Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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 $G$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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 $R+\frac{4\pi}{3}$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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 $\frac{\sqrt{30}}{6}|C-A|$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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 $m + n$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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 $m + n$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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 \(a\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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 $n = 5$

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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 $y=100$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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 $n = 5$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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 \(a + b\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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:
(1) $A_{1}^{2}=A_{2}^{2}=\ldots=A_{k}^{2}=0$,
(2) $A_{i} A_{j}=A_{j} A_{i}$ for all $1 \leq i, j \leq k$, and
(3) $A_{1} A_{2} \ldots A_{k} \neq 0$.
Please provide the sum of the smallest $n_k$ values for $k = 2$ and $k = 3$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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 \(m + n\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

$n^{2}(n \geqslant 4)$ 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。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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  $\qquad$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

$x$ 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。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

(i) A circle of radius $a$ rolls on the inner side of the circumference of a circle of radius $3a$. Find the area contained within the closed curve generated by a point on the circumference of the rolling circle. Let a=2，The original answer is in the format $m \pi$, Find the value of m.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Evaluate in closed form

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

Note:

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

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A player throwing a die scores as many points as on the top face of the die and is to play until his score reaches or passes a total $n$. Denote by $p(n)$ the probability of making exactly the total $n$, and find the value of $\lim _{n\to\infty} p(n)$. The original answer is in \frac{m}{n} format, please give the value of m + n.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

For a fixed positive integer $n$ let $x_{1}, x_{2}, \ldots, x_{n}$ be real numbers satisfying $0 \leq x_{k} \leq 1$ for $k=1,2, \ldots, n$. Determine the maximum value, as a function of $n$, of the sum of the $n(n-1) / 2$ terms:
$$
\sum_{\substack{i, j=1 \\ i<j}}^{n}\left|x_{i}-x_{j}\right|.
$$
Please provide the value of this sum when n = 3.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Find the volume of the four-dimensional hypersphere $x^{2}+y^{2}+z^{2}+t^{2}=r^{2}$, and also the hypervolume of its interior $x^{2}+y^{2}+z^{2}+t^{2}<r^{2}$. If \(r = 2\), given that the sum of these two parts can be written in the form of \(k\pi^{2}\), find the value of \(k\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在平面直角坐标系中, 设点集 \( \{P_{1}, P_{2}, \cdots, P_{4 n+1}\} \) = \{(x, y) \mid x, y \text{ 为整数, }|x| \leqslant n,|y| \leqslant n, x y=0\}, 其中， \(n \in \mathbf{N}_{+}\)。求 \( \left(P_{1} P_{2}\right)^{2}+\left(P_{2} P_{3}\right)^{2}+\cdots+\left(P_{n} P_{4 n+1}\right)^{2}+\left(P_{4 n+1} P_{1}\right)^{2} \) 的最小值。请给出 n = 1 时的结果。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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 $n_{1},\cdots, n_k$ 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 $G$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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. The answer is in the form $r + s\sqrt{t}$ with $r$, $s$, and $t$ as positive integers. Please provide the value of $r + s + t$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In how many ways can the integers from 1 to n 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? Please provide the number of ways for $n = 6$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $n$ be a positive integer. Let $a,b,x$ be real numbers, with $a \neq b$. and let $M_{n}$ denote the $2n \times 2n$ matrix whose $(i,j)$ entry $m_{ij}$ is given by $$\left.m_{ij}=\left\{\begin{array}{llll}x&\text{if}\quad i=j,\\a&\text{if}\quad i\neq j&\text{and}&i+j\text{is even},\\b&\text{if}\quad i\neq j&\text{and}&i+j\text{is odd}.\end{array}\right.\right.$$ Thus, for example, $M_2=\begin{pmatrix}x&b&a&b\\b&x&b&a\\a&b&x&b\\b&a&b&x\end{pmatrix}.$ Express $\lim_{x\rightarrow a}\det M_n/(x-a)^{2n-2}$ as a polynomial in $a,b$, and $n.$ where $\det M_n$ denotes the determinant of $M_n$. Please provide the value of this expression when $n = 3$, $a = 2$, and $b = 1$. The answer should be a single integer.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $P$ be a non-self-intersecting closed polygon with $n$ sides. Let its vertices be $P_1$, $P_{2},\cdots,P_{n}.$ Iet $m$ other points$,Q_1,Q_{2},\cdots,Q_{m}$ interior to $P$ be given. Let the figure be triangulated. This means that certain pairs of the $(n+m)$ points $P_1,\cdots,Q_m$ are connected by line segments such that (i) the resulting figure consists exclusively of a set $T$ of triangles, (ii) if two different triangles in $T$ have more than a vertex in common then they have exactly a side in common, and (iii) the set of vertices of the triangles in $T$ is precisely the set of ($n+m)$ points $P_{1},\cdots,Q_{m}.$ How many triangles in $T?$ Please provide the number of triangles when n = 5 and m = 4.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Evaluate the following limit: \lim _{n \rightarrow \infty} \int_{0}^{1} \int_{0}^{1} \cdots \int_{0}^{1} \cos ^{2}\left\{\frac{\pi}{2 n}\left(x_{1}+x_{2}+\cdots x_{n}\right)\right\} d x_{1} d x_{2} \cdots d x_{n}. The original answer is in the format $\frac{m}{n}$, please give the value of $m + n$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let R be the region consisting of all triples $(x,y,z)$ of nonnegative real numbers satisfying $x + y + z\leqslant 1$. Let $w=1-x-y-z$. Express the value of the triple integral  $$\iiint\limits_{R}x^{1}y^{9}z^{8}w^{4}dx dy dz$$  in the form $\frac{a!b!c!d!}{n!}$, where a, b, c, d, and n are positive integers. Please provide the integer result of this expression.Find the value of $a+b+c+d+n$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

