sungyub/dapo-math-17k-verl huggingface dataset:
user: 设 $R$ 为平面上以 $A(4,1) 、 B(-1,-6) 、 C(-3,2)$ 三点为顶点的三角形区域(包括三角形的边界). 试求当 $(x, y)$ 在 $R$ 上变动时，函数 $4 x-3 y$ 的极大值和极小值之差。

Answer: -4
user: 把半径为 1 的四个小球叠成两层放在桌面上：下层三个，上层一个，两两相切，求上层小球最高点离桌面的高度与$\frac{2 \sqrt{6}}{3}$的差。

Answer: 2
user: 解方程组 $\begin{cases}x-y+z=1, & \text { (1) } \\ y-z+u=2, & \text { (2) } \\ z-u+v=3, & \text { (3) } \\ u-v+x=4, & \text { (4) } \\ v-x+y=5, & \text { (5) }\end{cases}$ 请给出 x, y, z, u, v 的和。

Answer: 17
user: 解方程: $5 x^{2}+x-x \sqrt{5 x^{2}-1}-2=0$.的两根之和

Answer: 0
user: 已知复数数列 $\left\{z_{n}\right\}$ 满足 $z_{1}=1, z_{n+1}=\overline{z_{n}}+1+n \mathrm{i}$ ，其中 $n=1,2, \cdots$ ，其中 i 是虚数单位, $\overline{z_{n}}$ 表示 $z_{n}$ 的共轭复数, 则 $z_{2015}$ 的值中实数部分为 $\qquad$。

Answer: 2015
user: 设 $a_{1}, a_{2}, a_{3}, a_{4}$ 是 4 个有理数, 使得
$$
\left\{a_{i} a_{j} \mid 1 \leqslant i<j \leqslant 4\right\}=\left\{-24,-2,-\frac{3}{2},-\frac{1}{8}, 1,3\right\}
$$
求 $a_{1}+a_{2}+a_{3}+a_{4}$ 的所有可能值之和。

Answer: 0
user: 已知复数数列 $\left\{z_{n}\right\}$ 满足 $z_{1}=1, z_{n+1}=\overline{z_{n}}+1+n \mathrm{i}$ ，其中 $n=1,2, 3\cdots$ ，其中 i 是虚数单位, $\overline{z_{n}}$ 表示 $z_{n}$ 的共轭复数, 则 $z_{2015}$ 的实部值为 $\qquad$。

Answer: 2015
user: 求最大的实数 $\lambda$的八倍， 使得不等式
$$
\sum_{k=1}^{n} x_{k}^{3}\left(x_{k}-x_{k-1}\right) \geq \frac{1}{4}+\frac{\lambda}{n}
$$
对任意正整数 n 以及任意实数 $0=x_{0} \leq x_{1} \leq x_{2} \leq \cdots \leq x_{n}=1$ 均成立.

Answer: 3
user: In how many ways can the integers from 1 to $5$ be ordered subject to the condition that, except for the first integer on the left, every integer differs by 1 from some integer to the left of it?

Answer: 16
user: A game of solitaire is played as follows. After each play, according to the outcome, the player receives either $a$ or $b$ points ( $a$ and $b$ are positive integers with $a$ greater than $b$ ), and his score accumulates from play to play. It has been noticed that there are thirty-five non-attainable scores and that one of these is 58. Find the sum of $a$ and $b$.

Answer: 19
user: Evaluate$$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{n}\left(\left[\frac{2n}{k}\right]-2\left[\frac{n}{k}\right]-\ln4\right)$$
and express your answer in the form $\log a-b$, with $a$ and $b$ positive integers. 
Here $[x]$ is defined to be the integer such that $[x]\leqq x<[x]+1$ and $\log x$ is the logarithm of $x$ to base $e$.

