sungyub/dapo-math-17k-verl huggingface dataset:
user: Determine all real constants $t$ such that whenever $a, b, c$ are the lengths of the sides of a triangle, then so are $a^{2}+b c t, b^{2}+c a t, c^{2}+a b t$.If the length of the interval in which \(t\) lies is \(\frac{a}{b}\), find \(a + b\).

Answer: 7
user: A domino is a $2 \times 1$ or $1 \times 2$ tile. Determine in how many ways exactly $n^{2}$ dominoes can be placed without overlapping on a $2 n \times 2 n$ chessboard so that every $2 \times 2$ square contains at least two uncovered unit squares which lie in the same row or column. Please provide the value when n = 1.

Answer: 4
user: Let $m$ be a positive integer. Consider a $4 m \times 4 m$ array of square unit cells. Two different cells are related to each other if they are in either the same row or in the same column. No cell is related to itself. Some cells are coloured blue, such that every cell is related to at least two blue cells. Determine the minimum number of blue cells. Please provide the answer when $m=1$.

Answer: 6
user: The $n$ contestants of an EGMO are named $C_{1}, \ldots, C_{n}$. After the competition they queue in front of the restaurant according to the following rules. 
- The Jury chooses the initial order of the contestants in the queue. 
- Every minute, the Jury chooses an integer $i$ with $1 \leq i \leq n$. 
- If contestant $C_{i}$ has at least $i$ other contestants in front of her, she pays one euro to the Jury and moves forward in the queue by exactly $i$ positions. 
- If contestant $C_{i}$ has fewer than $i$ other contestants in front of her, the restaurant opens and the process ends. 
Determine for every $n$ the maximum number of euros that the Jury can collect by cunningly choosing the initial order and the sequence of moves.
Please provide the value when n = 9.

Answer: 502
user: Determine whether for every real number $t$ such that $0<t<\frac{1}{2}$ there exists an infinite set $S$ of positive integers such that 
$$ 
|x-m y|>t y 
$$ 
for every pair of different elements $x$ and $y$ of $S$ and every positive integer $m$ (i.e. $m>0$ ).The original answer is in the form of $t<\left\{\frac{s_{n-1}}{s_{n}}\right\}<\frac{a}{b}$. Find the value of $a+b$.

Answer: 3
user: Let $n$ be a positive integer. Dominoes are placed on a $2 n \times 2 n$ board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each $n$, determine the largest number of dominoes that can be placed in this way. (A domino is a tile of size $2 \times 1$ or $1 \times 2$. Dominoes are placed on the board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap. Two cells are said to be adjacent if they are different and share a common side.) Please provide the largest number of dominoes when n = 6.

Answer: 21
user: Let $n$ be a positive integer and fix $2 n$ distinct points on a circumference. Split these points into $n$ pairs and join the points in each pair by an arrow (i.e., an oriented line segment). The resulting configuration is good if no two arrows cross, and there are no arrows $\overrightarrow{A B}$ and $\overrightarrow{C D}$ such that $A B C D$ is a convex quadrangle oriented clockwise. Determine when n=6， the number of good configurations.

Answer: 924
user: 设 $O$ 为 $\triangle A B C$ 的外心, 且满足 $\overrightarrow{O A}+\overrightarrow{O B}=\overrightarrow{O C}.$ 则 $\frac{3}{\pi}\angle A C B=

Answer: 2
user: 设 $f(x), g(x)$ 分别是定义在 $\mathbf{R}$ 上的奇函数和偶函数, 且 $f(x)+g(x)=2^{x}$. 若对 $x \in[1,2]$, 不
等式 $a f(x)+g(2 x) \geqslant 0$ 恒成立, 则实数 $a$ 的最小值的$6$是多少 $\qquad$.

Answer: -17
user: 设 $x, y$ 均为非零实数, 且满足 $\frac{x \sin \frac{\pi}{5}+y \cos \frac{\pi}{5}}{x \cos \frac{\pi}{5}-y \sin \frac{\pi}{5}}=\tan \frac{9 \pi}{20}$. 求 $\frac{y}{x}$ 的值;

Answer: 1
user: 袋中装有 2 个红球、 3 个白球、 4 个黄球，从中任取 4 个球，则其中三种颜色的球都有的概率的7倍是 $\qquad$.

