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 $Answer is the answer to the problem.\n\n设实数 x,y,z\\in[0,1]，记M=\\min\\{x-xy-xz+yz,y-yx-yz+xz,z-zx-zy+xy\\},求 M 的最大可能值.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\nR... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n一个骰子连续掷两次，得到的点数依次为 a 和 b，则使得关于 x 的三次方程 x^3-(3a+1)x^2+(3a+2b)x-2b=0 有三个互不相等的实数根的概率 P=__________.（用数字作答）原始的答案是\\frac{m}{n}的形式，其中... | 7 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若复数 z 满足 \\left|\\df{z^2+1}{z+i}\\right|+\\left|\\df{z^2+4i-3}{z-i+2}\\right|=4，则 |z-1| 的最小值为__________.原始的答案是k\\sqrt{m}+n的形式，... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 x 是一个锐角，那么 \\df{8}{\\sin x}+\\df{1}{\\cos x} 的最小值是__________.原始的答案是k\\sqrt{m}+n的形式，请给出k + m + n的值。\n\nRemember to put your ... | 10 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n将一枚均匀的骰子独立投掷三次，所得的点数依次记为 x,y,z，则事件\"\\text{C}_7^x<\\text{C}_7^y<\\text{C}_7^z\"发生的概率为__________.原始的答案是\\frac{m}{n}的形式，其中m、n是互质... | 28 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若平面上非零向量 \\overrightarrow{\\alpha},\\overrightarrow{\\beta},\\overrightarrow{\\gamma} 满足 \\overrightarrow{\\alpha}\\bot\\overr... | 5 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n从 1,2,\\cdots,9 中任选 3 个数，它们两两互质的概率为__________.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\nRemember to put your answer on i... | 121 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设函数 f(x)=|x\\log_{2023}x|，实数 a<b<c 满足 f(a)=f(b)=f(c). 若 a, b, c 构成等比数列，则其公比为__________.原始的答案是k\\sqrt{m}+n的形式，请给出k + m + n的值。\n... | 4 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在平面直角坐标系中，点 A,B,C 在双曲线 xy=1 上，满足 \\triangle ABC 为等腰直角三角形. 求 \\triangle ABC 的面积的最小值.原始的答案是k\\sqrt{m}+n的形式，请给出k + m + n的值。\n\nRe... | 6 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在平面直角坐标系 xOy 中，点集 K=\\{(x,y)\\ |\\ x,y=-1,0,1\\}. 在 K 中随机取出三个点，则这三个点两两之间的距离均不超过 2 的概率为________.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的... | 19 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n正实数 x,y,z 满足 x+2y^2+4x^2y^2z^2=8，则 \\log_4x+\\log_2y+\\log_8z 的最大值为__________.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\n... | 4 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知直三棱柱 ABC-A_1B_1C_1 的底面为直角三角形，\\angle ACB=90^\\circ，AC=6，BC=CC_1=\\sqrt{2}. P 是 BC_1 上一动点，则 CP+PA_1 的最小值是__________.原始的答案是k\\... | 7 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求最小常数 \\lambda，使不等式xy(x^2+y^2)+yz(y^2+z^2)+zx(z^2+x^2)\\leq \\lambda(x+y+z)^4对所有非负实数 x,y,z 都成立.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的... | 9 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在正四面体 P-ABC 中，点 E,F 分别在棱 PB, PC 上. 若 PE\\neq PF，且 AE=AF=2, EF=\\sqrt{3}，则四面体 P-AEF 的体积为__________.原始的答案是\\frac{m}{n}的形式，其中m、n是... | 9 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在平面直角坐标系 xOy 中，A 为椭圆 \\df{x^2}{4}+y^2=1 上一点，M 为线段 OA 上的动点. 过 M 作直线与椭圆交于 P, Q 两点，若 \\overrightarrow{PM}=2\\overrightarrow{MQ}，求... | 5 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 x,y,z 是正实数，求2\\sqrt{(x+y+z)\\left(\\df{1}{x}+\\df{1}{y}+\\df{1}{z}\\right)}-\\sqrt{\\left(1+\\df{x}{y}\\right)\\left(1+\\df{... | 5 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知非负实数 a_i,b_i(1\\leq i\\leq 2024) 满足 \\ds{i=1}{2024}a_i=\\ds{i=1}{2024}b_i=1. 