Answer: -1
user: Suppose that $G$ is a group generated by elements $A$ and $B$, that is, every element of $G$ can be written as a finite "word" $A^{n_1}B^{n_2}A^{n_3}\cdots B^{n_k}$, where $n_1,\ldots,n_k$ are any integers, and $A^0=B^0=1$ as usual. Also, suppose that $A^{4}=B^{7}=ABA^{-1}B=1,A^{2}\neq1$, and $B\neq1$.
 How many elements of $G$ are of the form $C^{2}$ with $C$ in $G$?

Answer: 8
user: A particle moving on a straight line starts from rest and attains a velocity $2$ after traversing a distance $2$. If the motion is such that the acceleration was never increasing, find the maximum time for the traverse.

Answer: 2
user: Evaluate
$$
\sum_{k=1}^{n}(-1)^{k}\binom{n}{k}(x-k)^{n}
$$，when $n=2$

Answer: 2
user: Let $p$ be a prime number. Let $J$ be the set of all $2 \times 2$ matrices $\binom{c b}{c d}$ whose entries are chosen from $\{0,1,2, \cdots, p-1\}$ and satisfy the conditions $a+d \equiv 1(\bmod p), a d-b c \equiv 0(\bmod p)$. Determine how many members $J$ has. Please provide the number of members when $p=2$.

Answer: 6
user: For a set with $n$ elements, how many subsets are there whose cardinality (the number of elements in the subset) is respectively $\equiv 0(\bmod 3), \equiv 1(\bmod 3), \equiv 2(\bmod 3)$? In other words, calculate $$ s_{(, n}=\sum_{k-1(\text { mod })}\binom{n}{k} \quad \text { for } i=0.1,2 $$ Your result should be strong enough to permit direct evaluation of the numbers $s_{\text {th }}$ and to show clearly the relationship of $s_{0, n}$ and $s_{1, n}$ and $s_{2, n}$ to each other for all positive integers $n$. In particular, show the relationships among these three sums for $n=1000$. The original answer is in \frac{m}{n} format, please give the value of m + n.

Answer: 667
user: The lengths of the hour hand and the minute hand of an accurate clock are 3 and 4 respectively. Find the square of the distance between the tips of the hour hand and the minute hand when the rate of increase of this distance is the fastest.

Answer: 7
user: Find the area of a convex octagon that is inscribed in a circle and has four consecutive sides of length 3 units and the remaining four sides of length 2 units. Give the answer in the form $r+s\sqrt{t}$ with $r$, $s$, and $t$ positive integers，writing r.

Answer: 13
user: Let $v$ be a vertex (corner) of a cube $C$ with edges of length 4. Let $S$ be the largest sphere that can be inscribed in $C$. Let $R$ be the region consisting of all points $p$ between $S$ and $C$ such that $p$ is closer to $v$ than to any other vertex of the cube. Find the volume of $R+\frac{4\pi}{3}$.

Answer: 8
user: Find the value of \(3(A + B)\) where \(A\) and \(B\) are constants such that \(f(x)\) is a solution to the differential equation.

Answer: 4
user: Let $P$ be a polynomial, with real coefficients, in three variables and $F$ be a function of two variables such that 
$P(u x, u y, u z)=u^{2} F(y-x, z-x)$ for all real $x, y, z, u$,

and such that $P(1,0,0)=4, P(0,1,0)=5$, and $P(0,0,1)=6$. Also let $A, B, C$ be complex numbers with $P(A, B, C)=0$ and $|B-A|=10$. Find $\frac{\sqrt{30}}{6}|C-A|$.

Answer: 3
user: For each positive integer $n$, let $a(n)$ be the number of zeros in the base $3$ representation of $n$. For which positive real numbers $x$ does the series $$ \sum_{n=1}^{\infty} \frac{x^{a(n)}}{n^{3}} $$ converge? Please provide the sum of the bounds of the interval where x satisfies the condition.

Answer: 25
user: Develop necessary and sufficient conditions which ensure that $r_{1},r_{2},r_{3}$ and $r_{1}{}^{2},r_{2}{}^{2}, r_{3}{}^{2}$ are simultaneously roots of the equation $x^{3}+ax^{2}+bx+c=0$.How many possible situations are there in total?

