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 $Answer is the answer to the problem.\n\nA four-term sequence is formed by adding each term of a four-term arithmetic sequence of positive integers to the correspondin... | 206 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIsosceles trapezoid \\(ABCD\\) has parallel sides \\(\\overline{AD}\\) and \\(\\overline{BC},\\) with \\(BC < AD\\) and \\(AB ... | 4 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $P(x)$ be a polynomial with rational coefficients such that when $P(x)$ is divided by the polynomial $x^2 + x + 1$, the re... | 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 $S$ be the set of circles in the coordinate plane that are tangent to each of the three circles with equations $x^{2}+y^{2... | 136 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSteph scored $15$ baskets out of $20$ attempts in the first half of a game, and $10$ baskets out of $10$ attempts in the secon... | 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 $Answer is the answer to the problem.\n\nAnt Amelia starts on the number line at \\(0\\) and crawls in the following manner. For \\(n=1,2,3,\\) Amelia chooses a time d... | 5 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA bus takes $2$ minutes to drive from one stop to the next, and waits $1$ minute at each stop to let passengers board. Zia tak... | 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\nThe figure below depicts a regular $7$-gon inscribed in a unit circle.\n[asy] import geometry; unitsize(3cm); draw(circle((0,0... | 147 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $R$, $S$, and $T$ be squares that have vertices at lattice points (i.e., points whose coordinates are both integers) in th... | 337 |
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"content": "Solve the following math problem step by 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 cricket randomly hops between $4$ leaves, on each turn hopping to one of the other $3$ leaves with equal probability. After ... | 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 $Answer is the answer to the problem.\n\nMrs. Jones is pouring orange juice into four identical glasses for her four sons. She fills the first three glasses completely... | 7 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nJanet rolls a standard $6$-sided die $4$ times and keeps a running total of the numbers she rolls. Find the probability that a... | 265 |
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"content": "Solve the following math problem step by 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 digital display shows the current date as an $8$-digit integer consisting of a $4$-digit year, followed by a $2$-digit month... | 9 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA square of area \\(2\\) is inscribed in a square of area \\(3\\), creating four congruent triangles, as shown below. Find the... | 5 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the maximum area of an isosceles trapezoid that has legs of length \\(1\\) and one base twice as long as the other? Ex... | 5 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many three-digit positive integers $N$ satisfy the following properties? \nThe number $N$ is divisible by $7$. \nThe num... | 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\nFor complex number $u = a+bi$ and $v = c+di$ (where $i=\\sqrt{-1}$), define the binary operation $u \\otimes v = ac + bdi$. \... | 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\nHow many complex numbers satisfy the equation $z^5=\\overline{z}$, where $\\overline{z}$ is the conjugate of the complex numbe... | 7 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor how many ordered pairs $(a,b)$ of integers does the polynomial $x^3+ax^2+bx+6$ have $3$ distinct integer roots?\n\nRemembe... | 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 $Answer is the answer to the problem.