Answer: 4
user: 已知函数 $f(x)=-x^{3}+a x^{2}+b x+c(a, b, c \in \mathbf{R})$ 在 $(-\infty, 0)$ 上单调递减, 在 $(0,1)$ 上单调递增,且 $f(x)$ 在 $\mathbf{R}$ 上有三个零点, 1 是其中的一个零点.求 $f(2)$ 最小值的 $2 $倍是多少

Answer: -5
user: 已知 $a$ 为常数, 函数
$f(x)=\ln \frac{1-x}{1+x}-a x$.
 若 $a=-\frac{8}{3}$, 求 $f(x)$ 的取极大值时的x值的2倍为多少.

Answer: 1
user: 已知对一切的 $x \in \mathbf{R}$ 恒有 $3 \sin ^{2} x-\cos ^{2} x+4 a \cos x+a^{2} \leqslant 31$. 求实数 $a$ 的取值范围. 请给出区间的端点和.

Answer: 0
user: 设等比数列 $\left\{a_{n}\right\}$ 的前 $n$ 项和为 $S_{n}, S_{n}=2^{n}+r$ ( $r$ 为常数),记 $b_{n}=2\left(1+\log _{2} a_{n}\right)\left(n \in \mathbf{N}^{*}\right)$.若对于任意的正整数 $n$, 都有 $\frac{1+b_{1}}{b_{1}} \cdot \frac{1+b_{2}}{b_{2}} \cdots \frac{1+b_{n}}{b_{n}} \geqslant k \sqrt{n+1}$ 成立, 求实数 $k$ 的最大值的$2\sqrt{2}$倍是多少.

Answer: 3
user: 在三棱锥 $P-A B C$ 中, 三条棱 $P A$ 、 $P B 、 P C$ 两两垂直，且 $P A=1, P B=P C=2$. 若 $Q$ 为三棱锥 $P-A B C$ 外接球的球面上任一点, 则 $Q$ 到面 $A B C$ 距离的最大值减去$\frac{\sqrt{6}}6$的结果的2倍为 $\qquad$.

Answer: 3
user: 在 $\triangle A B C$ 中, $\cos B=\frac{1}{4}$. 则 $\frac{1}{\tan A}+\frac{1}{\tan C}$的最小值的$\frac{\sqrt{15}}3$倍为 $\qquad$.

Answer: 2
user: Let $a_{1}, a_{2} \ldots, a_{51}$ be non-zero elements of a field. We simultaneously replace each element with the sum of the 50 remaining ones. In this way we get a sequence $b_{1} \ldots, b_{51}$. If this new sequence is a permutation of the original one, what can be the characteristic of the field? Please provide the sum of the possible characteristics.

Answer: 9
user: Let $A$ be a real $4 \times 2$ matrix and $B$ be a real $2 \times 4$ matrix such that $$ AB=\left(\begin{array}{cccc} 1 & 0 & -1 & 0 \\ 0 & 1 & 0 & -1 \\ -1 & 0 & 1 & 0 \\ 0 & -1 & 0 & 1 \end{array}\right) $$ Find $B A.$ Provide the sum of all elements in the resulting matrix.

Answer: 4
user: How many nonzero coefficients can a polynomial $P(z)$ have if its coefficients are integers and $|P(z)| \leq 2$ for any complex number $z$ of unit length? Please provide the sum of all possible numbers of nonzero coefficients.

Answer: 3
user: For each positive integer $k$, find the smallest number $n_{k}$ for which there exist real $n_{k} \times n_{k}$ matrices $A_{1}, A_{2}, \ldots, A_{k}$ such that all of the following conditions hold: 
$A_{1}^{2}=A_{2}^{2}=\ldots=A_{k}^{2}=0$, 
$A_{i} A_{j}=A_{j} A_{i}$ for all $1 \leq i, j \leq k$, and 
$A_{1} A_{2} \ldots A_{k} \neq 0$. 
Please provide the sum of the smallest $n_k$ values for $k = 2$ and $k = 3$.

Answer: 12
user: Compute the sum of the series \( \sum_{k=0}^{\infty} \frac{1}{(4 k+1)(4 k+2)(4 k+3)(4 k+4)}=\frac{1}{1 \cdot 2 \cdot 3 \cdot 4}+\frac{1}{5 \cdot 6 \cdot 7 \cdot 8}+\cdots \). The original answer is in a special format：$\frac{\ln m}n-\frac\pi{p}.$ provide the sum of the denominators in the expression.