Answer: 8
user: In a solid sphere of radius $R$ the density $\rho$ is a function of $r$, the distance from the center of the sphere. If the magnitude of the gravitational force of attraction due to the sphere at any point inside the sphere is $k r^{2}$, where $k$ is a constant, find $\rho$ as a function of $r$. Find also the magnitude of the force of attraction at a point outside the sphere at a distance $r$ from the center. (Assume that the magnitude of the force of attraction at a point $P$ due to a thin spherical shell is zero if $P$ is inside the shell, and is $m / r^{2}$ if $P$ is outside the shell, $m$ being the mass of the shell, and $r$ the distance of $P$ from the center.)The answer can be written as $\frac{k R^{m}}{r^{n}}$ for $r > R$. Find the value of $m + n$.

Answer: 6
user: Let $\min (x, y)$ denote the smaller of the numbers $x$ and $y$. For what $\lambda$'s does the equation 
$$\int_{0}^{1} \min (x, y) f(y) d y=\lambda f(x)$$ 
have continuous solutions which do not vanish identically in ( 0,1$)$? The largest $\lambda$ can be expressed in the form of $\frac{m}{n\pi^{2}}$. Find the value of $m + n$.

Answer: 5
user: Find the equations of the two straight lines each of which cuts all the four straight lines 
$x=1, y=0 ; \quad y=1, z=0 ; \quad z=1, x=0 ; \quad x=y=-6 z$.
If \(L_{1}\) is given by
$$y = a_{1}x + b_{1} = c_{1}z + d_{1}$$
and \(L_{2}\) is given by
$$y = a_{2}x + b_{2} = c_{2}z + d_{2}$$
Find the expression of \(a_{1} + b_{1} + c_{1}z + d_{1} + a_{2} + b_{2} + c_{2}z + d_{2}\).

Answer: 1
user: In assigning dormitory rooms, a college gives preference to pairs of students in this order: $$AA, AB, AC, BB, BC, AD, CC, BD, CD, DD$$ in which $AA$ means two seniors, $AB$ means a senior and a junior, etc. Determine numerical values to assign to $A, B, C, D$ so that the set of numbers $A+A, A+B, A+C, B+B$, etc., corresponding to the order above will be in descending magnitude. Find the general solution and the solution in least positive integers. Please provide the sum of A, B, C, and D in the solution with the least positive integers.

Answer: 23
user: Given the parabola $y^{2}=2 m x$, what is the length of the shortest chord that is normal to the curve at one end?Write down the value when \(m = \sqrt{3}\).

Answer: 9
user: Find the equation of the parabola which touches the $x$-axis at the point $(1,0)$ and the $y$-axis at the point $(0,2)$. Find the coordinates of its vertex.If the coordinates of the vertex are $(\frac{a}{b},\frac{c}{d})$, find the value of $a + b + c + d$.

Answer: 68
user: A particle moves on a straight line, the only force acting on it being a resistance proportional to the velocity. If it started with a velocity of $1,000 \mathrm{ft}$. per sec. and had a velocity of 900 ft. per sec. when it had travelled $1,200 \mathrm{ft}$., calculate to the nearest hundredth of a second the time it took to travel this distance.The answer is written in the form of \(a\times 10^{-2}\). Find the value of \(a\).

Answer: 126
user: Find the shortest distance between the plane $A x+B y+C z+1=0$ and the ellipsoid $x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2}=1$. (For brevity, let
$$
h=1 / \sqrt{A^{2}+B^{2}+C^{2}} \text { and } m=\sqrt{a^{2} A^{2}+b^{2} B^{2}+c^{2} C^{2}} \text {.) }
$$
State algebraically the condition that the plane shall lie outside the ellipsoid.Write down the values when \(m\geqslant1\).

Answer: 0
user: Let \alpha and \beta be given positive real numbers, with \alpha<\beta. If two points are selected at random from a straight line segment of length \beta, what is the probability that the distance between them is at least \alpha? Please provide the value of the answer in the form simplest fraction by assuming \alpha = 1 and \beta = 2, and then giving the sum of the numerator and denominator.