\n\nUsain is walking for exercise by zigzagging across a \\(100\\)-meter by \\(30\\)-meter rectangular field, beginning at point \... | 11 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nConsider the set of complex numbers $z$ satisfying $|1+z+z^{2}|=4$. The maximum value of the imaginary part of $z$ can be writ... | 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\nIn the state of Coinland, coins have values $6,10,$ and $15$ cents. Suppose $x$ is the value in cents of the most expensive it... | 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 $Answer is the answer to the problem.\n\nCircle \\(C_1\\) and \\(C_2\\) each have radius \\(1\\), and the distance between their centers is \\(\\frac{1}{2}\\). Circle ... | 31 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the product of all solutions to the equation \n$$\\log_{7x}2023\\cdot \\log_{289x}2023=\\log_{2023x}2023?$$\n\nRememb... | 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\nFour congruent semicircles are drawn on the surface of a sphere with radius $2$, as shown, creating a close curve that divides... | 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\nRows $1, 2, 3, 4,$ and $5$ of a triangular array of integers are shown below. \n$$\\begin{array}{cccccccc}&&&&1&&&&\\\\&&&1&&... | 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\nAlina writes the numbers $1, 2, \\dots , 9$ on separate cards, one number per card. She wishes to divide the cards into $3$ gr... | 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 $Answer is the answer to the problem.\n\nCircle \\(C_1\\) and \\(C_2\\) each have radius \\(1\\), and the distance between their centers is \\(\\frac{1}{2}\\). Circle ... | 31 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many distinct values of $x$ satisfy \n$\\lfloor{x}\\rfloor^2-3x+2=0$, where $\\lfloor{x}\\rfloor$ denotes the largest inte... | 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\nAn arithmetic sequence of positive integers has $n \\ge 3$ terms, initial term $a$, and common difference $d > 1$. Carl wrote ... | 20 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA regular pentagon with area \\(\\sqrt{5}+1\\) is printed on paper and cut out. The five vertices of the pentagon are folded i... | 6 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFifteen integers $a_1, a_2, a_3, \\dots, a_{15}$ are arranged in order on a number line. The integers are equally spaced and h... | 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\nAaliyah rolls two standard $6$-sided dice. She notices that the product of the two numbers rolled is a multiple of $6$. Find t... | 6 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nOn Monday, Taye has $2. Every day, he either gains $3 or doubles the amount of money he had on the previous day. How many diff... | 6 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA group of frogs (called an army) is living in a tree. A frog turns green when in the shade and turns yellow when in the sun. ... | 24 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nRodrigo has a very large sheet of graph paper. First he draws a line segment connecting point $(0,4)$ to point $(2,0)$ and col... | 7000 |
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"content": "Solve the following math problem step by 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已知两个实数集合 $A=\\left\\{a_{1}, a_{2}, a_{3}, \\cdots, a_{100}\\right\\}$ ，与 $B=\\left\\{b_{1}, b_{2}, \\cdots, b_{50}\\right\\}$ ... | 148 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若 \\(\\left(x^{2}-2 x+m\\right)\\left(x^{2}-2 x+n\\right)=0\\) 的 \\(4\\) 个根组成首项为 \\(\\frac{1}{4}\\) 的等差数列，则 \\(|m-n|\\) 的值为多少？... | 3 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$x 、 y$ 均为负数, 且 $x+y=-1$, 求 $x y+\\frac{1}{x y}$ 的最小值。答案的形式为$\\frac{k}{m}$，请给出k和m的和，其中k和m是最简分数形式下的整数。