | 121 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe radius of Earth at the equator is approximately 4000 miles. Suppose a jet flies once around Earth at a speed of 500 miles ... | 50 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA circle has an inscribed triangle whose sides are $5\\sqrt{3}$, $10\\sqrt{3}$, and $15$. If the measure of the angle subtende... | 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 $Answer is the answer to the problem.\n\nA point \\(P\\) lies in the same plane as a given square with side length \\(1\\). The vertices of the square, taken countercl... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAt summer camp, there are $20$ campers in each of the swimming class, the archery class, and the rock climbing class. Each cam... | 12 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf Scott rolls four fair six-sided dice, find the probability that he rolls more 2's than 1's. The probability is given in the... | 1717 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf \\(\\frac{x}{x-1} = \\frac{y^2+2y-1}{y^2-2y-2}\\), find the value of \\(x\\). The original answer can be expressed in the f... | 2 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf \\( |x - \\log y| = x + \\log y \\) where \\( x \\) and \\( \\log y \\) are real, find the expression that represents the r... | 2 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the first step in finding the product \\((3x + 2)(x - 5)\\) by use of the distributive property in the form \\(a(b + c) =... | 0 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe first $2007$ positive integers are each written in base $3$. How many of these base-$3$ representations are palindromes? (... | 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 $Answer is the answer to the problem.\n\nIn \\(\\triangle ABC\\), point \\(D\\) is on \\(AC\\) and point \\(F\\) is on \\(BC\\). It is given that \\(AB \\perp AC\\), \... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor how many ordered pairs $(a,b)$ of integers does the polynomial $x^3 + ax^2 + bx + 6$ have 3 distinct integer roots?\n\nRem... | 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\nHalfway through a $100$-shot archery tournament, Chelsea leads by $50$ points. For each shot, a bullseye scores $10$ points, w... | 42 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEquilateral $\\triangle ABC$ is inscribed in a circle. A second circle is tangent internally to the circumcircle at $T$ and ta... | 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\nFrom a two-digit number $N$, we subtract the number with the digits reversed and find that the result is a positive perfect cu... | 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\nIf $\\frac{8}{19}$ of the product of the largest two elements of a positive integer set is not greater than the sum of the oth... | 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 $Answer is the answer to the problem.\n\nLet \\(ABCD\\) be a regular tetrahedron and let \\(E\\) be a point inside the face \\(ABC\\). Denote by \\(s\\) the sum of the... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCircles with centers \\(P\\), \\(Q\\), and \\(R\\), having radii \\(1\\), \\(2\\), and \\(3\\), respectively, lie on the same ... | 8 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA 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 |
[
{
"content": "Solve the following math problem step by 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 solutions does the equation $\\tan{(2x)} = \\cos{\\left(\\frac{x}{2}\\right)}$ have on the interval $[0, 2\\pi]$? Wri... | 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\nA rectangle is inscribed in a triangle such that its base coincides with the base $b$ of the triangle. If the altitude of the ... | 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 $Answer is the answer to the problem.\n\nIf \\(x \\in \\left[-\\frac{5\\pi}{12}, -\\frac{\\pi}{3}\\right]\\), find the maximum value of \\(y = \\tan\\left(x + \\frac{2... | 20 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf $n$ is any whole number, find the largest number by which $n^2(n^2 - 1)$ is always divisible.\n\nRemember to put your answe... | 12 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor every \\( x, y \\in \\mathbb{R}^+ \\), the function \\( f: \\mathbb{R}^+ \\to \\mathbb{R} \\) satisfies the condition \\( ... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor each $x$ in $[0,1]$, define\n\\[ f(x) = \\begin{cases} 2x, & \\text{if } 0 \\leq x \\leq \\frac{1}{2}; \\\\\n2 - 2x, & \\t... | 2007 |
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"content": "Solve the following math problem step by 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 this diagram, $AB$ and $AC$ are the equal sides of an isosceles triangle $ABC$, in which an equilateral triangle $DEF$ is i... | 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 $Answer is the answer to the problem.\n\nOn a standard die, one of the dots is removed at random, with each dot equally likely to be chosen. The die is then rolled. Wh... | 32 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA planet orbits around a star $S$, as shown in the figure. The semi-major axis of the orbit is $a$. The perigee, namely the sh... | 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 $Answer is the answer to the problem.\n\nIn \\(\\triangle ABC\\), \\(\\angle C = 90^\\circ\\), \\(AC = 6\\), and \\(BC = 8\\). Points \\(D\\) and \\(E\\) are on \\(\\o... | 23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the $xy$-plane, consider the L-shaped region bounded by horizontal and vertical segments with vertices at $(0,0)$, $(0,3)$,... | 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\nHow many ordered integer pairs $(x,y)$ are there such that $xy = 4(y^2 + x)$?