Answer: 5
user: 若不等式 $\sqrt{x}+\sqrt{y} \leqslant k \sqrt{5 x+y}$ 对任意的正实数 $x 、 y$ 都成立, 则实数 $k$ 的最小值为$\frac{\sqrt{m}}{n}$的格式，请给出$m + n$的值。

Answer: 35
user: 已知 a 、 b 、 c 分别为 $\triangle A B C$ 的三个内角 A 、 B 、 C 的对边长， $a=2$ ，且 $(b+2) \sin \frac{C}{2} \cdot \sin \frac{A-B}{2}=c \cos \frac{A}{2} \cdot \cos \frac{B-C}{2}$. 若 D 是 B C 的中点,求 A D 长的最小值。原始答案为 $\frac{\sqrt{m}}{n}$ 的格式，请给出 $m + n$ 的值。

Answer: 6
user: 已知复数 $z$ 满足 $|z|=1$, 则 $\left|z^{3}+z^{2}-5 z+3\right|$ 的最大值为 ? 原始答案为 $\frac{m\sqrt{3}}{n}$ 的格式，请给出 $m + n$ 的值。

Answer: 155
user: 设 $x_{1}=3, x_{n+1}=\sqrt{x_{n}+14}-\sqrt{x_{n}+2}\left(n \in N^{*}\right)$, 求当$n=100$时$\left[x_{1}+x_{2}+\cdots+x_{n}\right]$ 的值。

Answer: 200
user: Find all positive integers that are within $250$ of exactly $15$ perfect squares. Please provide the sum of the endpoints of the intervals that contain these integers.

Answer: 1002
user: For which real numbers $c$ is there a straight line that intersects the curve 
$$ 
y=x^{4}+9 x^{3}+c x^{2}+9 x+4 
$$ 
in four distinct points? The original answer is in the format $c < \frac{m}{n}$. Please provide the value of $m + n$.

Answer: 251
user: Let $I_{m} = \int_{0}^{2 \pi} \cos (x) \cos (2 x) \cdots \cos (m x) d x$. For which integers $m, 1 \leqslant m \leqslant 10$, is $I_{m} \neq 0$? Please provide the sum of all such integers $m$.

Answer: 22
user: Define a polynomial $f_{n}(x)$, where $n \geqslant 0$ is $f_{0}(x)=1$, $n \geqslant 1$ is $f_{n}(0)=0$, and $n \geqslant 0$ is
$$
\frac{d}{d x}\left(f_{n+1}(x)\right)=(n+1) f_{n}(x+1)
$$
. Clearly decompose $f_{100}(1)$ into powers of different prime numbers, $a^{b}$, and find the value of $a+b$.

Answer: 200
user: Let $C$ be the unit circle $x^{2}+y^{2}=1$. A point $p$ is chosen randomly on the circumference $C$ and another point $q$ is chosen randomly from the interior of $C$ (these points are chosen independently and uniformly over their domains). Let $R$ be the rectangle with sides parallel to the $x$-and $y$-axes with diagonal $p q$. What is the probability that no point of $R$ lies outside of $C$? The original answer is in $\frac{n}{\pi^m}$format, please give the value of $m + n$.

Answer: 6
user: Evaluate $\int_{0}^{\infty} t^{-1 / 2} e^{-1985\left(t+t^{-1}\right)} d t$. You may assume that $\int_{-\infty}^{\infty} e^{-x^{2}} d x=\sqrt{\pi}$. The original answer is in the form $\frac{\sqrt{\pi}}{\sqrt{m}} e^{-2 \times m}$ . Please provide the value of $m + 1$.