\n\nRemember to put your ... | 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\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 a... | 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空间中点集定义如下: $A_{n}=\\left\\{\\left.(x, y, z) \\in \\mathbf{R}^{3}|3| x\\right|^{n}+|8 y|^{n}+|z|^{n} \\leq 1\\right\\}, A=\\big... | 1 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 $x+y+z=2016, \\frac{1}{x}+\\frac{1}{y}+\\frac{1}{z}=\\frac{1}{2016}$, 求 $(x-2016)(y-2016)(z-2016)$ 的值。\n\nRemember to put y... | 0 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n正方体的 $8$ 个顶点中任取 $3$ 个构成三角形，求三角形是等腰三角形的概率。答案的形式为\\frac{k}{m}，请给出k+m的值。\n\nRemember to put your answer on its own line after \"A... | 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用 $a_{n}$ 表示距离 $\\sqrt{n}$ 最近的正整数, 已知 $\\frac{1}{a_{1}}+\\frac{1}{a_{2}}+\\cdots+\\frac{1}{a_{n}}=2016$, 求 $n$ 的值。\n\nRemember... | 1017072 |
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"content": "Solve the following math problem step by 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 $5+6 i$,则 $b$ 等于多少？\n\nRemember to put your answer on it... | 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 $Answer is the answer to the problem.\n\n在正方体的八个顶点中任取 3 个构成三角形，求所构成的三角形是等腰三角形的概率。答案的形式为\\(\\frac{k}{m}\\)，请给出k+m的值。\n\nRemember to put your answer on its own line afte... | 11 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若方程 $x^{2}-3 x-1=0$ 的根也是方程 $x^{4}+a x^{2}+b x+c=0$ 的根, 则 $a+b-2 c$ 的值为多少？\n\nRemember to put your answer on its own line after... | -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 $Answer is the answer to the problem.\n\n已知 \\(k \\neq 1\\)，则等比数列 \\(a+\\log _{2} k, a+\\log _{4} k, a+\\log _{8} k\\) 的公比为多少？答案应为 \\(\\frac{m}{n}\\) 的形式，请给出 \\(m+n\\)... | 4 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$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。原始答案的形式为 $k\\sqrt{m}$，请给出 $k + m$ 的值。\n\nRemembe... | 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函数 $f(x)=x(x+1)(x+2)(x+3)$ 的最小值为多少？\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}... | -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整数 $p, q$ 满足 $p+q=218, x^{2}+p x+q=0$ 有整数根，求满足这样条件的整数对 $(p, q)$ 的个数。\n\nRemember to put your answer on its own line after \"An... | 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已知正整数 $n$ 满足 $n \\neq 2017$ ，且 $n^{n}$ 与 $2017^{2017}$ 有相同的个位数字，则 $|2017-n|$ 的最小值为多少？\n\nRemember to put your answer on its ow... | 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已知存在正整数 $a, b, c$ 满足 $a+b+c=407,10^{n} \\mid a b c$ ，则 $n$ 的最大值是多少？\n\nRemember to put your answer on its own line after \"Ans... | 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若 $a, b, c$ 和 $\\left(a-\\frac{1}{b}\\right)\\left(b-\\frac{1}{c}\\right)\\left(c-\\frac{1}{a}\\right)$ 均为正整数，求 $2 a+3 b+5 c$ ... | 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一群学生参加学科夏令营，每名同学至少参加数学、物理、化学中的一门学科考试. 已知有 $100$ 名学生参加了数学考试，$50$名学生参加了物理考试， $48$ 名学生参加了化学考试.若学生总数是参加至少两门考试的学生的 $2$ 倍，也是参加三门考试学生... | 108 |
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{
"content": "Solve the following math problem step by 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若实数 $x, y$ 满足 $\\frac{x^{2}}{4}+y^{2}=1$ ，则 $|3 x+4 y-12|$ 的取值范围是 $[c, d]$。请分别求出 $c$ 和 $d$ 的值，并给出 $c + d$ 的值。\n\nRemember to p... | 24 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知三个复数 $z_{1}=(5-a)+(6-4 b) \\mathrm{i}, z_{2}=(2+2 a)+(3+b) \\mathrm{i}, z_{3}=(3-a)+(1+3 b) \\mathrm{i}(a, b \\in$ $\\mathbf... | 40 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若数列 $\\left\\{a_{n}\\right\\}$ 满足 $a_{1}=12, a_{n+1}=\\frac{1}{4}\\left(3+a_{n}+\\right.$ $\\left.3 \\sqrt{1+2 a_{n}}\\right)\... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若复数 \\(z\\) 满足 \\(\\frac{z}{2}\\) 与 \\(\\frac{2}{z}\\) 的实部、虚部均在区间 \\([-1,1]\\) 上取值，则 \\(z\\) 在复平面上形成轨迹的面积是多少？答案的形式为 \\(a-b\\pi... | 14 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n用 $[x]$ 表示不超过 $x$ 的最大整数，比如 [1.2] $=1$, $[-1.2]=-2$. 若 $\\alpha=\\frac{1+\\sqrt{5}}{2}$, 求 $\\left[\\alpha^{12}\\right]$ 的值。\n\... | 321 |
