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函数 $f(x)=\\frac{5-4 x+x^{2}}{2-x}$ 在 $(-\\infty, 2)$ 上的最小值是多少？\n\nRemember to put your answer on its own line after \"Answer:\... | 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 $Answer is the answer to the problem.\n\n已知函数 \\(f(x)=x^{2}-2 t x+t\\)。当 \\(x \\in [-1,1]\\) 时，记 \\(f(x)\\) 的最小值为 \\(m\\)。求 \\(m\\) 的最大值。原始答案的形式为\\(\\frac{k}{m}\\)，请给出... | 5 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n对任意正整数 $n$ 与 $k(k \\leqslant n), f(n, k)$ 表示不超过 $\\left[\\frac{n}{k}\\right]$ （ $[x]$ 表示不超过实数 $x$ 的最大整数）且与 $n$ 互素的正整数的个数。求 $f(... | 14 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在 $(x+y+z)^{8}$ 的展开式中，求所有形如 $x^{2} y^{a} z^{b}(a, b \\in \\mathbf{N})$ 的项的系数之和。\n\nRemember to put your answer on its own line... | 1792 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在 \\(\\triangle A B C\\) 中, \\(\\overrightarrow{A B} \\cdot \\overrightarrow{B C}=3 \\overrightarrow{C A} \\cdot \\overrightar... | 3 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在 $(x+y+z)^{8}$ 的展开式中，所有形如 $x^{2} y^{a} z^{b}(a 、 b \\in \\mathbf{N})$ 的项的系数之和为多少？\n\nRemember to put your answer on its own l... | 1792 |
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"content": "Solve the following math problem step by 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 $F_{1} 、 F_{2}$ 为椭圆 $\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1(a>b>0)$ 的左、右焦点，该椭圆上存在两点 $A 、 B$ ，使得 $\\overrightarrow{F_{1}... | 4 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知数列 $\\left\\{a_{n}\\right\\}$ 满足\n$a_{n}=(\\sqrt{2}+1)^{n}-(\\sqrt{2}-1)^{n}(n \\in \\mathbf{N})$.\n记 $[x]$ 表示不超过实数 $x$ 的最大整... | 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已知方程 $1+x-\\frac{x^{2}}{2}+\\frac{x^{3}}{3}-\\frac{x^{4}}{4}+\\cdots-\\frac{x^{2018}}{2018}=0$ 的所有实数根都在区间 $[a, b]$ 内（其中 $a, b ... | 3 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$A B$ 为过抛物线 $y^{2}=4 x$ 焦点 $F$ 的弦, $O$ 为坐标原点, 且 $\\angle O F A=135^{\\circ}, C$ 为抛物线准线与 $x$ 轴的交点，求 $\\angle A C B$ 的正切值。答案的形式为... | 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设复数 $z_{1}, z_{2}$ 在复平面内对应的点分别为 $Z_{1}, Z_{2}, O$ 为坐标原点, 若 $\\left|z_{1}\\right|=1$, $5 z_{1}^{2}+z_{2}^{2}-2 z_{1} z_{2}=0$, ... | 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 $Answer is the answer to the problem.\n\n求值: \\(\\lim _{n \\rightarrow+\\infty}\\left(\\sum_{k=1}^{n} \\arctan \\frac{2}{k^{2}}\\right)=(\\quad)\\pi\\)。如果答案是\\(\\frac{... | 7 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设实数 $x_{1}, x_{2}, \\cdots, x_{21}$ 满足 $0 \\leqslant x_{i} \\leqslant 1(1,2, \\cdots, 21)$, 求 $\\sum_{i=1}^{21} \\sum_{k=1}^{2... | 220 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设实数 $x_{1}, x_{2}, \\cdots, x_{21}$ 满足 $0 \\leq x_{i} \\leq 1(1,2, \\cdots, 21)$ ，求 $\\sum_{i=1}^{21} \\sum_{k=1}^{21}\\left|x... | 220 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设复数 $z$ 满足 $|3 z-7 \\mathrm{i}|=3$, 求 $\\left|\\frac{z^{2}-2 z+2}{z-1+\\mathrm{i}}\\right|$ 的最大值。答案的形式为 $\\frac{k}{m}$，请给出 $k ... | 11 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 $A, B, C$ 是集合 $\\{1,2, \\cdots, 2020\\}$ 的子集，且满足 $A \\subseteq B \\subseteq C$ ，求这样的有序组 $(A, B, C)$ 的总数。