给定一个 $2008 \times 2008$ 的棋盘，棋盘上每个小方格的颜色均不相同。在棋盘的每一个小方格中填入 $C 、 G 、 M 、 O$ 这 4 个字母中的一个, 若棋盘中每一个 $2 \times 2$ 的小棋盘中都有 $C 、 G 、 M 、 O$ 这 4 个字母, 则称这个棋盘为"和谐棋盘"。和谐棋盘有$a\times b^{c}-d$，求$a+b+c+d$的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 $S_{n}=1+2+3+\cdots+n, n \in \boldsymbol{N}^{*}$, 求 $f(n)=\frac{S_n}{(n^2+32)S_n^{+}1}$的最大值。原始答案是 $\frac{m}{n}$ 的形式，请给出 $m + n$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

for all polygons $P$, where the double integral is taken over $H(P)$. The original answer is in $\frac{m}{n}$ format, please give the value of $m + n$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Assume $F$ is a finite field with an odd number of elements $m$. Assume $p(x)$ is an irreducible (i.e., non-factorizable) polynomial in $F$ of the form
$$
x^{2}+b x+c, \quad b, c \in F
$$
For how many elements $k$ in $F$, $p(x)+k$ is irreducible in $F$? How many elements are there when $m=7$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A point P moves inside a unit square in a straight line at unit speed. When it meets a corner it escapes. When it meets an edge its line of motion is reflected so that the angle of incidence equals the angle of reflection. Let $N(T)$ be the number of starting directions from a fixed interior point $P_{0}$ for which P escapes within T units of time. Find the least constant a for which constants b and c exist such that $$ N(T) \leq a T^{2}+b T+c $$ for all $T>0$ and all initial points $P_{0}$. The original answer is in \(\pi\) format, please give the integer part of the answer.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $p$ be a prime number. Let $J$ be the set of all $2 \times 2$ matrices $\binom{a b}{c d}$ 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. The original answer is in the format of $p^2 + p$. Let p=5，Determine how many members $J$ has.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Three numbers are chosen independently at random, one from each of the three intervals $[0, L_i](i=1, 2,3)$. If the distribution of each random number is uniform with respect to length in the interval it is chosen from, determine the expected value of the smallest of the three numbers chosen. The original answer is in the format $\frac{a}{b} L_{1}^{2} L_{2} L_{3}-\frac{c}{d} L_{1}^{3}\left(L_{2}+L_{3}\right)+\frac{e}{f} L_{1}^{4}$, please give the value of a+b+c+d+e+f.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $F(x)$ be a real valued function defined for all real $x$ except for $x=0$ and $x=1$ and satisfying the functional equation $F(x)+F\{(x-1) / x\}=1+x$. Find all functions $F(x)$ satisfying these conditions. The original answer is in the format \frac{m}{n}, please provide the value of m + n when x = 2.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A particle moving on a straight line starts from rest and attains a velocity $v_0$ after traversing a distance $s_0$. If the motion is such that the acceleration was never increasing, find the maximum time for the traverse. 假设So=2,Vo=1,The original answer is in the format \frac{m}{n}, please give the value of m+n.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Determine the minimum value of
$$(r-1)^{2}+\left(\frac{s}{r}-1\right)^{2}+\left(\frac{t}{s}-1\right)^{2}+\left(\frac{4}{t}-1\right)^{2}$$
for all real numbers r, s, t with $1 \leq r \leq s \leq t \leq 4$.The original answer form is $a-b \ sqrt {c} $, find the value of a+b+c