求\\ds{i=1}{2024}a_ib_{i+1}b_{i+2}b_{i+3}a_{i+4}... | 109 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n将 1,2,3,4,5,6 随机排成一行，记为 a, b, c, d, e, f，则 abc+def 是偶数的概率为________.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\nRemember to... | 19 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设复数 z 满足 |z|=1，且使得关于 x 的方程 zx^2+2\\bar{z}x+2=0 有实根. 则这样的 z 的和为________.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\nRemembe... | -1 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n将一枚硬币和一个骰子同时投掷，硬币出现正面记为 2、出现反面记为 1. 此数与骰子的点数之积记为 x（例如硬币出现正面，骰子点数为 3，则 x=2\\times3=6），那么 x 的数学期望是__________.原始的答案是\\frac{m}{n}的... | 25 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 x,y\\in[0,+\\infty)，且满足 x^3+y^3+6xy=8. 则 2x^2+y^2 的最小值为__________.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\nRemember ... | 11 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 x_1,x_2,\\cdots,x_{50} 是实数，且 \\ds{i=1}{50}\\df{x_i}{k+i}=\\df{2021}{2k+1} 对任意 k\\in\\{1,2,\\cdots,50\\} 成立. 求 \\ds{i=1}{50}... | 12751 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知实数 x,y 满足 x<y,|x|<3,|y|<3，且关于 t 的方程 (x^3-y^3)t^4+(3x+y)t^2+\\df{1}{x-y}=0 有实根. 则点 (x,y) 构成的区域的面积为__________.原始的答案是\\frac{m}{... | 14 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在平面直角坐标系 xOy 中，抛物线 C:y^2=4x 的焦点为 F，过 C 上一点 P（异于 O）作 C 的切线与 y 轴交于点 Q. 若点 Q 关于直线 PF 的对称点在 x 轴上，则 \\triangle PQF 的面积为__________.原... | 5 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n盒子中有大小、形状完全相同的 3 个红球和 3 个白球. 现抛掷一枚质地均匀的骰子，掷出几点就从盒子中取出几个球. 则取出的球中红球个数大于白球个数的概率为_________.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m... | 79 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知双曲线 C:\\df{x^2}{a^2}-\\df{y^2}{b^2}=1(a>0,b>0) 的离心率为 \\df{\\sqrt{17}}{3}，F 为右焦点，点 A,B 在右支上. 设 D 为 A 关于原点 O 的对称点，且 DF\\bot AB... | 3 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设实数 a,b,c,d 满足 a^2+b^2+c^2+d^2=1. 求 (a-b)(b-c)(c-d)(d-a) 的最小值，并求达到最小值时所有的 (a,b,c,d).原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n... | 7 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知函数 f(x)=\\cos^4x+\\sin^4x+a\\sin4x-b，且 f(x+\\frac{\\pi}{6}) 为奇函数. 若方程 f(x)+m=0 在 [0,\\pi] 上有四个不同的实数解 x_1, x_2, x_3, x_4，则 f(... | 13 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n从面积为 1 的正 7 边形的 7 个顶点中随机选出 4 个不同的顶点，它们构成的四边形的面积大于 \\df{1}{2} 的概率是__________.（王正供题）原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值... | 9 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设复数 x,y,z 满足：|x+2y+3z|=1. 求|x|^2+|y|^2+|z|^2+|x^2+y^2+z^2|的最小值.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\nRemember to put... | 8 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n点 M,N 分别在 \\triangle ABC 的边 AB,AC 上，且\\overrightarrow{AM}=x\\overrightarrow{AB},\\quad \\overrightarrow{AN}=y\\overrightarrow{... | 94 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若 \\triangle ABC 的内角 A,B,C 满足 \\sin A=\\cos B=\\tan C，求 \\cos^3A+\\cos^2A-\\cos A 的值.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + ... | 3 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n直角三角形 DEF 的三个顶点分别在等边三角形 ABC 的边 AB, BC, CA 上，且 \\angle DEF=90^\\circ, \\angle EDF=30^\\circ. 