\n\nRemember to put your answer on its own line a... | 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\nA positive integer $n$ has 60 divisors and $7n$ has 80 divisors. What is the greatest integer $k$ such that $7^k$ divides $n$?... | 2 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLily has an unfair coin that has a \\(\\frac{2}{3}\\) probability of showing heads and a \\(\\frac{1}{3}\\) probability of sho... | 11 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nJamal wants to store $30$ computer files on floppy disks, each of which has a capacity of $1.44$ megabytes (MB). Three of his ... | 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\nThe base of a triangle is $15$ inches. Two lines are drawn parallel to the base, terminating in the other two sides, and divid... | 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\nOn an auto trip, the distance read from the instrument panel was $450$ miles. With snow tires on for the return trip over the ... | 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\nFor all integers $n$ greater than $1$, define $a_n = \\frac{1}{\\log_n 2002}$. Let $b = a_2 + a_3 + a_4 + a_5$ and $c = a_{10}... | -1 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA ball of radius \\(R\\) and mass \\(m\\) is magically put inside a thin shell of the same mass and radius \\(2R\\). The syste... | 3 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA subset of the integers $1, 2, \\ldots, 100$ has the property that none of its members is 3 times another. What is the larges... | 76 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$N$ is a $50$-digit number (in the decimal scale). All digits except the $26^{\\text{th}}$ digit (from the left) are $1$. If $... | 3 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA rectangular floor measures $a$ by $b$ feet, where $a$ and $b$ are positive integers with $b > a$. An artist paints a rectang... | 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\nUsing a table of a certain height, two identical blocks of wood are placed as shown in Figure 1. The length $r$ is found to be... | 30 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe graphs of $2x + 3y - 6 = 0$, $4x - 3y - 6 = 0$, $x = 2$, and $y = \\frac{2}{3}$ intersect in how many points?\n\nRemember ... | 1 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $p$ and $q$ be two consecutive terms of the sequence of odd primes. Find the minimum number of positive divisors of $p + q... | 4 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat are the last three digits of $49^{303} \\cdot 3993^{202} \\cdot 39^{606}$?\n\nRemember to put your answer on its own line... | 561 |
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"content": "Solve the following math problem step by 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 base-10 equation, each letter represents a unique digit: $AM \\cdot PM = ZZZ$. Find the sum of $A + M + P + Z... | 21 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the product of all odd positive integers less than 10000? Express your answer in the form of a factorial expression di... | 20000 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the area of the region enclosed by the graph of the equations $x^2 - 14x + 3y + 70 = 21 + 11y - y^2$ that lies below t... | 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 point $(-1, -2)$ is rotated $270^{\\circ}$ counterclockwise about the point $(3, 1)$. What are the coordinates of its new ... | 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 $Answer is the answer to the problem.\n\nMy frisbee group often calls \"best of five\" to finish our games when it's getting dark, since we don't keep score. The game ... | 5 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTwo circles are externally tangent. Lines \\(\\overline{PAB}\\) and \\(\\overline{PA'B'}\\) are common tangents with \\(A\\) a... | 2 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$AB$ is the diameter of a circle centered at $O$. $C$ is a point on the circle such that angle $\\angle BOC$ is $60^\\circ$. I... | 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\nFor how many integers $n$, with $2 \\leq n \\leq 80$, is $\\frac{(n-1)n(n+1)}{8}$ equal to an integer? Provide the number of s... | 49 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nArithmetic sequences $(a_n)$ and $(b_n)$ have integer terms with $a_1 = b_1 = 1 < a_2 \\le b_2$ and $a_nb_n = 2010$ for some $... | 8 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $ABC$ be a triangle with $|AB|=18$, $|AC|=24$, and $m(\\widehat{BAC}) = 150^\\circ$. Let $D$, $E$, $F$ be points on sides ... | 96 |
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{
"content": "Solve the following math problem step by 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 distinct residues of the number $2012^n + m^2 \\pmod{11}$ where $m$ and $n$ are positive integers.\n\nRemember... | 39 |
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{
"content": "Solve the following math problem step by 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 triangle have lengths $11$, $15$, and $k$, where $k$ is an integer. Find the number of integer values of $k$ fo... | 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 $Answer is the answer to the problem.\n\nA unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure 1. The cube is then c... | 13 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven $a_1 = 1$, $a_{n+1} = \\frac{a_n}{\\sqrt{1 + 4a_n^2}}$ for $n \\ge 1$, find the least $k$ such that $a_k < 10^{-2}$.\n\n... | 2501 |
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{
"content": "Solve the following math problem step by 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 solutions does the equation $\\sin \\left( \\frac{\\pi}{2} \\cos x\\right)=\\cos \\left( \\frac{\\pi}{2} \\sin x\\rig... | 2 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many different solutions does the congruence $x^3 + 3x^2 + x + 3 \\equiv 0 \\pmod{25}$ have? Provide your answer as an int... | 6 |
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