[
{
"content": "Solve the following math problem step by 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若二次函数 $f(x)$ 满足 $\\forall x \\in \\mathbf{R}, 2 x \\leqslant f(x) \\leqslant \\frac{x^{2}+4}{2}$ 及 $f(-2)=0$ ，求 $f(10)$ 的值。\n\... | 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\n满足正整数 $n \\leqslant 2022$ 且 $2n+1$ 与 $3n+1$ 均为完全平方数的 $n$ 的个数是 _______.\n\nRemember to put your answer on its own line after \"... | 1 |
[
{
"content": "Solve the following math problem step by 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 $\\triangle A B C$ 的三边长分别是 $2,3,4$, 其外心为 $O$, 求 $\\overrightarrow{O A} \\cdot \\overrightarrow{A B}+\\overrightarrow{O B} \\... | -27 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在长方体 $A B C D-A_{1} B_{1} C_{1} D_{1}$ 中，$A B=2, A D=A A_{1}=1$，求点 $A$ 到平面 $A_{1} B D$ 的距离。答案的形式为\\frac{k}{m}，请给出k+m的值。\n\nRem... | 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若从 $0,1,2, \\cdots, 9$ 中选取 $5$ 个两两互异的数字依次排成一个五位数（包括 $0$ 在首位的五位数，其大小就是把 $0$ 去掉后的四位数），求它能被 $396$ 整除的概率。答案的形式为\\frac{a}{b}，请给出a+b... | 316 |
[
{
"content": "Solve the following math problem step by 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若空间向量 $\\boldsymbol{a} ,\\boldsymbol{b} ,\\boldsymbol{c}$ 满足 $|\\boldsymbol{a}| \\leqslant 1,|\\boldsymbol{b}| \\leqslant 1, \... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若 $x^{2}+y^{2} \\leqslant 1(x y \\in \\mathbf{R})$, 求 $x^{2}+x y-y^{2}$ 的最大值与最小值所组成的区间。答案形如 \\left[-\\frac{\\sqrt{k}}{m}, \\fr... | 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若 $A, B, C \\subseteq\\{1,2,3, \\cdots, 2020\\}, A \\subseteq C, B \\subseteq C$ ，求有序集合组 $(A, B, C)$ 的组数。原始答案的形式为 $k^m$，请给出 $k... | 2025 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEvaluate \\(\\lim \\limits_{n \\rightarrow \\infty} \\sum\\limits_{k=1}^{n} \\arctan \\frac{2}{k^{2}}\\). The answer is in the... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n函数 $f(x)=\\frac{2 e^{x}}{e^{x}+e^{-x}}+\\sin x(-2 \\leqslant x \\leqslant 2)$ 的最大值与最小值之和是多少？\n\nRemember to put your answer on... | 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已知 $P(X=k)=\\frac{1}{2^{k}}(k=1,2,3, \\cdots)$，若 $Y$ 为 $X$ 除以 $3$ 所得的余数，则求随机变量 $Y$ 的期望。原始答案的形式为 \\frac{k}{m}，请给出 $k + m$ 的值。\n... | 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 $Answer is the answer to the problem.\n\n若某个正四棱锥的相邻两个侧面所成二面角的大小为 \\(\\alpha\\)，侧棱与底面所成线面角的大小为 \\(\\beta\\)，求满足条件的表达式的值。答案应为 \\(k \\sec \\alpha + m \\tan^2 \\beta = n\\... | 2 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若函数 $f(t)=t^{2}+2 t$ ，则点集 $\\left\\{(x, y) \\left\\lvert\\,\\left\\{\\begin{array}{l}f(x)+f(y) \\leqslant 2 \\\\ f(x) \\geqsla... | 2 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若 $a^{2}(b+c)=b^{2}(a+c)=1(a \\neq b)$ ，则 $(a+b) c^{2}-a b c$ 的值为多少？\n\nRemember to put your answer on its own line after \"An... | 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顶点为 $P$ 的圆锥的轴截面是等腰直角三角形， $A$ 是底面圆周上的点， $B$ 是底面圆内的点， $O$ 为底面圆圆心， $A B \\perp O B$ ，垂足为 $B, O H \\perp P B$ ，垂足为 $H$ ，且 $P A=4, ... | 11 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求不等式 $\\sqrt{\\log _{2} x-1}+\\frac{1}{2} \\log _{\\frac{1}{2}} x^{3}+2>0$ 的解集。解集的形式为 [a, b)，请给出 a + b 的值。\n\nRemember to put ... | 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记集合 $T=\\{0,1,2,3,4,5,6\\}, M=\\left\\{\\left.\\frac{a_{1}}{7}+\\frac{a_{2}}{7^{2}}+\\frac{a_{3}}{7^{3}}+\\frac{a_{4}}{7^{4}} ... | 6 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n使关于 $x$ 的不等式 $\\sqrt{x-3}+\\sqrt{6-x} \\geqslant k$ 有解的实数 $k$ 的最大值是多少？请以$\\sqrt{m}$的形式给出答案，并求出m的值。\n\nRemember to put your ans... | 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设实数 $a$ 使得不等式 $|2 x-a|+|3 x-2 a| \\geqslant a^{2}$ 对任意实数 $x$ 恒成立，求满足条件的 $a$ 所组成的集合的区间端点。如果区间为 $[m, n] = \\left[\\frac{k}{l}, \... | 6 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n以 $\\sqrt{2}$ 和 $1-\\sqrt[3]{2}$ 为两个根的有理系数多项式的次数最低是____。\n\nRemember to put your answer on its own line after \"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 $x, y \\in \\mathbf{Z}$ ，若 $\\left(x^{2}+x+1\\right)^{2}+\\left(y^{2}+y+1\\right)^{2}$ 为完全平方数，则数对 $(x, y)$ 的对数是多少？\n\nRemem... | 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$1 \\times 1!+2 \\times 2!+\\cdots+672 \\times 672!$ 被 $2019$ 除的余数是多少？\n\nRemember to put your answer on its own line after \"... | 2018 |