Answer: 30
user: (a) A sequence $x_{1}, x_{2}, \ldots$ of real numbers satisfies $$ x_{n+1}=x_{n} \cos x_{n} \quad \text { for all } \quad n \geq 1 $$ Determine if this sequence converges for all initial values $x_{1}$. (b) A sequence $y_{1}, y_{2}, \ldots$ of real numbers satisfies $$ y_{n+1}=y_{n} \sin y_{n} \quad \text { for all } \quad n \geq 1 $$ Determine if this sequence converges for all initial values $y_{1}$. If the answer to (a) is NO and (b) is YES, provide the sum of their binary representations as a single integer.

Answer: 1
user: Given an integer $n>1$, let $S_{n}$ be the group of permutations of the numbers $1,2, \\ldots, n$. Two players, A and B, play the following game. Taking turns, they select elements (one element at a time) from the group $S_{n}$. It is forbidden to select an element that has already been selected. The game ends when the selected elements generate the whole group $S_{n}$. The player who made the last move loses the game. The first move is made by A. For which values of n does player A have a winning strategy? Please provide the sum of these values of n.

Answer: 5
user: Consider the following sequence \(\left(a_{n}\right)_{n=1}^{\infty}=(1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1, \ldots)\). Find all pairs \((\alpha, \beta)\) of positive real numbers such that \(\lim \limits_{n \rightarrow \infty} \frac{\sum\limits_{k=1}^{n} a_{k}}{n^{\alpha}}=\beta\). The original answer is in \(\frac{\sqrt{m}}{n}\) format, please give the value of \(m + n\).

Answer: 5
user: We say that a subset of $\mathbb{R}^{n}$ is $k$-almost contained by a hyperplane if there are less than $k$ points in that set which do not belong to the hyperplane. We call a finite set of points $k$-generic if there is no hyperplane that $k$-almost contains the set. For each pair of positive integers $k$ and $n$, find the minimal number $d(k, n)$ such that every finite $k$-generic set in $\mathbb{R}^{n}$ contains a $k$-generic subset with at most $d(k, n)$ elements. Calculate the value of $d(k, n)$ when $k = 2$ and $n = 2$, and provide the result of k + n + d(k, n).

Answer: 8
user: Let $F(0)=0, F(1)=\frac{3}{2}$, and $F(n)=\frac{5}{2} F(n-1)-F(n-2)$ for $n \geq 2$. Determine whether or not $\sum\limits_{n=0}^{\infty} \frac{1}{F\left(2^{n}\right)}$ is a rational number. Please provide the integer value of the sum.

Answer: 1
user: An $n \times n$ complex matrix $A$ is called $t$-normal if $A A^{t}=A^{t} A$ where $A^{t}$ is the transpose of $A$. For each $n$, determine the maximum dimension of a linear space of complex $n \times n$ matrices consisting of t-normal matrices. Please provide the value of the dimension when n = 11. The answer should be in the format \frac{m}{n}, directly give the value of m + n.

Answer: 134
user: Determine all complex numbers $\lambda$ for which there exist a positive integer $n$ and a real $n \times n$ matrix $A$ such that $A^{2}=A^{T}$ and $\lambda$ is an eigenvalue of $A$. The answer is $\frac{m \pm \sqrt{n} i}{p}$ ,please give the sum of m,n and p.

Answer: 4
user: Determine all rational numbers $a$ for which the matrix
$$
\left(\begin{array}{cccc}
a & -a & -1 & 0 \\
a & -a & 0 & -1 \\
1 & 0 & a & -a \\
0 & 1 & a & -a
\end{array}\right)
$$
is the square of a matrix with all rational entries. Provide the integer value of $a$ that satisfies the condition.

Answer: 0
user: For $R>1$ let $\mathcal{D}_{R}=\left\{(a, b) \in \mathbb{Z}^{2}: 0<a^{2}+b^{2}<R\right\}$. Compute 
$$ 
\lim _{R \rightarrow \infty} \sum_{(a, b) \in \mathcal{D}_{R}} \frac{(-1)^{a+b}}{a^{2}+b^{2}} 
$$ 
The original answer is in the format $m\pi \log n$. Please provide the integer value of m and n.