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A sequence of convex polygons $\{P_n\}$, $n\geqslant0$, is defined inductively as follows. $P_0$ is an equilateral triangle with sides of length 1. Once $P_n$ has been determined, its sides are trisected; the vertices of $P_{n+1}$ are the interior trisection points of the sides of $P_n$. Thus, $P_{n+1}$ is obtained by cutting corners off $P_n$, and $P_n$ has $3\cdot2^n$ sides. ($P_1$ is a regular hexagon with sides of length $1/3$.) Express $\lim_{n\rightarrow\infty}Area(P_n)$ in the form $\sqrt{a}/b$, where a and b are positive integers. Please provide the value of a + b.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $V$ be the region in the cartesian plane consisting of all points $(x, y)$ satisfying the simultaneous conditions $$|x|\leqslant y\leqslant|x|+3\text{ and }y\leqslant4.$$ Find the centroid $(\bar{x},\bar{y})$ of $V$. The original answer for $\bar{y}$ is in the form \(\frac{m}{n}\). Please provide the value of  m + n$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Evaluate $$\lim\limits_{x\to1^-}\prod\limits_{n=0}^\infty\left(\frac{1+x^{n+1}}{1+x^n}\right)^{x^n}.$$ Please provide the value of a-b, where the original answer is in the form of \ln a-b.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $S={(a, b) \mid a=1,2,..., n, b=1,2,3}$. A rook tour of $S$ is a polygonal path made up of line segments connecting points $p_1, p_2,\dots,p_{3n}$ in sequence such that  (i) $p_i \in S$,  (ii) $p_i$ and $p_{i+1}$ are a unit distance apart, for $1 \le i< 3n$,  (iii) for each $p\in S$ there is a unique i such that $p_i =p$. How many rook tours are there that begin at $(1,1)$ and end at $(n,1)$? Please provide the value of the number of rook tours when n = 5.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Four points are chosen uniformly and independently at random in the interior of a given circle. Find the probability that they are the vertices of a convex quadrilateral. The original answer is in the format $1-\frac{m}{n\pi^2}$, please provide the value of m + n.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

For each continuous function $f:[0,1]\to\mathbb{R}$, let $I(f)=$ $\int _{0}^{1}x^{2}f( x)$ $dx$ and $J( x) = \int _0^{1}x\left ( f( x) \right ) ^{2}$ $dx.$ Find the maximum value of $I(f)-J(f)$ over all such functions $f.$ The original answer is in the format \frac{m}{n}, please give the value of m + n.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $k$ be an integer greater than 1. Suppose $a_0>0$, and define $$a_{n+1}=a_n+\frac{1}{\sqrt[k]{a_n}}$$ for $n>0.$ Evaluate $$\lim_{n\to\infty}\frac{a_n^{k+1}}{n^k}.$$ The original answer is in \left(\frac{k+1}{k}\right)^{k} format, please give the value of m + n where m/n is the simplified fraction form of the original answer when k = 1.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