求 \\df{S_{\\triangle DEF}}{S_{\\tr... | 17 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求具有下述性质的最小正数 c：对任意整数 n\\geq4，以及集合 A\\subseteq\\{1,2,\\cdots,n\\}，若 |A|>cn，则存在函数 f:A\\rightarrow\\{-1,1\\}，满足\\left|\\ds{a\\in ... | 5 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求最大的实数 k，使得对任意正实数 a,b，均有(a+b)(ab+1)(b+1)\\geq kab^2.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\nRemember to put your answe... | 31 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n将一个正方形分割成有限个白色和红色矩形，其边均与正方形的边平行. 在每个白色矩形内，写下它的宽与高的比值；在每个红色矩形内，写下它的高与宽的比值. 我们计算出所有矩形内写的数的总和 x. 如果两种颜色矩形的总面积相同，求 x 的最小可能值.原始的答案是... | 7 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 y^2=4x 的三条切线的斜率为 1,2,3，则三条切线围成的三角形的外接圆直径为__________.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\nRemember to put your ans... | 8 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若平面向量 \\overrightarrow{a}=(2^m,-1) 与 \\overrightarrow{b}=(2^m-1,2^{m+1}) 垂直，其中 m 为实数，则 \\overrightarrow{a} 的模为________.原始的答案是k... | 11 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 x\\in[-\\frac{\\pi}{6},\\frac{\\pi}{6}]，则函数 f(x)=\\sin^2x-2\\sin x 的最大值为__________.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + ... | 9 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设实数 a_1,a_2,a_3,a_4,a_5 满足，对 k=1,2,3,4,5，\\df{a_1}{k^2+1}+\\df{a_2}{k^2+2}+\\df{a_3}{k^2+3}+\\df{a_4}{k^2+4}+\\df{a_5}{k^2+5}=... | 6932047 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设非负实数 x_1,x_2,\\cdots,x_{2018} 满足 \\ds{1\\leq i<j\\leq 2018}{}x_ix_j=1. 求 \\ds{i=1}{2018}\\df{1}{s-x_i} 的最小值，其中 s=\\ds{i=1}{20... | 38 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设函数 f(x)=A\\sin(\\omega x+\\varphi) (A\\neq0,\\omega>0,0<\\varphi<\\df{\\pi}{2}). 若 f(\\df{5\\pi}{6})+f(0)=0，则 \\omega 的最小值为__... | 11 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知正实数 a,b 满足 a^b=(8a)^{9b}，则 \\log_a(4a^2) 的值为__________.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\nRemember to put your ... | 65 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n某校三个年级举办乒乓球比赛，每个年级选派 4 名选手参加比赛. 组委会随机将这 12 名选手分成 6 组，每组 2 人，则在上述分组方式中每组的 2 人均来自不同年级的概率为__________.原始的答案是\\frac{m}{n}的形式，其中m、n是... | 449 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设实数 a,b,c\\in(0,2]，且 b\\geq 3a 或 a+b\\leq\\df{4}{3}，则 \\max\\{b-a,c-b,4-2c\\} 的最小值为__________.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。... | 5 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在椭圆 \\Gamma 中，A 为长轴的一个端点，B 为短轴的一个端点，F_1,F_2 为两个焦点. 若 \\overrightarrow{AF_1}\\cdot\\overrightarrow{AF_2}+\\overrightarrow{BF_1}... | 4 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求最大的实数 c，使得对平面上任意两个单位向量 \\overrightarrow{v_1}, \\overrightarrow{v_2}，均存在单位向量 \\overrightarrow{v}，满足|\\overrightarrow{v}\\cdot\... | 3 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设集合 S=\\{1,2,3,4,5\\}. 若 S 的子集 A 满足：若 x\\in A，则 6-x\\in A，则称子集 A 具有性质 p. 现从 S 的所有非空子集中，等可能地取出一个，则所取出的非空子集具有性质 p 的概率为__________... | 38 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设非负实数 a,b,c,x,y,z 满足 a+b+c=x+y+z=1. 求 (a-x^2)(b-y^2)(c-z^2) 的最大值.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\nRemember to p... | 17 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设正实数 u,v,w 满足 u\\sqrt{vw}+v\\sqrt{wu}+w\\sqrt{uv}\\geq1. 求 u+v+w 的最小值.原始的答案是k\\sqrt{m}+n的形式，请给出k + m + n的值。\n\nRemember to put... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n函数 y=\\df{1}{3}\\cos2x+\\sin x-\\cos x 在区间 \\left[-\\df{\\pi}{2},\\df{\\pi}{2}\\right] 上的最大值为__________.原始的答案是\\frac{m}{n}的形式，... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在平面直角坐标系中，甲从原点出发，到点 (4,4)；乙从点 (2,0) 出发，到点 (6,4). 已知两人同时出发，且每次向上或向右走一步. 则甲、乙的行走路线没有公共点的概率为__________.原始的答案是\\frac{m}{n}的形式，其中m、... | 19 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求最小的实数 C，使得对任意正实数 a_1, a_2, a_3, a_4, a_5（允许相同），总可以选择不同的下标 i,j,k,l，满足 \\left|\\df{a_i}{a_j}-\\df{a_k}{a_l}\\right|\\leq C.原始的答... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求最大的实数 a，使得对任意正整数 n 及任意实数 0=x_0<x_1<\\cdots<x_n，均有\\ds{i=1}{n}\\df{1}{x_i-x_{i-1}}\\geq a\\ds{i=1}{n}\\df{i+1}{x_i}.原始的答案是\\fr... | 13 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设非负实数 a,b,c,d 满足 a+b+c+d=4. 求\\df{a}{b^3+4}+\\df{b}{c^3+4}+\\df{c}{d^3+4}+\\df{d}{a^3+4}的最小值.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若无穷等比数列 \\{a_n\\} 的各项和为 1，各项的绝对值之和为 2，则首项 a_1 的值为__________.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\nRemember to put yo... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在 1,2,\\cdots,10 中随机选出三个不同的数，它们两两互素的概率为__________.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\nRemember to put your answer ... | 27 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 a,b 是从集合 \\{1,2,3,4,5\\} 中随机选取的数，则直线 y=ax+b 与圆 x^2+y^2=2 有公共点的概率是__________.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\n... | 44 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 n 是正整数，A_1,A_2,\\cdots,A_n 是 n 个不同的非空有限集，满足对任意 1\\leq i,j,k\\leq n，若 A_i\\cap A_j=\\emptyset，则存在 1\\leq l\\leq n，使得(A_i\\cup... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 f(x) 是定义在 \\bR 上的函数，且对任意实数 x，均有 2f(x)+f(x^2-1)=1，则 f(\\sqrt{2}) 的值为__________.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n数列 \\{a_n\\} 满足 a_1=1,\\df{a_{n+1}-a_n}{a_n}=\\df{a_{n+2}-a_{n+1}}{a_{n+2}} (n\\in\\bN^\\ast). 若 a_1a_2+a_2a_3+\\cdots+a_6a_7=... | 2035 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n一珍稀物种出现在地球，对每个珍稀生物，每天有如下事件发生：有 p (0\\leq p\\leq 1) 的概率消失，有 \\df{1-p}{3} 的概率保持不变，有 \\df{1-p}{3} 的概率分裂成两个，有 \\df{1-p}{3} 的概率分裂成三... | 22 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在矩形 $A B C D$ 中, $A B=2, A D=1$, 边 $D C$ 上（包含点 $D 、 C$ ）的动点 $P$ 与 $C B$ 延长线上 (包含点 $B$ ) 的动点 $Q$ 满足 $|\\overrightarrow{D P}|=|\... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n袋子 $A$ 中装有 2 张 10 元纸币和 3 张 1 元纸币，袋子 $B$ 中装有 4 张 5 元纸币和 3 张 1元纸币。现随机从两个袋子中各取出两张纸币，则 $A$ 中剩下的纸币面值之和大于 $B$ 中剩下的纸币面值之和的概率为 $\\qqua... | 44 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在平面直角坐标系 $x O y$ 中，点集 $K=\\{(x, y) \\mid x, y=-1,0,1\\}$ 。在 $K$ 中随机取出三个点，则这三点中存在两点之间距离为 $\\sqrt{5}$ 的概率为 $\\qquad$.\n原始的答案是\\f... | 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\n设数列 $\\left\\{a_{n}\\right\\}$ 是等差数列，数列 $\\left\\{b_{n}\\right\\}$ 满足 $b_{n}=a_{n+1} a_{n+2}-a_{n}^{2}, n=1,2, \\cdots$ 。\n（1）... | 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在平面直角坐标系 $x O y$, 点集 $K=\\{(x, y) \\mid x, y \\in-1,0,1\\}$ 。在 $K$ 中随机取出三个点,则这三个点两两之间的距离均不超过 2 的概率是 $\\qquad$.\n原始的答案是\\frac{m... | 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将 $1,2,3,4,5,6$ 随机排成一行, 记为 $a, b, c, d, e, f$, 则 $a b c+def$ 是偶数的概率为 $\\qquad$.