Answer: 1
user: Let $\Omega=\left\{(x, y, z) \in \mathbb{Z}^{3}: y+1 \geq x \geq y \geq z \geq 0\right\}$. A frog moves along the points of $\Omega$ by jumps of length 1. For every positive integer $n$, determine the number of paths the frog can take to reach $(n, n, n)$ starting from $(0,0,0)$ in exactly $3 n$ jumps. The original answer is in the format \(\frac{\binom{3 n}{n}}{2 n+1}\), please provide the value of \(\binom{3 n}{n} + (2 n+1)\) when \(n = 3\).

Answer: 843
user: Evaluate the product
$$
\prod_{n=3}^{\infty} \frac{\left(n^{3}+3 n\right)^{2}}{n^{6}-64}
$$
The original answer is in \(\frac{m}{n}\) format, please give the value of m + n.

Answer: 79
user: A four-digit number $Y E A R$ is called very good if the system $$ \begin{aligned} & Y x+E y+A z+R w=Y \\ & R x+Y y+E z+A w=E \\ & A x+R y+Y z+E w=A \\ & E x+A y+R z+Y w=R \end{aligned} $$ of linear equations in the variables $x, y, z$ and $w$ has at least two solutions. Find all very good YEARs in the 21st century and provide the sum of all these years.

Answer: 12309
user: Find all twice continuously differentiable functions $f: \mathbb{R} \rightarrow(0,+\infty)$ satisfying 
$$ 
f^{\prime \prime}(x) f(x) \geq 2\left(f^{\prime}(x)\right)^{2} 
$$ 
for all $x \in \mathbb{R}$. Please provide the sum of the solutions $f^{\prime}$ and $f^{\prime \prime}$.

Answer: 0
user: Let $p$ be a prime number and let $k$ be a positive integer. Suppose that the numbers $a_{i}=i^{k}+i$ for $i=0,1, \\ldots, p-1$ form a complete residue system modulo $p$. What is the set of possible remainders of $a_{2}$ upon division by $p$? Please provide the sum of all possible remainders.

Answer: 5
user: 一个正数，若其小数部分、整数部分和其自身成等比数列，则该数为 $\qquad$. 原始答案为\frac{m+\sqrt{n}}{k}的格式，请给出m + n + k的值。

Answer: 8
user: 设整数 $a_{1}, a_{2}, \cdots, a_{2019}$ 满足 $1=a_{1} \leqslant a_{2} \leqslant \cdots \leqslant a_{2019}=99$. 记 $f=\left(a_{1}^{2}+a_{2}^{2}+\cdots+a_{2019}^{2}\right)-\left(a_{1} a_{3}+a_{2} a_{4}+a_{3} a_{5}+\cdots+a_{2017} a_{2019}\right)$. 求 $f$ 的最小值 $f_{0}$. 并确定使 $f=f_{0}$ 成立的数组 $\left(a_{1}, a_{2}, \cdots, a_{2019}\right)$ 的个数。标准答案形式为$C_{a}^{b}$，计算出a+b的值。

Answer: 2016
user: 设 $f(x)=\frac{1}{x^{3}-x}$, 求满足不等式 $f(2)+f(3)+\cdots+f(n)>\frac{499}{2020}$ 的最小正整数 $n$. 请给出这个整数n的值。

Answer: 13
user: Determine all pairs $(a, b) \in \mathbb{C} \times \mathbb{C}$ satisfying
$$
|a|=|b|=1 \quad \text { and } \quad a+b+a \bar{b} \in \mathbb{R}.
$$
(proposed by Mike Daas, Universiteit Leiden)
Please provide the sum of all possible values of a and b.

Answer: 0
user: 8 个女孩与 25 个男孩围成一圈，任何两个女孩之间至少站两个男孩，则共有多少种不同的排列方法？答案标准形式为$C^{a}_{b} \cdot c! \cdot d!$计算出a+b+c+d的值。

Answer: 55
user: Find a necessary and sufficient condition on the positive integer $n$ that the equation

$$x^{n}+(2+x)^{n}+(2-x)^{n}=0$$

has a rational root. Please provide the integer value of $n$ that satisfies this condition.