答案的形式为 $x^y$，请给出x和y... | 2024 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere exist positive integers $A$, $B$, and $C$, with no common factor greater than 1, such that:\n\\[A \\log_{200} 5 + B \\lo... | 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\nFive friends sat in a movie theater in a row containing $5$ seats, numbered $1$ to $5$ from left to right. (The directions \"l... | 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 $Answer is the answer to the problem.\n\nSix distinct integers are picked at random from \\(\\{1,2,3,\\ldots,10\\}\\). What is the probability that, among those select... | 4 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTriangle \\( ABC \\) has side lengths in arithmetic progression, and the smallest side has length \\( 6 \\). If the triangle h... | 18 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet the sequence \\((a_n)\\) be defined as \\(a_1=\\frac{1}{3}\\) and \\(a_{n+1}=\\frac{a_n}{\\sqrt{1+13a_n^2}}\\) for every \... | 193 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the figure, $ABCD$ is a $2 \\times 2$ square, $E$ is the midpoint of $\\overline{AD}$, and $F$ is on $\\overline{BE}$. If $... | 16 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe cost $C$ of sending a parcel post package weighing $P$ pounds, where $P$ is an integer, is $10$ cents for the first pound ... | 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 $Answer is the answer to the problem.\n\nA circular piece of metal of maximum size is cut out of a square piece, and then a square piece of maximum size is cut out of ... | 3 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet \\((1+x+x^2)^n = a_0 + a_1x + a_2x^2 + \\ldots + a_{2n}x^{2n}\\) be an identity in \\(x\\). If we let \\(s = a_0 + a_2 + a... | 4 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLine $\\ell$ in the coordinate plane has the equation $3x - 5y + 40 = 0$. This line is rotated $45^{\\circ}$ counterclockwise ... | 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\n定义序列 $\\{a_n\\}_{n\\geq 1}$ 如下：\n- 当 $n \\leq 2$ 时，$a_n = n - 1$\n- 对于所有 $n \\geq 2$，$a_n$ 是 $a_{n-1} + a_{n-2}$ 除以 $3$ 的余数\n\... | 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\nLet $P(x) = 1 - x + x^2 - x^3 + \\dots + x^{18} - x^{19}$ and $Q(x) = P(x - 1)$. Find the coefficient of $x^2$ in polynomial $... | 1140 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSquare $BDEC$ with center $F$ is constructed on the outside of triangle $ABC$ such that $\\angle A = 90^\\circ$, $|AB| = \\sqr... | 11 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many ways are there to make change for $55$ cents using any number of pennies, nickels, dimes, and quarters?\n\nRemember t... | 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 $Answer is the answer to the problem.\n\nA six-digit number (base 10) is considered \"squarish\" if it satisfies the following conditions:\n\n1. None of its digits are... | 2 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSquares $ABCD$ and $EFGH$ are congruent, $AB=10$, and $G$ is the center of square $ABCD$. Find the area of the region in the p... | 175 |
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"content": "Solve the following math problem step by 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 pyramid of non-negative integers is constructed as follows:\n\n1. The first row consists of only $0$.\n2. The second row con... | 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\nA privateer discovers a merchantman $10$ miles to leeward at 11:45 a.m. and with a good breeze bears down upon her at $11$ mph... | 35 |