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\... | 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将 $1,2,3,4,5,6$ 随机排成一行, 记为 $a, b, c, d, e, f$, 则 $a b c+d e f$ 是奇数的概率为 $\\qquad$.\n原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的... | 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\n在 $\\triangle A B C$ 中, $\\sin A=\\frac{3}{5}, \\cos B=\\frac{5}{13}$, 那么 $\\cos C$ 的值等于 $\\qquad$.原始的答案是\\frac{m}{n}的形式，其中m、n... | 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\n在平面直角坐标系中, 点 $A, B, C$ 在双曲线 $x y=1$ 上, 满足 $\\triangle A B C$ 为等腰直角三角形。求 $\\triangle A B C$ 的面积的最小值.\n原始的答案是k\\sqrt{m}+n的形式，请给出... | 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在凸四边形 $A B C D$ 中, $\\overrightarrow{B C}=2 \\overrightarrow{A D}$. 点 $P$ 是四边形 $A B C D$ 所在平面上一点, 满足\n$$\n\\overrightarrow{P A... | 2358 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$x_{1}, x_{2}, x_{3}, x_{4}$ 为互不相等的正实数, $x_{i 1}, x_{i 2}, x_{i 3}, x_{i 4}$ 为 $x_{1}, x_{2}, x_{3}, x_{4}$ 的任意顺序排列, $X=\\max ... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在凸四边形 $A B C D$ 中, $\\overrightarrow{B C}=2 \\overrightarrow{A D}$. 点 $P$ 是四边形所在平面上一点, 满足 $\\overrightarrow{P A}+2020 \\overri... | 2358 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n数列 $\\left\\{a_{n}\\right\\}$ 满足 $a_{1}=1, a_{n}=\\frac{1}{4 a_{n-1}}+\\frac{1}{n}(n \\geq 2)$, 则 $a_{100}$ 的值为 $\\qquad$.原始的答... | 301 |
[
{
"content": "Solve the following math problem step by 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在平面直角坐标系 $x O y$ 中， $\\Gamma_{1}$ 是以 $(2,1)$ 为圆心的单位圆， $\\Gamma_{2}$ 是以 $(10,11)$为圆心的单位圆。过原点 $O$ 作一条直线 $l$, 使得 $l$ 与 $\\Gamma_{... | 87 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n等差数列 $\\left\\{a_{n}\\right\\}$ 满足 $a_{2021}=a_{20}+a_{21}=1$, 则 $a_{1}$ 的值为 $\\qquad$.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m ... | 5982 |
[
{
"content": "Solve the following math problem step by 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 $m$ 为实数, 复数 $z_{1}=1+2 \\mathrm{i}, z_{2}=m+3 \\mathrm{i}$ (这里 i 为虚数单位), 若 $z_{1} \\cdot \\overline{z_{2}}$ 为纯虚数，则 $\\left|z... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $\\alpha, \\beta \\in \\mathbf{R}$, 若 $\\tan (\\alpha+\\beta)=2, \\tan (\\alpha+2 \\beta)=3$, 则 $\\tan \\alpha$ 的值为 $\\qquad... | 22 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在某次数学竞赛小组交流活动中，四名男生与三名女生按随机次序围坐一圈，则三名女生两两不相邻的概率为 $\\qquad$.\n原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\nRemember to put y... | 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在 $1,2, \\cdots, 10$ 中随机选出三个不同的数, 它们两两互素的概率为 $\\qquad$.\n原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\nRemember to put your ... | 27 |
[
{
"content": "Solve the following math problem step by 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 $y=f(x)$ 的图像既关于点 $(1,1)$ 中心对称, 又关于直线 $x+y=0$轴对称。若 $x \\in(0,1)$ 时, $f(x)=\\log _{2}(x+1)$, 则 $f\\left(\\log _{2} 10\\righ... | 22 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在平面直角坐标系中, 双曲线 $\\Gamma: x^{2}-y^{2}=1$ 的右顶点为 $A$ 。将圆心在 $y$ 轴上，且与 $\\Gamma$ 的两支各恰有一个公共点的圆称为 \"好圆\"。若两个好圆外切于点 $P$ ，圆心距为 $d$ ，求 ... | 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一只青蛙在正方形 $A B C D$ 的四个顶点间跳跃, 每次跳跃总是等可能地跳至与当前所在顶点相邻的两个顶点之一，且各次跳跃是独立的. 