Answer: 1
user: For $n=1,2, \ldots$ let
$$
S_{n}=\log \left(\sqrt[n^{2}]{1^{1} \cdot 2^{2} \cdot \ldots \cdot n^{n}}\right)-\log (\sqrt{n})
$$
where $\log$ denotes the natural logarithm. Find $\lim\limits _{n \rightarrow \infty} S_{n}$. The original answer is in $-\frac{m}{n}$ format, please give the value of m + n.

Answer: 5
user: We throw a dice (which selects one of the numbers $1,2, \ldots, 6$ with equal probability) $n$ times. What is the probability that the sum of the values is divisible by 5? What is the value of the probability when n = 1/6?

Answer: 0
user: Let $n$ be a positive integer. Consider an $n \times n$ matrix with entries $1,2, \ldots, n^{2}$ written in order starting top left and moving along each row in turn left-to-right. We choose $n$ entries of the matrix such that exactly one entry is chosen in each row and each column. What are the possible values of the sum of the selected entries? Please provide the value when n = 2.

Answer: 5
user: 设 $A(2,0)$ 为平面上一定点, $P\left(\sin \left(2 t-60^{\circ}\right), \cos \left(2 t-60^{\circ}\right)\right)$ 为动点, 则当 $t$ 由 $15^{\circ}$ 变到 $45^{\circ}$ 时, 线段 $A P$ 扫过的面积是 $\frac{m}{n}\pi$ 形式，请给出 m + n 的值。

Answer: 7
user: 设 $n=1990$, 则 
$$ 
\frac{1}{2^{n}}\left(1-3 C_{n}^{2}+3^{2}{ C_n ^4}-3^{3} C_{n}^{6}+\cdots+3^{994} C_{n}^{1998}-3^{995} C_{n}^{1990}=\right. \qquad. 
$$ 
The original answer is in $-\frac{m}{n}$ format, please give the value of m + n.

Answer: 3
user: 设棱锥 $M-A B C D$ 的底面为正方形, 且 $M A=M D, M A \perp A B$, 如果 $\triangle A M D$ 的面积为 1, 试求能够放入这个棱锥的最大球的半径。已知原始答案为\sqrt{m}-n的形式，请给出m+n的值。

Answer: 3
user: $n^{2}$ 个正数排成 $n$ 行 $n$ 列，满足每一行的数成等差数列，每一列的数成等比数列，并且所有公比相等。已知 $a_{24}=1, a_{42}=\frac{1}{8}, a_{43}=\frac{3}{16}$ ，求 $a_{11}+a_{22}+\cdots \cdots+a_{n n}$。当n=2时，给出 $S$ 的值。

Answer: 1
user: 设 $n=1990$, 则 $\frac{1}{2^{n}}\left(1-3 C_{n}^{2}+3^{2} C_{n}^{4}-3^{3} C_{n}^{6}+\cdots+3^{994} C_{n}^{1988}-3^{995} C_{n}^{1990}\right) = \qquad$. 该表达式的原始答案是$-\frac{m}{n}$的形式，请给出m + n的值。

Answer: 3
user: 8 个女孩与 25 个男孩围成一圈，任何两个女孩之间至少站两个男孩，那么，共有多少种不同的排列方法？标准答案形式为$\frac{a!\cdot b!}{c!}$，计算出a+b+c的值。

Answer: 50
user: $n^{2}(n \geqslant 4)$ 个正数排成 $n$ 行 $n$ 列, $$ \begin{array}{llllll} a_{11} & a_{12} & a_{13} & a_{14} & \cdots \cdots & a_{1 n} \\ a_{21} & a_{22} & a_{23} & a_{24} & \cdots \cdots & a_{2 n} \\ a_{31} & a_{32} & a_{33} & a_{34} & \cdots \cdots & a_{3 n} \\ a_{41} & a_{42} & a_{43} & a_{44} & \cdots \cdots & a_{4 n} \\ \cdots & \cdots & \cdots & \cdots & \cdots \cdots & \cdots \\ a_{n 1} & a_{n 2} & a_{n 3} & a_{n 4} & \cdots \cdots & a_{n n} \end{array} $$ 其中每一行的数成等差数列，每一列的数成等比数列，并且所有公比相等. 已知 $a_{24}=1, a_{42}=\frac{1}{8}, a_{43}=\frac{3}{16}$, 求 $a_{11}+a_{22}+\cdots+a_{n n}$. 请提供 n = 2 时的值。