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"content": "Solve the following math problem step by 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 men, Alpha, Beta, and Gamma, working together, do a job in $6$ hours less time than Alpha alone, in $1$ hour less time t... | 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 $Answer is the answer to the problem.\n\nAn observer stands on the side of the front of a stationary train. When the train starts moving with constant acceleration, it... | 81 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA box contains 11 balls, numbered 1, 2, 3, ..., 11. If 6 balls are drawn simultaneously at random, find the probability that t... | 349 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf two poles $20''$ and $80''$ high are $100''$ apart, find the height (in inches) of the intersection of the lines joining th... | 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 $f(x) = \\sqrt{\\sin^4 x + 4\\cos^2 x} - \\sqrt{\\cos^4 x + 4\\sin^2 x}$. Find an equivalent form of $f(x)$ in the format ... | 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 $Answer is the answer to the problem.\n\nHow many integer pairs \\((x, y)\\) are there such that:\n\\[0 \\leq x < 165, \\quad 0 \\leq y < 165 \\text{ and } y^2 \\equiv... | 99 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn a class, some pupils learn German, while others learn French. The number of girls learning French and the number of boys le... | 23 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $A$, $B$ be the number of digits of $2^{1998}$ and $5^{1998}$ in the decimal system. Find the value of $A + B$.\n\nRemembe... | 1999 |
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"content": "Solve the following math problem step by 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 probability of having $2$ adjacent white balls or $2$ adjacent blue balls in a random arrangement of $3$ red, $2$ ... | 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 a ten-mile race, First beats Second by $2$ miles and First beats Third by $4$ miles. If the runners maintain constant speed... | 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\nVessel $A$ contains liquids $X$ and $Y$ in the ratio $X:Y = 8:7$. Vessel $B$ holds a mixture of $X$ and $Y$ in the ratio $X:Y ... | 37 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nJoe is right at the middle of a train tunnel and he realizes that a train is coming. The train travels at a speed of $50$ mile... | 144 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe addition below is incorrect. What is the largest digit that can be changed to make the addition correct?\n\n\\[\n\\begin{a... | 7 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f(x)=x^3+7x^2+9x+10$. Find the value of $p$ such that for every integer $a,b$, the statement\n\\[\nf(a) \\equiv f(b) \\ (... | 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\nIf $\\frac{m}{n} = \\frac{4}{3}$ and $\\frac{r}{t} = \\frac{9}{14}$, find the value of $\\frac{3mr - nt}{4nt - 7mr}$. The orig... | 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 watch loses $2\\frac{1}{2}$ minutes per day. It is set right at $1$ P.M. on March 15. Let $n$ be the positive correction, in... | 359 |