Answer: 1986
user: Given positive integers $m$ and $n \geq m$, determine the largest number of dominoes ( $1 \times 2$ or $2 \times 1$ rectangles) that can be placed on a rectangular board with $m$ rows and $2 n$ columns consisting of cells ( $1 \times 1$ squares) so that: 
$\mathrm{(i)}$ each domino covers exactly two adjacent cells of the board; 
$\mathrm{(ii)}$ no two dominoes overlap; 
$\mathrm{(iii)}$no two form a $2 \times 2$ square; and 
$\mathrm{(iv)}$the bottom row of the board is completely covered by $n$ dominoes.
Please calculate the number of dominoes when $m = 6$ and $n = 5$

Answer: 27
user: Find (in closed form) the difference between the number of positive integers at most $2^{2017}$ with even weight and the number of positive integers at most $2^{2017}$ with odd weight. The original answer is in the format $2^m$. Please provide the value of m + 1.

Answer: 1010
user: Let $n$ be an integer greater than 1, and let $X$ be a set of $n$ elements. If the union $A_{1} \cup \cdots \cup A_{k}$ is a proper subset of $X$, and no element of $X$ is in exactly one of $A_{i}$, then the set of non-empty subsets $A_{1}, \ldots, A_{k}$ of $X$ is compact. Find the maximum cardinality of the set of proper non-empty subsets of $X$ such that no non-empty subset is compact, and compute its value for $n=100$.

Answer: 198
user: Fix an integer $n \geq 2$. An $n \times n$ sieve is an $n \times n$ array with $n$ cells removed so that exactly one cell is removed from each row and column. For any positive integer $k$, a stick is a $1 \times k$ or $k \times 1$ array. For any sieve $A$, let $m(A)$ denote the minimum number of sticks required to partition $A$. Find all possible values of $m(A)$, since $A$ is different in all possible $n \times n$ sieves, for $n=100$.

Answer: 198
user: Let $\mathbb{N}$ denote the set of all positive integers. Find a specific function $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for $x = 1$ and $y = 1$, the expression $x^{2}-y^{2}+2 y(f(x)+f(y))$ becomes a square of an integer. Provide the value of $f(8)$.

Answer: 8
user: Find all pairs of positive real numbers $(x, y)$ such that $x y$ is an integer and
$$
x+y=\left\lfloor x^{2}-y^{2}\right\rfloor.
$$
Calculate the value of $x$ when $y=100$.

Answer: 101
user: Find all positive integers $a, b, n$ and prime numbers $p$ that satisfy the following conditions
$$
a^{2013}+b^{2013}=p^{n}
$$
Calculate the value of $n$ when $a=2$.

Answer: 2014
user: Olja writes down $n$ positive integers $a_{1}, a_{2}, \ldots, a_{n}$ smaller than $p_{n}$ where $p_{n}$ denotes the $n$-th prime number. Oleg can choose two (not necessarily different) numbers $x$ and $y$ and replace one of them with their product $x y$. If there are two equal numbers Oleg wins. Can Oleg guarantee a win? Please provide the smallest integer n for which Oleg can guarantee a win.

Answer: 2
user: Let $n$ be a positive integer. Let $B_{n}$ be the set of all binary strings of length $n$. For a binary string $s_{1} s_{2} \ldots s_{n}$, we define its twist in the following way. First, we count how many blocks of consecutive digits it has. Denote this number by $b$. Then, we replace $s_{b}$ with $1-s_{b}$. A string $a$ is said to be a descendant of $b$ if $a$ can be obtained from $b$ through a finite number of twists. A subset of $B_{n}$ is called divided if no two of its members have a common descendant. Find the largest possible cardinality of a divided subset of $B_{n}$. Please provide the value when $n = 4$. The answer should be an integer.

Answer: 4
user: We are given a combination lock consisting of 6 rotating discs. Each disc consists of digits $0,1,2, \ldots, 9$, in that order (after digit 9 comes 0 ). Lock is opened by exactly one combination. A move consists of turning one of the discs one digit in any direction and the lock opens instantly if the current combination is correct. Discs are initially put in the position 000000, and we know that this combination is not correct. What is the least number of moves necessary to ensure that we have found the correct combination?

Answer: 999998
user: We call a sequence of $n$ digits one or zero a code. Subsequence of a code is a palindrome if it is the same after we reverse the order of its digits. A palindrome is called nice if its digits occur consecutively in the code. (Code (1101) contains 10 palindromes, of which 6 are nice.) 
What is the least number of palindromes in a code when $n=3$?