若青蛙第一次跳跃前位于顶点 $A$ ，则它第 6 次跳跃后恰好仍位于顶点 $A$ 的概率为\n原始的答案是\\fr... | 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 $\\triangle A B C$ 中, 已知\n$\\cos C=\\frac{\\sin A+\\cos A}{2}=\\frac{\\sin B+\\cos B}{2}$, 求 $\\sin C$ 的值\n原始的答案是\\frac{m}{n... | 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\n在矩形 $A B C D$ 中, $A B=2, A D=1$, 在 $D C$ 边上（包含点 $D 、 C$ ）的动点 $P$ 与 $C B$延长线上（包含 $B$ ）的动点 $Q$ 满足 $|\\overrightarrow{D P}|=|\\ov... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$x_{1}, x_{2}, x_{3}, x_{4}$ 为互不相等的正实数, $x_{i 1}, x_{i 2}, x_{i 3}, x_{i 4}$ 为 $x_{1}, x_{2}, x_{3}, x_{4}$ 的任意顺序排列, $X=\\max ... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n定义 $x * y=\\frac{x+y}{1+x y}$, 则 $(\\cdots((2 * 3) * 4) \\cdots) * 21=\\qquad$.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\... | 231 |
[
{
"content": "Solve the following math problem step by 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如图 1，抛物线 $y^{2}=2 p x(p>0)$ 的焦点为 $F$, 准线为 $l$, 过点 $F$ 的直线与抛物线交于 $A 、 B$ 两点, 且 $|A B|=3 p$. 设点 $A 、 B$ 在 $l$ 上的射影分别为 $A^{\\prim... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $\\triangle A B C$ 的内角 $\\angle A 、 \\angle B 、 \\angle C$ 的对边分别为 $a 、 b 、 c$, 且 $\\angle A-\\angle C=\\frac{\\pi}{2}, a 、 b... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n有 20 张卡片上分别写有数字 1,2, $\\cdots, 20$, 将它们放入一个盒子内。有 4 个人从中不放回地各抽取一张卡片，抽到两个较小数字的两人在同一组，抽到两个较大数字的两人在同一组。现其中有两人抽到 $5 、 14$, 则此两人在同一组... | 58 |
[
{
"content": "Solve the following math problem step by 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在正三棱锥 $P-A B C$ 中, 已知侧棱与底面所成的角为 $45^{\\circ}$ 。则相邻两侧面所成角的余弦值为 $\\qquad$.\n#*原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\nRe... | 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 $Answer is the answer to the problem.\n\n在一个圆上随机取三点, 则以此三点为顶点的三角形是锐角三角形的概率为\n$\\qquad$.\n#*原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\nRemember to put your answer ... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知正四面体的四个表面上分别写有数字 $1 、 2 、 3 、 4$, 将四个这样的密度均匀的正四面体同时投郑于桌面上，与桌面接触的四个面上的四个数的和能被 4 整除的概率为\n$\\qquad$.\n原始的答案是\\frac{m}{n}的形式，其中m... | 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已知 $f(x)$ 是定义在 $\\mathbf{R}$ 上的函数, 且对任意实数 $x$, 均有 $2 f(x)+f\\left(x^{2}-1\\right)=1$, 则 $f(\\sqrt{ } 2)$ 的值为 $\\qquad$。\n原始的答案... | 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已知抛物线 $y^{2}=2 p x(p>0), A, B$ 是抛物线上不同于顶点 $O$ 的两个动点. 记 $\\angle A O B=\\theta\\left(\\theta \\neq 90^{\\circ}\\right)$. 若 $S_{... | 0 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n如图，设椭圆 $\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1(a>b>0)$ 的左、右焦点分别为 $F_{1} 、 F_{2}$ ，过点 $F_{2}$ 的直线交栯圆于 $A\\left(x_{1}, y_{1... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $a, b, c$ 均大于 1, 满足 $\\left\\{\\begin{array}{l}\\lg a+\\log _{b} c=3, \\\\ \\lg b+\\log _{a} c=4.\\end{array}\\right.$ 求 $\\... | 19 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求具有下述性质的最小正数 $c$ ：对任意整数 $n \\geq 4$,以及集合 $A \\subseteq\\{1,2, \\cdots, n\\}$, 若 $|A|>c n$, 则存在函数 $f: A \\rightarrow\\{1,-1\\}$... | 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设 $x \\neq y$, 且两数列 $x, a_{1}, a_{2}, a_{3}, y$ 和 $b_{1}, x, b_{2}, b_{3}, y, b_{4}$ 均为等差数列, 那么 $\\frac{b_{4}-b_{3}}{a_{2}-a_{... | 11 |
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