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"content": "Solve the following math problem step by 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 $E(n)$ denote the sum of the even digits of $n$. For example, $E(5681) = 6 + 8 = 14$. Find the value of $E(1) + E(2) + E(3... | 400 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知四面体 \\(D-A B C\\) 中, \\(A C=B C=A D=B D=1\\)，求 \\(D-A B C\\) 体积的最大值。答案的形式为 \\(\\frac{k \\sqrt{n}}{m}\\)，请给出k+m+n的值。\n\nRemem... | 32 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若正整数 $(m, n)$ 的最大公约数、最小公倍数分别是 $10!, 50!$ ，则数对 $(m, n)$ 的对数是多少？\n\nRemember to put your answer on its own line after \"Answer:\... | 32768 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在平面直角坐标系中，$O$ 是坐标原点，若两定点 $A, B$ 满足 $|\\overrightarrow{O A}|=|\\overrightarrow{O B}|=\\overrightarrow{O A} \\cdot \\overrightar... | 7 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$a, b, c$ 是等差数列，点 $P$ 到直线 $l: a x+b y+c=0$ 的垂足为 $M, P(-1,0), N(2,1)$，求 $M N$ 的最大值。答案的形式为 $k \\sqrt{2}$，请给出 $k$ 的值。\n\nRemember... | 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 $Answer is the answer to the problem.\n\n已知 2023 可以拆分为几个正整数之和，所有整数中的最大值和最小值相差不超过 1 的情况有多少种可能？\n\nRemember to put your answer on its own line after \"Answer:\".",
"... | 2023 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在正四面体 $A B C D$ 中，棱长为 $2 \\sqrt{2}$。点 $P$ 满足 $|\\overrightarrow{P A}+\\overrightarrow{P B}|=2$。求 $\\overrightarrow{A P} \\cdot... | 6 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n从棱长为 1 个单位长度的正方体的底面一顶点 $A$ 出发，每次均随机沿一条棱行走一个单位长度。进行 2 次这样的操作回到 $A$ 的概率是多少？答案的形式为\\(\\frac{k}{m}\\)，请给出k+m的值。\n\nRemember to put... | 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若 $F\\left(\\frac{1-x}{1+x}\\right)=x$ ，则满足该条件的等式可以表示为 F(kx+m) = n-F(x)。请计算k+m+n的值。\n\nRemember to put your answer on its own ... | -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若 $F\\left(\\frac{1-x}{1+x}\\right)=x$ ，求出满足该条件的$F(x)$的性质，并填入下列等式中正确的表达式。原始答案的形式为F(-a-x) = -b - F(x)，请给出a + b的值。\n\nRemember t... | 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设实数 $a 、 b 、 c$ 满足\n$$\n\\left\\{\\begin{array}{l}\na^{2}-b c-8 a+7=0 \\\\\nb^{2}+c^{2}+b c-6 a+6=0\n\\end{array}\\right.\n$$\... | 10 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设有三个函数，第一个是 $y=\\varphi(x)$ ，它的反函数是第二个函数，而第三个函数的图象与第二个函数的图象关于 $x+y=0$ 对称。求第三个函数的表达式。答案的形式为 $y = a\\varphi(bx)$，请给出 $a + b$ 的值。... | -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已知原点在粗圆 $k^{2} x^{2}+y^{2}-4 k x+2 k y+k^{2}-1=0$ 的内部，那么参数 $k$ 的取值范围是$-a < k < 0$ 或 $0 < k < b$。请计算a和b的绝对值之和。\n\nRemember to 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\n已知 n 为不超过 2015 的正整数且 $1^{n}+2^{n}+3^{n}+4^{n}$ 的个位数为 0，求满足条件的正整数 n 的个数。\n\nRemember to put your answer on its own line after \... | 1512 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n三角形 \\(A B C\\) 的三个顶点分别对应复数 \\(z_{1}, z_{2}, z_{3}\\)，已知 \\(\\frac{z_{2}-z_{1}}{z_{3}-z_{1}}=1+2 i\\)，求三角形 \\(A B C\\) 的面积与其最长... | 6 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n实系数方程 $x^{4}+a x^{3}+b x^{2}+c x+d=0$ 的根都不是实数，其中两个根的和为 $2+i$，另两根的积为 $5+6i$，求 $b$ 的值。