[
{
"content": "Solve the following math problem step by 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$ 是任意自然数，$1^{3}+2^{3}+3^{3}+\\cdots+n^{3}$ 的个位数不可能是______。\n\nRemember to put your answer on its own line after \"Answer:\".... | 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 $Answer is the answer to the problem.\n\n已知 \\(x, y, z>0\\)，则 \\(f(x, y, z)=\\frac{x y z}{(4 x+1)(9 x+y)(4 y+z)(z+9)}\\) 的最大值为多少？答案的形式为 \\(\\frac{k}{m^2}\\)，请给出 \\(k +... | 25 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$f(x)=\\frac{x}{\\sqrt{1+x^{2}}}+\\frac{1-x^{2}}{1+x^{2}}$ 的取值范围是 $(-2, y]$，其中$y$的原始答案是以\\frac{k}{m}的形式给出，请计算k+m的值。\n\nRemembe... | 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设非负实数 $x, y$ 满足 $2 x+y=1$, 求 $x+\\sqrt{x^{2}+y^{2}}$ 的最小值。答案的形式为\\frac{k}{m}，请给出k+m的值。\n\nRemember to put your answer on its o... | 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若 $a_{1}, a_{2}, a_{3}, a_{4} \\in\\{1,2,3,4\\}, N\\left(a_{1}, a_{2}, a_{3}, a_{4}\\right)$ 为 $a_{1}, a_{2}, a_{3}, a_{4}$ 中不... | 239 |
[
{
"content": "Solve the following math problem step by 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, 若 $\\triangle A B C$ 是边长为 3 的正三角形, 且 $P A=3, P B=4$, $P C=5$ ，求该四面体的体积。答案的形式为\\(\\sqrt{k}\\)，请给出k的值。\n\nReme... | 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 $Answer is the answer to the problem.\n\n在 \\(\\triangle ABC\\) 中，求 \\(\\sin A + \\sin B \\sin C\\) 的最大值。答案的形式为 \\(\\frac{k + \\sqrt{m}}{n}\\)，请计算k + m + n的值。\n\nRemem... | 8 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n不定方程 $x+2 y+3 z=100$ 的非负整数解的组数是多少？\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
... | 884 |
[
{
"content": "Solve the following math problem step by 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 $\\omega=\\cos \\frac{2 \\pi}{5}+\\operatorname{isin} \\frac{2 \\pi}{5}, f(x)=x^{2}+x+2$, 求 $f(\\omega) f\\left(\\omega^{2}... | 11 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 \\(z, w \\in \\mathbf{C},|z+w|=1,\\left|z^{2}+w^{2}\\right|=4\\)，求 \\(|z w|\\) 的最大值。原始答案的形式为 \\(\\frac{k}{m}\\)，请给出\\(k+m\\... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设复数 \\(z, w\\) 满足: \\(|w+z|=1,\\left|w^{2}+z^{2}\\right|=4\\)，求 \\(|w z|\\) 的最小值。答案应以 \\(\\frac{k}{m}\\) 的形式给出，其中 \\(\\frac{k}... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $A, B, C$ 是三角形的三个内角，求 $\\sin A+\\sin B \\sin C$ 的最大值。答案的形式为 \\(\\frac{k+\\sqrt{n}}{m}\\)，请给出k + n + m的值。\n\nRemember to put ... | 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设 $\\vec{e}_{1}, \\vec{e}_{2}$ 是两个单位向量, $x, y$ 是实数. 若 $\\left\\langle\\vec{e}_{1}, \\vec{e}_{2}\\right\\rangle=\\frac{\\pi}{3}... | 8 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知集合 $M=\\{-1,0,1\\}, N=\\{2,3,4,5,6\\}$ 。设映射 $f: M \\rightarrow N$ 满足：对任意的 $x \\in M, x+f(x)+x f(x)$ 是奇数。求这样的映射 $f$ 的个数。\n\nR... | 50 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在 Rt \\(\\triangle A B C\\) 中, \\(\\angle A B C=\\frac{\\pi}{2}, A B=\\sqrt{3}, B C=1, \\frac{\\overrightarrow{P A}}{|\\overri... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设复数 \\(z\\) 满足 \\(|3z-7i|=3\\)，求 \\(\\left|\\frac{z^{2}-2z+2}{z-1+i}\\right|\\) 的最大值。答案的形式为 \\(\\frac{k}{m}\\)，请给出k+m的值。\n\nRe... | 11 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n将号码分别为 \\(1, 2, \\cdots, 9\\) 的九个小球放入一个袋中, 这些小球仅号码不同, 其余完全相同。甲从袋中摸出一个小球，其号码为 \\(a\\)，放回后，乙从此袋中再摸出一个球，其号码为 \\(b\\)。求使得不等式 \\(a-... | 142 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n甲、乙两人进行乒乓球比赛，约定每局胜者得 1 分，负者得 0 分，比赛进行到有一人比对方多 2 分或打满 6 局时停止。设甲在每局中获胜的概率为 \\(\\frac{2}{3}\\)，乙在每局中获胜的概率为 \\(\\frac{1}{3}\\)，且各局... | 347 |
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