What is the largest possible radius of a circle contained in a 4-dimensional hypercube of side length 1? The original answer is in the format $\frac{\sqrt{m}}{n}$, please provide the value of m + n.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设实数 $x_{1}, x_{2}, \cdots, x_{22}$ 满足对任意 $1 \leqslant i \leqslant$ 22, 有 $2^{i-1} \leqslant x_{i} \leqslant 2^{i}$ 。 求
$$
\left(x_{1}+x_{2}+\cdots+x_{22}\right)\left(\frac{1}{x_{1}}+\frac{1}{x_{2}}+\cdots+\frac{1}{x_{22}}\right)
$$
的最大值。原始答案为$(2^{12}-1-\frac{1}{2^{11}})^{2}$，请给出近似整数值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

给定整数 $n(n \geqslant 3)$ ，设 $A_{1}, A_{2}, \cdots, A_{2 n}$是集合 $\{1,2, \cdots, n\}$ 的两两不同的非空子集，记 $A_{2 n+1}=A_{1}$. 求  当$n=7$时，
$$
\sum_{i=1}^{2 n} \frac{\left|A_{i} \cap A_{i+1}\right|}{\left|A_{i}\right| \cdot\left|A_{i+1}\right|}
$$
的最大值.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

给定正整数 $n(n \geqslant 3)$ 。对于 $1,2, \cdots, n$的任意一个排列 $P=\left(x_{1}, x_{2}, \cdots, x_{n}\right)$, 若 $i<j<k$, 则称 $x_{j}$ 介于 $x_{i}$ 和 $x_{k}$ 之间（如在排列（ $1,3,2,4$ ）中，3 介于 1 和 4 之间， 4 不介于 1 和 2 之间）。设集合 $S=\left\{P_{1}, P_{2}, \cdots, P_{m}\right\}$的每个元素 $P_{i}$ 是 $1,2, \cdots, n$ 的排列. 已知 \{1,2, \cdots, n\} 的任意三个不同数中都有一个数，它在每个 $P_{i}(1 \leqslant i \leqslant m)$ 中都不介于另外两个数之间。求 $m$ 的最大值。请给出当 $n = 5$ 时，$m$ 的最大值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

定义 $n$ 元整数组的一次变换为
$$
\begin{aligned}
& \left(a_{1}, a_{2}, \cdots, a_{n-1}, a_{n}\right) \\
& \rightarrow\left(a_{1}+a_{2}, a_{2}+a_{3}, \cdots, a_{n-1}+a_{n}, a_{n}+a_{1}\right)
\end{aligned}
$$
求所有的正整数对 $(n, k)(n 、 k \geqslant 2)$ ，满足对任意的 $n$ 元整数数组 $\left(a_{1}, a_{2}, \cdots, a_{n}\right)$ ，在有限次变换后所得数组中每一个数均为 $k$ 的倍数。请给出最小的n和k的和。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

将关于 $x$ 的多项式 $f(x)=1-x+x^{2}-x^{3}+\cdots-x^{19}+x^{20}$ 表为关于 $y$ 的多项式 $g(y)=a_{0}+a_{1} y+a_{2} y^{2}+\cdots$ $+a_{19} y^{19}+a_{20}{ }^{20}$, 其中 $y=x-4$, 则 $6(a_{0}+a_{1}+\cdots+a_{20})=$ $\qquad$

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知过点 $(0,1)$ 的直线 $l$ 与曲线 $C: y=x+\frac{1}{x}(x>0)$ 交于两个不同点 $M$ 与 $N$, 求曲线 $C$ 在点 $M 、 N$ 处切线的交点轨迹。请提供这两个交点坐标的所有整数部分之和。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知 $p, q(q \neq 0)$ 是实数, 方程 $x^{2}-p x+q=0$ 有两个实根 $\alpha, \beta$,数列 $\left\{a_{n}\right\}$ 满足 $a_{1}=p, a_{2}=p^{2}-q, \quad a_{n}=p a_{n-1}-q a_{n-2}(n=3,4, \cdots)$\n若 $p=1, q=\frac{1}{4}$, 求 $\left\{a_{n}\right\}$ 的前 $n$ 项和。请给出 $s_n$ 当 $n=2$ 时的值，并计算出结果为 $\frac{m}{n}$ 格式，给出 $m+n$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