Answer: 1
user: 设棱锥 $M-A B C D$ 的底面是正方形，且 $M A=M D, M A \perp A B$ ，如果 $\triangle A M D$ 的面积为 1, 试求能够放入这个棱锥的最大球的半径。原始答案为$\sqrt{m}-n$，请给出m+n的值。

Answer: 3
user: 设 $S=\{1,2, \ldots, n\}$ ， $A$ 为至少含有两项的公差为正的等差数列，其项都在 $S$ 中，且添加 $S$ 的其他元素于 $A$ 后不能构成与 $A$ 有相同公差的等差数列. 求这种 $A$ 的个数（这里只有两项的数列也看作等差数列). 请提供 n = 16 时的值。

Answer: 64
user: 设正三棱雉 $P-A B C$ 的高为 $P O, M$ 为 $P O$ 的中点, 过 $A M$ 作与棱 $B C$ 平行的平面, 将三棱雉截为上、下两部分，试求此两部分的体积比。请给出比值中两个数之和的值。

Answer: 25
user: 设 $O$ 为抛物线的顶点, $F$ 为焦点, 且 $P Q$ 为过 $F$ 的弦. 已知 $|O F|=a,|P Q|=$ $b$, 求 $\triangle O P Q$ 的面积。请提供答案的格式为 a \sqrt{ab}，当a=b=1时，给出 a +ab 的值。

Answer: 2
user: 设 $z_{1}, z_{2}$ 都是复数, 且 $\left|z_{1}\right|=3,\left|z_{2}\right|=5\left|z_{1}+z_{2}\right|=7$, 则 $\arg \left(\frac{z_{2}}{z_{1}}\right)^{3}$ 的值是 $\qquad$. 请给出原始答案中\pi前的常数部分值。

Answer: 1
user: 设 $x, y, z$ 是实数, $3 x, 4 y, 5 z$ 成等比数列, 且 $\frac{1}{x}, \frac{1}{y}, \frac{1}{z}$ 成等差数列, 则 $\frac{x}{z}+\frac{z}{x}$ 的值是以\frac{m}{n}的形式表示，请给出m + n的值。

Answer: 49
user: 二次方程 $(1-i) x^{2}+(\lambda+i) x+(1+i \lambda)=0(i$ 为虚数单位, $\lambda \in R)$ 有两个虚根的充分必要条件是 ，$\lambda$ 不可能的的取值为 $\qquad$.

Answer: 2
user: 若 $z \in C, \arg \left(z^{2}-4\right)=\frac{5 \pi}{6}, \arg \left(z^{2}+4\right)=\frac{\pi}{3}$, 则 $z$ 的值是 $\qquad$.已知答案格式为\(\pm(m+\sqrt{n} i)\)，求\(m+n\)的值。

Answer: 4
user: 设正数列 $a_{0} 、 a_{1} 、 a_{2} 、 \\ldots 、 a_{n} 、 \\ldots$ 满足
$$
\\sqrt{a_{n} a_{n-2}}-\\sqrt{a_{n-1} a_{n-2}}=2 a_{n-1}, \\quad(n \\geq 2)
$$
且 $a_{0}=a_{1}=1$, 计算前三项 $a_0, a_1, a_2, a_3$ 的和。

Answer: 60
user: 设 $0<a<b$, 过两定点 $A(a, 0)$ 和 $B(b, 0)$ 分别引直线 $l$ 和 $m$, 使与抛物线 $y^{2}=x$ 有四个不同的交点, 当这四点共圆时，求这种直线 $l$ 与 $m$ 的交点 $P$ 的轨迹。请给出满足条件的方程x系数和的值。

Answer: 2
user: 若 $z \in \mathbf{C}, \arg \left(z^{2}-4\right)=\frac{5 \pi}{6}, \arg \left(z^{2}+4\right)=\frac{\pi}{3}$, 请提供 $z$ 的两个解中实部和虚部之和。

Answer: 0
user: 已知有向线段 $P Q$ 的起点 $P$ 和终点 $Q$ 的坐标分别为 $(-1,1)$ 和 $(2,2)$, 若直线 $l: x+m y+m=0$ 与 $P Q$的延长线相交，则 $m$ 的取值范围是 。请提供区间端点的积。

Answer: 2
user: 已知一平面与一正方体的 12 条棱的夹角都等于 $\alpha$, 则 $\sin \alpha$ 的原始答案是 \frac{\sqrt{m}}{n} 的形式，请给出 m + n 的值。