Answer: 4
user: Find all pairs $(x, y)$ of integers that satisfy the equation $$ x^{2} y+y^{2}=x^{3}. $$ Please provide the value of x + y for the pair that satisfies the equation.

Answer: -12
user: A friendly football match lasts 90 minutes. In this problem, we consider one of the teams, coached by Sir Alex, which plays with 11 players at all times.\na) Sir Alex wants for each of his players to play the same integer number of minutes, but each player has to play less than 60 minutes in total. What is the minimum number of players required?\nb) For the number of players found in a), what is the minimum number of substitutions required, so that each player plays the same number of minutes?\nRemark: Substitutions can only take place after a positive integer number of minutes, and players who have come off earlier can return to the game as many times as needed. There is no limit to the number of substitutions allowed.\nPlease provide the sum of the answers for parts a) and b).

Answer: 35
user: Find all polynomials $P$ with integer coefficients such that $P(0) \neq 0$ and 
$$ 
P^{n}(m) \cdot P^{m}(n) 
$$ 
is a square of an integer for all nonnegative integers $n, m$. 
Remark : For a nonnegative integer $k$ and an integer $n$, $P^k(n)$ is defined as follows: $P^k(n) = n\text{ if }k = 0$ and $P^k(n)=P(P^{k-1}(n))$ if $k>0.$
Please provide the sum of the coefficients of the polynomial.

Answer: 2
user: 已知矩阵
$$
A=\left[\begin{array}{ccc}
2 & 0 & 0 \\
1 & 3 & 1 \\
-1 & 1 & 3
\end{array}\right], A^{10}=\left[\begin{array}{lll}
b_{11} & b_{12} & b_{13} \\
b_{21} & b_{22} & b_{23} \\
b_{31} & b_{32} & b_{33}
\end{array}\right] \text {, 求 } b_{22}.
$$

Answer: 524800
user: 正三棱雉 $P-A B C$ 中, $A B=1, A P=2$, 过 $A B$ 的平面 $\alpha$ 将其体积平分, 则棱 $P C$与平面 $\alpha$ 所成角的余弦值为。原始答案的形式为\frac{m\sqrt{n}}{k}，请给出m + n + k的值。

Answer: 18
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$n$的值。

Answer: 13
user: A positive integer $k \geqslant 3$ is called fibby if there exists a positive integer $n$ and positive integers $d_{1}<d_{2}<\ldots<d_{k}$ with the following properties: - $d_{j+2}=d_{j+1}+d_{j}$ for every $j$ satisfying $1 \leqslant j \leqslant k-2$, - $d_{1}, d_{2}, \ldots, d_{k}$ are divisors of $n$, - any other divisor of $n$ is either less than $d_{1}$ or greater than $d_{k}$. Find the largest fibby number.

Answer: 4
user: 设 $A$ 为三元集合（三个不同实数组成的集合），集合 $B=\{x+y \mid x, y \in A, x \neq y\}$ ，若 $B=$ $\left\{\log _{2} 6, \log _{2} 10, \log _{2} 15\right\}$ ，则集合 $A=$ $\qquad$.若集合\A={a,\log_b,\log_c\},其中$a,b,c$为正整数，求$a+b+c$.

Answer: 49
user: 已知函数 $f(x)=(\sqrt{1+x}+\sqrt{1-x}-3)\left(\sqrt{1-x^{2}}+1\right)$ 的最小值为 $m$, 最大值为 $M$, 则 $\frac{M}{m}$ 的原始答案为 $\frac{a-b\sqrt{c}}{d} $的形式。请给出 $a + b + c + d$ 的值。

Answer: 8
user: 已知向量 $\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}$ 满足 $|\boldsymbol{a}|:|\boldsymbol{b}|:|\boldsymbol{c}|=1: k: 3\left(k \in \mathbf{Z}_{+}\right)$, 且 $\boldsymbol{b}-\boldsymbol{a}=2(\boldsymbol{c}-\boldsymbol{b})$, 若 $\alpha$ 为 $\boldsymbol{a}, \boldsymbol{c}$ 的夹角，则原始答案为$-\frac{m}{n}$格式，请给出$m + n$的值。