函数 $f(x)=a^{2 x}+3 a^{x}-2(a>0, a \neq 1)$ 在区间 $x \in[-1,1]$ 上的最大值为 8, 则它在这个区间上的最小值是 。已知原始答案的形式为$\frac{m}{n}$，请给出m + n的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知 $x>0, y>0$, 且满足 \(\left\{\begin{array}{l}\cos ^{2}(\pi x)+2 \sin (\pi y)=1 &\quad①\\ \sin (\pi x)+\sin (\pi y)=0&\quad ②, \\ x^{2}-y^{2}=12&\quad③\end{array}\right.\)
求有序数对 \((x, y)\) 中的 x + y 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知函数 $f(x)=a \sin x-\frac{1}{2} \cos 2 x+a-\frac{3}{a}+\frac{1}{2}, a \in \mathbf{R}, a \neq 0$.\n（1）若对任意 $x \in \mathbf{R}$, 都有 $f(x) \leqslant 0$, 求 $a$ 的取值范围;\n（2）若 $a \geqslant 2$ ，且存在 $x \in \mathbf{R}$ ，使得 $f(x) \leqslant 0$ ，求 $a$ 的取值范围。\n请提供所有区间端点的和。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

复平面上，非零复数 $z_{1}, z_{2}$ 在以 $i$ 为圆心， 1 为半径的圆上， $\overline{z_{1}} z_{2}$ 的实部为零， $z_{1}$ 的辐角主值为 $\frac{\pi}{6}$ ，则 $z_{2}=$。请注意，$z_{2}$ 的原始答案为$-\frac{\sqrt{3}}{2}+\frac{3}{2}i$，请给出虚部分子和分母之和。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在 $\triangle A B C$ 中, $A B=1, A C=2, \cos B=2 \sin C$, 则 $B C$ 的长为。原始答案的格式为 $\frac{\sqrt{m}+\sqrt{n}}{k}$，请给出 m + n + k 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 \(\alpha, \beta \in \mathbf{R}\), 若 \(\tan (\alpha+\beta)=2, \tan (\alpha+2 \beta)=3\), 则 \(\tan \alpha\) 的值为\(\frac{m}{n}\)的格式，请给出\(m + n\)的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

给定实数 $r$ ，甲、乙两人玩如下的游戏. 黑板上写着一个含有三个绝对值的算式： $$ S=|\square-\square|+|\square-\square|+|\square-\square|, $$ 其中 6 个空格 "$\square$"中尚未填数.每一回合，甲选取区间 $[0,1]$ 中的一个实数（不同回合中可以选相同的数），乙将该数填在某个空格之中.经过六个回合之后所有 6 个空格中均填了数， $S$ 的值也随之确定.若 $S \geq r$ ，则甲胜，否则乙胜. 求所有的实数 $r$ ，使得甲有获胜策略. 原始答案为 $\frac{m}{n}$ 的格式，请给出 m + n 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知数列 $\left\{a_{n}\right\}$ 的各项均为非负实数, 且满足：对任意整数 $n \geq 2$, 均有 $a_{n+1}=a_{n}-a_{n-1}+n$. 若 $a_{2} a_{2022}=1$, 求 $a_{1}$ 的最大可能值。原始答案为\frac{m}{n}格式，请给出m + n的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若正数 $a, b$ 满足 $\log _{2} a+\log _{4} b=8, \log _{4} a+\log _{8} b=2$, 则 $\log _{8} a+\log _{2} b$ 的值为。请给出原始答案中分子与分母绝对值之和的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 $a$ 为实数. 若存在实数 $t$, 使得 $\frac{a-\mathrm{i}}{t-\mathrm{i}}+\mathrm{i}$ 为实数 ( $\mathrm{i}$为虚数单位), 则 $a$ 的取值范围是 $\qquad$. 求出4a的最大值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设函数 $f(x)=\frac{x^{2}+x+16}{x}(2 \leq x \leq a)$, 其中实数 $a>2$. 若 $f(x)$ 的值域为 $[9,11]$, 则 $a$ 的取值范围是一个区间$[p, q]$. 请给出$p + q$的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