Answer: 6
user: $x$ 的二次方程 $x^{2}+z_{1} x+z_{2}+m=0$ 中, $z_{1}, z_{2}, m$ 均是复数，且 $z{ }_{1}^{2}-4 z_{2}=16+20 i$ ，设这个方程的两个根 $\alpha 、 \beta$, 满足 $|\alpha-\beta|=2 \sqrt{7}$, 求 $|m|$ 的最大值和最小值的和。

Answer: 14
user: 设 $\left\{a_{n}\right\}$ 为等差数列, $\left\{b_{n}\right\}$ 为等比数列, 且 $b_{1}=a_{1}^{2}, b_{2}=a_{2}^{2}, b_{3}=a_{3}^{2}\left(a_{1}<a_{2}\right)$,又 $\lim _{n \rightarrow \infty}\left(b_{1}+b_{2}+\cdots+b_{n}\right)=\sqrt{2}+1$. 试求 $\left\{a_{n}\right\}$ 的首项的两倍与公差的和。

Answer: -2
user: 椭圆 $\rho=\frac{1}{2-\cos \theta}$ 的短轴长等于多少？原始答案为$\frac{m \sqrt{n}}{k}$格式，请给出m+n+k的值。

Answer: 8
user: 若复数 $z_{1} 、 z_{2}$ 满足 $\left|z_{1}\right|=2,\left|z_{2}\right|=3,3 z_{1}-2 z_{2}=\frac{3}{2}-\mathrm{i}$, 则 $z_{1} \cdot z_{2}=$ 以形式 $-\frac{a}{b} + \frac{c}{d} \mathrm{i}$ 表示，给出 a + b + c + d 的值。

Answer: 128
user: 正方体 $A B C D-A_{1} B_{1} C_{1} D_{1}$ 的棱长为 1. 则直线 $A_{1} C_{1}$ 与 $B D_{1}$ 的距离是\qquad$。原始答案为 \frac{\sqrt{m}}{n} 的形式，请给出 m + n 的值。

Answer: 12
user: 已知点 $A(0,2)$ 和抛物线 $y^{2}=x+4$ 上两点 $B, C$, 使得 $A B \perp B C$, 求点 $C$ 的纵坐标的取值范围。请提供该范围的除无穷以外的端点之和。

Answer: 4
user: 已知复数 $Z_{1} 、 Z_{2}$ 满足 $\left|Z_{1}\right|=2,\left|Z_{2}\right|=3$, 若它们所对应的向量的夹角为 $60^{\circ}$, 则 $\left|\frac{Z_{1}+Z_{2}}{Z_{1}-Z_{2}}\right|=$ $\qquad$ ; 题目的原始答案为分数形式 $\frac{\sqrt{m}}{n}$, 请给出 m + n 的值。

Answer: 140
user: 实数 $a, b, c$ 和正数 $\lambda$ 使得 $f(x)=x^{3}+a x^{2}+b x+c$ 有三个实根 $x_{1}, x_{2}, x_{3}$, 且满足\(1) $x_{2}-x_{1}=\lambda$;\(2) $x_{3}>\frac{1}{2}\left(x_{1}+x_{2}\right)$.求 $\frac{2 a^{3}+27 c-9 a b}{\lambda^{3}}$ 的最大值。原始答案为$\frac{m\sqrt{n}}{j}$，请给出m+n+j的和。

Answer: 8
user: 将八个半径都为 1 的球分放两层放置在一个圆柱内，并使得每个球都和其相邻的四个球相切，且与圆柱的一个底面及侧面都相切。假设原始答案为$m+\sqrt[n]{j}$的形式，请给出m+n+j的和。

Answer: 14
user: Determine the position of a normal chord of a parabola such that it cuts off of the parabola a segment of minimum area. Please provide the sum of the coordinates of the points, when a = 2.

Answer: 8
user: A penny in a corner. A circle moves so that it is continually in contact with all three coordinate planes of an ordinary rectangular system. Find the locus of the center of the circle. Please provide the num of the possible condition that $x_{i}^{2} \leq r^{2}$.

Answer: 3
user: Find a necessary and sufficient condition on the positive integer $\boldsymbol{n}$ that the equation $$x^{n}+(2+x)^{n}+(2-x)^{n}=0$$ has a rational root. Please provide the integer value of $n$ that satisfies this condition.