\n\nRemember to put your answer on its own... | 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\n设 $A$ 是以 $B C$ 为直径的圆上的一点, $D, E$ 是线段 $B C$ 上的点, $F$ 是 $B C$ 延长线上的点, 已知 $B F=4, B D=2, B E=5, \\angle B A D=\\angle A C D, \\an... | 11 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $A B, C D$ 是圆 $O$ 的两条垂直直径, 弦 $D F$ 交 $A B$ 于点 $E, D E=24, E F=18$, 求 $O E$的值。答案的形式是$k \\sqrt{m}$，请给出k + m的值。\n\nRemember to ... | 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 $Answer is the answer to the problem.\n\n已知正方形 \\(A B C D\\) 的边长为 \\(1\\)，\\(P_{1}, P_{2}, P_{3}, P_{4}\\) 是正方形内部的 4 个点，使得 \\(\\triangle A B P_{1}, \\triangle B C P_{2... | 6 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n正整数 $9+95+995+\\cdots+\\underbrace{99 \\cdots 9}_{2016} 5$ 的十进制表示中数值 1 的个数为多少？\n\nRemember to put your answer on its own line ... | 2014 |
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{
"content": "Solve the following math problem step by 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将等差数列 $1,5,9,13, \\ldots, 2017$ 排成一个大数 $15913 \\ldots 2017$, 则该数被 9 除的余数为多少？\n\nRemember to put your answer on its own line af... | 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已知存在正整数 a, b, c 满足 $a+b+c=407,10^{n} \\mid a b c$ ，求 n 的最大值。\n\nRemember to put your answer on its own line after \"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 $\\triangle A B C$ 中, $A=90^{\\circ}, A B=1, A C=\\sqrt{3}$, 点 P 满足 $\\frac{\\overrightarrow{P A}}{|\\overrightarrow{P A}|}+... | 120 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 \\(\\alpha, \\beta\\) 为锐角, 且 \\(\\cos (\\alpha+\\beta)=\\frac{\\sin \\alpha}{\\sin \\beta}\\), 求 \\(\\tan \\alpha\\) 的最大值。答案... | 6 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $A, B$ 分别是 $x$ 轴, $y$ 轴上的动点, 若以 $A B$ 为直径的圆 $C$ 与直线 $2x+y-4=0$ 相切, 则圆 $C$ 面积的最小值为 $\\frac{k\\pi}{m\\pi}$，请给出 $k + m$ 的值。\n\n... | 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 $Answer is the answer to the problem.\n\n已知数列 \\(\\left\\{a_{n}\\right\\}\\) 的前 \\(n\\) 项和 \\(S_{n}=(-1)^{n} \\cdot a_{n}+\\frac{1}{2^{n}}+n-3\\)，且实数 \\(t\\) 满足 \\(\\l... | 16 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n\\(\\lim _{n \\rightarrow+\\infty} \\sum_{k=1}^{n} \\arctan \\frac{2}{k^{2}}\\) 等于 \\(x \\pi\\)，其中答案以\\(\\frac{k}{m}\\)的最简分数形式... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n将函数 \\(y=\\sqrt{4+6 x-x^{2}}-2(x \\in[0,6])\\) 的图象逆时针方向旋转 \\(\\theta(0 \\leqslant \\theta \\leqslant \\alpha)\\), 得到曲线 \\(C\\)... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$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 $\\... | 5 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 \\(y^{2}=4x\\)，过 \\(A(-2,3)\\) 做抛物线两条切线，交 y 轴于 \\(B, C\\) 两点，求 \\(\\triangle ABC\\) 的外接圆方程。答案的形式为\\((x + \\frac{k}{2})^2 + ... | 15 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n恰有一个实数 $x$ 使得 $x^{3}-a x-1=0$ 成立，求实数 $a$ 的取值范围。答案的形式为 \\((-∞, \\frac{a \\sqrt[3]{b}}{c})\\)，请给出 \\(a+b+c\\) 的值。\n\nRemember to... | 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 $Answer is the answer to the problem.\n\n将函数 \\(y=\\sqrt{4+6 x-x^{2}}-2(x \\in[0, 6])\\) 的图象绕坐标原点逆时针方向旋转角 \\(\\theta(0 \\leqslant \\theta \\leqslant \\alpha)\\)，得到曲线 \... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知函数 \\(f(x)=\\sin x \\cos x+\\sin x+\\frac{2}{5} \\cos x\\left(0 \\leqslant x \\leqslant \\frac{\\pi}{2}\\right)\\) 的最大值与最小值分... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在四面体 \\(A B C D\\) 中，若 \\(A C = B C = A D = B D = 1\\)，则求四面体 \\(A B C D\\) 体积的最大值。原始答案为\\(\\frac{k \\sqrt{m}}{n}\\)的形式，请给出k+m+... | 32 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 2023 可以拆分为几个正整数之和，所有整数中的最大值和最小值相差不超过 1。有多少种可能？\n\nRemember to put your answer on its own line after \"Answer:\".",