不等式 $\frac{20}{x-9}>\frac{22}{x-11}$ 的解集为 $\qquad$，标准答案格式为$(-\infty,a) \cup(b,c)$请给出解集的区间端点a+b+c和的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设正数 $a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}$ 满足: $a_{1}, a_{2}, a_{3}$ 成公差为 $b_{1}$ 的等差数列, $b_{1}, b_{2}, b_{3}$ 成公比为 $a_{1}$ 的等比数列, 且 $a_{3}=b_{3}$. 求 $a_{3}$ 的最小值, 并确定当 $a_{3}$ 取到最小值时 $a_{2} b_{2}$ 的值。原始答案分别为$\frac{m\sqrt{n}}{k}$的形式，请给出m+n+k的和。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

一枚不均匀的硬币，若随机抛掷它两次均得到正面的概率为 \frac{1}{2} ，则随机抛掷它两次得到正面、反面各一次的概率的近似整数值是多少？

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知复数 $z$ 满足 $|z|=1$, 且 $\operatorname{Re} \frac{z+1}{\bar{z}+1}=\frac{1}{3}$, 请给出 \operatorname{Re} \frac{Z}{\bar{z}} 的值为 -\frac{m}{n} 时的 m+n 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若正四棱雉 $P-A B C D$ 的各条棱长均相等, $M$ 为棱 $A B$ 的中点, 则异面直线 $B P$ 与 $C M$ 所成的角的余弦值为 \qquad。已知原始答案为 \frac{\sqrt{m}}{n} 的格式，请给出 m + n 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

给定正实数 $a, b, a<b$. 设 $x_{1}, x_{2}, \cdots, x_{2022} \in[a, b]$,求
$$
\frac{\left|x_{1}-x_{2}\right|+\left|x_{2}-x_{3}\right|+\cdots+\left|x_{2021}-x_{2022}\right|+\left|x_{2022}-x_{1}\right|}{x_{1}+x_{2}+\cdots+x_{2022}}
$$
的最大值. 题目原始答案为 \frac{m}{n} 形式，当a=b=1,请给出 m +n的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

求出所有满足下面要求的不小于 1 的实数 $t$ ：对任意 $a, b \in[-1, t]$, 总存在 $c, d \in[-1, t]$, 使得 $(a+c)(b+d)=1$. 答案标准形式为\frac{a}{b},请给出满足条件的 $t$ 的区间两端点a+b之和。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

方程 $\sin x=\cos 2 x$ 的最小的 20 个正实数解之和的原始答案是 $m \pi$ 的形式，请给出 $m$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 $a=1+10^{-4}$ . 在 $2023 \times 2023$ 的方格表的每个小方格中填入区间 $[1, a]$ 中的一个实数. 设第 $i$ 行的总和为 $x_{i}$ ，第 $i$ 列的总和为 $y_{i}$ ， $1 \leq i \leq 2023$. 求 \frac{y_{1} y_{2} \cdots y_{2023}}{x_{1} x_{2} \cdots x_{2023}} 的最大值。原始答案为\frac{m}{n}的形式，当a=1时请给出m+n的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 $F_{1}, F_{2}$ 为椭圆 $\Omega$ 的焦点, 在 $\Omega$ 上取一点 $P$ (异于长轴端点), 记 $O$ 为 $\Delta P F_{1} F_{2}$ 的外心, 若 $\overrightarrow{P O} \cdot \overrightarrow{F_{1} F_{2}}=2 \overrightarrow{P F_{1}} \cdot \overrightarrow{P F_{2}}$, 则 $\Omega$ 的离心率的最小值为 $\frac{\sqrt{m}}{n}$ 的形式，请给出 $m + n$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设复数 $z$ 满足 $|z|=1$ ，使得关于 $x$ 的方程 $z x^{2}+2 \bar{z} x+2=0$ 有实根，则这样的复数 $z$ 的和为以分数形式表示，请给出分子与分母的和。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

$[2 \pi-6,4-\pi]$

设 $f(x)$ 是定义在 $\mathbf{R}$ 上的以 2 为周期的偶函数, 在区间 $[1,2]$ 上严格递减, 且满足 $f(\pi)=1, f(2 \pi)=0$, 则不等式组 $\left\{\begin{array}{l}0 \leqslant x \leqslant 1, \\ 0 \leqslant f(x) \leqslant 1\end{array}\right.$ 的解集为. 请给出区间左端点与两倍右端点之和的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在平面直角坐标系 $x O y$ 中, 圆 $\Omega$ 与抛物线 $\Gamma: y^{2}=4 x$ 恰有一个公共点, 且圆 $\Omega$ 与 $x$轴相切于 $\Gamma$ 的焦点 $F$ 。已知原始答案为 \( \frac{4 \sqrt{3}}{9} \) 的格式，请给出 $m + n + k$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