Answer: 1
user: Assuming that the roots of $x^{3}+p x^{2}+q x+r=0$ are all real and positive, find the relation between $p, q$, and $r$ which is a necessary and sufficient condition that the roots may be the cosines of the angles of a triangle. Given that the original answer is in the form of an equation $p^n-mq-jr=1$, provide the sum of $m$, $n$, and $j$ when they satisfy the equation.

Answer: 6
user: The flag of the United Nations consists of a polar map of the world, with the North Pole as center, extending approximately to $45^{\circ}$ South Latitude. The parallels of latitude are concentric circles with radii proportional to their co-latitudes. Australia is near the periphery of the map and is intersected by the parallel of latitude $30^{\circ} \mathrm{S}$. In the very close vicinity of this parallel, how much are East and West distances exaggerated as compared to North and South distances? The original answer is in the format $m\pi/n\sqrt{j}\sim2.42$, please provide the value of m + n + j.

Answer: 10
user: Let $f(x)$ be a cubic polynomial with roots $x_{1}, x_{2}$, and $x_{3}$. Assume that $f(2 x)$ is divisible by $f^{\prime}(x)$ and compute the sum of the absolute values of the ratios $x_{1}: x_{2}: x_{3}$.

Answer: 9
user: Find the length of the curve $y^{2}=x^{3}$ from the origin to the point where the tangent makes an angle of $45^{\circ}$ with the $x$-axis. The original answer involves an integral that evaluates to a form with square roots. The original answer form is $\ frac {a} {b} (c \ sqrt {d}-e ）Find the value of a+b+c+d+e.

Answer: 40
user: What is the maximum of $\left|z^{3}-z+2\right|$, where $z$ is a complex number with $|z|=1$? The original answer is in the format \sqrt{m}, please provide the value of m.

Answer: 13
user: Find the smallest volume bounded by the coordinate planes and by a tangent plane to the ellipsoid $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1. $$ The original answer is in the format $\frac{1}{2} \sqrt{3} abc$, Let abc=2 \ sqrt {3}, find the value of V.

Answer: 3
user: Find all solutions of $n^{m}=m^{n}$ in integers $n$ and $m(n\neq m)$. Prove that you have obtained all of them. Please provide the sum of all values of $m + n$ for each solution.

Answer: 0
user: The graph of the equation $x^{y}=y^{x}$ in the first quadrant (i.e., the region where $x>0$ and $y>0$) consists of a straight line and a curve. Find the sum of the coordinates of the intersection points of a straight line and a curve, rounded down.

Answer: 5
user: Evaluate in closed form

$$\sum_{k=1}^{n}\binom{n}{k} k^{2}$$

$$\binom{n}{k}=\frac{n(n-1) \cdots(n-k+1)}{1 \cdot 2 \cdots k}$$Please provide the value when \(n = 3\).

Answer: 24
user: Evaluate

$$\lim _{n \rightarrow \infty} \sum_{j=1}^{n^{2}} \frac{n}{n^{2}+j^{2}}$$

The original answer is in \frac{m\pi}{n} format. Please provide the value of m + n.

Answer: 3
user: A horizontal disc of diameter 3 inches is rotating at 4 revolutions per minute. A light is shining at a distant point in the plane of the disc. An insect is placed at the edge of the disc furthest from the light, facing the light. It at once starts crawling, and crawls so as always to face the light, at 1 inch per second. Set up the differential equation of motion, and find at what point the insect again reaches the edge of the disc. The original answer indicates that the insect leaves the disc at a point with a coordinate in the format \frac{m}{n},0. Please provide the value of m + n.

Answer: 5
user: Two uniform solid spheres of equal radii are so placed that one is directly above the other. The bottom sphere is fixed, and the top sphere, initially at rest, rolls off. At what point will contact between the two spheres be "lost"? Assume the coefficient of friction is such that no slipping occurs. The original answer is in the format \theta=\arccos (m / n), please give the value of m + n.

Answer: 27
user: If $x_{1}, \ldots, x_{n}$ denote the $n$th roots of unity, evaluate
$$
\pi\left(x_{i}-x_{j}\right)^{2} \quad(i<j)
$$What is the value when n=3。

Answer: -27