Answer: 13
user: 已知正四面体可容纳 10 个半径为 1 的小球，则正四面体棱长的最小值为？答案的格式为$a+b\sqrt{c}$，请给出$a+b+c$的值。

Answer: 12
user: 已知 $\triangle A B C$ 为椭圆 $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$ 的内接三角形，且 $A B$ 过点 $P(1,0)$ ，则 $\triangle A B C$ 的面积的最大值为 ? 请给出最大值的答案格式为 $\frac{m\sqrt{n}}{k} $的格式，请计算 $m + n+k$ 的值。

Answer: 21
user: 已知过点 $P(3,0)$ 斜率为 $k$ 的直线 $l$ 交双曲线 $C: x^{2}-\frac{y^{2}}{3}=1$ 右支于 $A, B$ 两点, $F$ 为双曲线 $C$ 的右焦点, 且 $|A F|+|B F|=16$, 求 $k$ 的值。请给出所有满足条件的|k|的和。

Answer: 6
user: 设实数 $t \in[0, \pi]$ 。若关于 $x$ 的方程 $\cos (x+t)=1-\cos x$ 有解，求 $t$ 的取值范围中包含几个整数？

Answer: 3
user: 设自然数 $n \geqslant 3$, 实数 $x_{1}, x_{2}, \cdots, x_{n}$ 满足 $x_{1}+x_{2}+\cdots+x_{n}=n, x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=n^{2}$, 求 $\sum_{i=1}^{n} x_{i}^{3}$ 的最小值 。请提供n=3时最小值的整数结果。

Answer: 23
user: 已知 $f(x)=2 m x^{2}-2 m x-8 x+9, g(x)=m x-m$ ，对任意 $x \in \mathbf{R}, f(x)$ 与 $g(x)$ 至少有一个为正数，则实数 m 的取值范围是. 请给出该区间的两个端点之和.

Answer: 8
user: 过抛物线 $E: y^{2}=2 x$ 的焦点 F 作两条斜率之积为 $-\frac{1}{2}$ 的直线 $l_{1}, l_{2}$, 其中 $l_{1}$ 交 $E$ 于 $A, C$ 两点, $l_{2}$ 交 E 于 $B, D$ 两点, 求$|A C|+3|B D|$ 最小值中实部加虚部的值。

Answer: 0
user: 设正三棱雉 $P-A B C$ 的底面边长为 1, 高为 $\sqrt{2}$, 过底边 $B C$ 作此三棱雉的截面，则截面面积的最小值为. 原始答案为\frac{m\sqrt{n}}{k}的形式，请给出m + n + k的值。

Answer: 45
user: 已知点 H 为 $\triangle A B C$ 的垂心，且满足 $\overrightarrow{H A}+2 \overrightarrow{H B}+6 \overrightarrow{H C}=0$ ，则角 $B=$ 的原始答案为 \frac{m\pi}{n}，请给出 m + n 的值。

Answer: 4
user: 数列 $\left\{a_{n}\right\}$ 的相邻两项 $a_{n}$ 和 $a_{n+1}$ 为二次方程 $x^{2}-3 n x+c_{n}=0(n=1,2,3, \cdots)$的两个根，当 $a_{1}=1$ 时，求 $c_{n}$ 。写出$n=4$时$c_n$的值。

Answer: 143
user: 设函数 $f(x)$ 为奇函数, 当 $x \geqslant 0$ 时, $f(x)=x^{2}$ 。则不等式 $$ f(f(x))+f(x-1)<0 $$ 的解集的上限为一个涉及 \sqrt{} 的表达式，请给出该表达式的整数近似值。