"rol... | 2023 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n圆周上 \\(A_{1}, A_{2}, \\cdots, A_{7}\\) 七个点两两相连。求任选两条线段，这两条线段无公共点的概率。答案的形式为\\(\\frac{k}{m}\\)，请计算k+m的值。\n\nRemember to put your... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设复数 $\\omega=\\left(\\frac{a+\\mathrm{i}}{1+\\mathrm{i}}\\right)^{2}$ ，其中 $a$ 为实数，若 $\\omega$ 的实部为 2 ，求 $\\omega$ 的虚部。答案的形式为 $... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在四棱锥 $P-A B C D$ 中, $B_{1}, D_{1}$ 分别为侧棱 $P B, P D$ 的中点。求四面体 $A B_{1} C D_{1}$ 的体积与四棱锥 $P-A B C D$ 的体积之比。假设比值的形式为k:m，请计算k+m的值。... | 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 $Answer is the answer to the problem.\n\n在 \\(\\triangle ABC\\) 中，三边长 \\(a, b, c\\) 满足 \\(a+c=3b\\)，求 \\(\\tan \\frac{A}{2} \\tan \\frac{C}{2}\\) 的值。答案的形式为\\(\\frac{k}... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设双曲线 $C_{1}: \\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{4}=k(a>2, k>0)$, 椭圆 $C_{2}: \\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{4}=1$, 若 $C_{2}... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若 $A+B=\\frac{2 \\pi}{3}$，求 $\\cos ^{2} A+\\cos ^{2} B$ 的最小值和最大值。答案分别为\\frac{k}{m}和\\frac{n}{p}的形式，请计算k+m+n+p的值。\n\nRemember t... | 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 $0<x<\\frac{\\pi}{2}$, 且 $\\frac{\\sin ^{4} x}{9}+\\frac{\\cos ^{4} x}{4}=\\frac{1}{13}$, 则 $\\tan x$ 的值可以表示为\\frac{k}{m}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已知 $0<k<4$, 直线 $l_{1}: k x-2 y-2 k+$ $8=0$ 和直线 $l_{2}: 2 x+k^{2} y-4 k^{2}-4=0$ 与两坐标轴围成一个四边形。求使这个四边形面积最小的 $k$ 值。原始答案是以\\(\\fra... | 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 $Answer is the answer to the problem.\n\n在 5 件产品中有 4 件正品、1 件次品。从中任取两件, 记其中含正品的个数为随机变量 \\(\\xi\\)。求 \\(\\xi\\) 的数学期望 \\(E(\\xi)\\)。答案应为\\(\\frac{k}{m}\\)的形式，请给出\\(k+m\\... | 13 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设正三角形 \\(\\triangle_{1}\\) 的面积为 \\(S_{1}\\)，作 \\(\\triangle_{1}\\) 的内切圆，再作内切圆的内接正三角形 \\(\\triangle_{2}\\)，面积为 \\(S_{2}\\)，如此下去... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知正三棱柱 \\(P-ABC\\) 的底面 \\(ABC\\) 为正三角形，该正三棱柱的外接球的球心 \\(O\\) 满足 \\(\\overrightarrow{OA}+\\overrightarrow{OB}+\\overrightarrow{O... | 6 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设方程\n$$\n1+x-\\frac{x^{2}}{2}+\\frac{x^{3}}{3}-\\frac{x^{4}}{4}+\\cdots-\\frac{x^{2018}}{2018}=0\n$$\n的所有实数根均在区间 $[a, b](a 、 b... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设实数 $a 、 b 、 c$ 满足\n$$\n\\left\\{\\begin{array}{l}\na^{2}-b c-8 a+7=0 \\\\\nb^{2}+c^{2}+b c-6 a+6=0\n\\end{array}\\right.\n$$\... | 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设实数 $a 、 b 、 c$ 满足 $\\left\\{\\begin{array}{l}a^{2}-b c-8 a+7=0, \\\\ b^{2}+c^{2}+b c-6 a+6=0,\\end{array}\\right.$ 求 $a$ 的取值范... | 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已知 $M=\\left\\{(x, y) \\mid y \\geqslant x^{2}\\right\\}, N=\\left\\{(x, y) \\mid x^{2}+(y-a)^{2} \\leqslant 1\\right\\}$, 找出使... | 125 |
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