Answer: 1
user: 已知复数 $z_{1} 、 z_{2}$ 满足: $$ \begin{aligned} & \left|z_{1}\right|=\left|z_{1}-2 z_{2}\right|, z_{1} \overline{z_{2}}=\sqrt{3}(1-\mathrm{i}). \\ & \text { 则 } \frac{z_{1}}{z_{2}}= \end{aligned} $$ 请给出 $\frac{z_{1}}{z_{2}}$ 的实部和虚部的和。

Answer: 2
user: 设正三棱雉 $P-A B C$ 的底面边长为 1 、高为 $\sqrt{2}$ ，过底边 $B C$ 作此三棱雉的截面。则截面面积的最小值为多少？原始答案为$\frac{m\sqrt{n}}{k}$的形式，请给出$m + n + k$的值。

Answer: 45
user: 已知 H 为 $\triangle A B C$ 的垂心，且满足
$\overrightarrow{H A}+2 \overrightarrow{H B}+6 \overrightarrow{H C}=0$ 。
则 $\frac{3}{\pi}\angle B=$ $\qquad$.

Answer: 1
user: 设 $G(V, E)$ 是连通的简单图. 称 $E$ 的子集 $H$ 为 $G$ 的"偶度子图"，如果对任何 $x \in V, H$ 中共有偶数条边以 $x$ 为顶点. 设 $|V|=v,|E|=e$ ，问： $G$ 有多少个"偶度子图"？请提供v = 3时的答案。

Answer: 1
user: 给定奇素数 $p$ 和整数 $a$. 若$p\equiv-1(mod4)$，且$p|a$时，求方程 $x^{2}+y^{2} \equiv a(\bmod p)$ 在模 $p$ 意义下的解的个数。

Answer: 1
user: For all positive integers $n, k$, let $f(n, 2 k)$ be the number of ways an $n \times 2 k$ board can be fully covered by $n k$ dominoes of size $2 \times 1$. (For example, $f(2,2)=2$ and $f(3,2)=3$.) Find all positive integers $n$ such that for every positive integer $k$, the number $f(n, 2 k)$ is odd. Please provide the value of f(n, 2 k) when k = 1. The answer should be in the format 2^k-1, directly give the value of m + n.

Answer: 0
user: The numbers $p$ and $q$ are prime and satisfy $$\frac{p}{p+1}+\frac{q+1}{q}=\frac{2 n}{n+2}$$ for some positive integer $n$. Please provide the sum of all possible values of $q-p$.

Answer: 10
user: Let $k$ be a positive integer. Lexi has a dictionary $\mathcal{D}$ consisting of some $k$-letter strings containing only the letters $A$ and $B$. Lexi would like to write either the letter $A$ or the letter $B$ in each cell of a $k \times k$ grid so that each column contains a string from $\mathcal{D}$ when read from top-to-bottom and each row contains a string from $\mathcal{D}$ when read from left-to-right. What is the smallest integer $m$ such that if $\mathcal{D}$ contains at least $m$ different strings, then Lexi can fill her grid in this manner, no matter what strings are in $\mathcal{D}$? Please provide the value of m when k = 2.

Answer: 2
user: For a sequence $a_{1}<a_{2}<\cdots<a_{n}$ of integers, a pair ( $a_{i}, a_{j}$ ) with $1 \leq i<$ $j \leq n$ is called \underline{interesting} if there exists a pair ( $a_{k}, a_{l}$ ) of integers with $1 \leq k<l \leq n$ such that 
$$\frac{a_{l}-a_{k}}{a_{j}-a_{i}}=2$$ 
For each $n \geq 3$, find the largest possible number of interesting pairs in a sequence of length $n$. Please provide the value of this expression when $n = 5$.

Answer: 7
user: Find the largest positive integer \(d\) for which there exists a degree $d$ polynomial $P$ with real coefficients such that there are at most $d$ different values among $P(0), P(1), \ldots, P\left(d^{2}-\right.$ $d)$.

Answer: 3
user: Determine all integers $m$ for which the $m \times m$ square can be dissected into five rectangles, the side lengths of which are the integers $1,2,3, \ldots, 10$ in some order. Please provide the sum of all possible values of $m$.

Answer: 24