称一个复数列 $\left\{z_{n}\right\}$ 是"有趣的"，若 $\left|z_{1}\right|=1$ ，且对任意正整数 $n$ ，均有 $4 z_{n+1}^{2}+$ $2 z_{n} z_{n+1}+z_{n}^{2}=0$ 。求最大的常数 $C$ ，使得对一切有趣的数列 $\left\{z_{n}\right\}$ 及任意正整数 $m$ ，均有 $\left|z_{1}+z_{2}+\cdots+z_{m}\right| \geqslant C$ 。原始答案为\frac{\sqrt{m}}{n}的形式，请给出m + n的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在平面直角坐标系中，设$\vec{e}$是一个单位向量，向量$\vec{a}$满足$\vec{a} \cdot \vec{e}=2$，并且$|\vec{a}|^{2} \leqslant 5|\vec{a}+t \vec{e}|$对任意实数$t$成立。请提供$|\vec{a}|$的取值范围左端点的两倍与右端点之差。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 $a>0$, 函数 $f(x)=x+\frac{100}{x}$ 在区间 $(0, a]$ 上的最小值为 $m_{1}$, 在区间 $[a,+\infty)$ 上的最小值为 $m_{2}$, 若 $m_{1} m_{2}=2020$, 则 $a$ 的值为。请提供两个可能值的和。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知集合 $A=\{1,5\}, B=\left\{x^{2}-3 x+1, x^{2}-4 x+5\right\}$ ，若集合 $A \cup B$ 的元素个数是 3 ，请给出所有可能的实数 $x$ 的和。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在 $\triangle A B C$ 中, $\cos A, \sin A, \tan B$ 依次构成以 $\frac{3}{4}$ 为公比的等比数列。请给出 $\cot C$ 的值，其原始答案为 \frac{m}{n} 格式，请给出 m + n 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在 $\triangle A B C$ 中, $a=B C, b=C A, c=A B$. 若 $a^{2}+b^{2}=6, \cos (A-B) \cos C=\frac{2}{3}$,求 $\triangle A B C$ 的外接圆半径。原始答案为 \frac{m\sqrt{n}}{p} 的形式，请给出 m + n + p 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 $a, b \in \mathbf{R}$ .若关于 $z$ 的方程
$$
\left(z^{2}+a z+b\right)\left(z^{2}+a z+2 b\right)=0
$$
有 4 个互不相等的复数根 $z_{1}, z_{2}, z_{3}, z_{4}$, 且它们在复平面上对应的点恰是一个边长为 1 的正方形的四个顶点，求 $\left|z_{1}\right|+\left|z_{2}\right|+\left|z_{3}\right|+\left|z_{4}\right|$ 的值。原始答案为\sqrt{m} + n\sqrt{p}的形式，请给出m + n + p的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设正四棱锥 $P-A B C D$ 的底面边长与高相等, 点 $G$ 为侧面 $\triangle P B C$ 的重心,则直线 $A G$ 与底面 $A B C D$ 所成角的正弦值的原始答案为 \(\frac{\sqrt{m}}{n}\) 的形式，请给出 m + n 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

复数 $z_{1}, z_{2}, \cdots, z_{100}$ 满足: $z_{1}=3+2 \mathrm{i}, z_{n+1}=\overline{z_{n}} \cdot \mathrm{i}^{n}(n=1,2, \cdots, 99)$ （ i 为虚数单位), 请给出 $z_{99}+z_{100}$ 的实部与虚部之和的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 $i=\sqrt{-1}$ 为虚数单位, 则 $i+2 i^{2}+3 i^{3}+\cdots+2013 i^{2013}$ 的实部与虚部的和为多少。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知正三棱锥 $P-A B C$ 的底面边长为 1, 高为 $\sqrt{2}$, 则其内切球半径是一个分数，形式为 \frac{m}{n}。请给出 m + n 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在 $\triangle A B C$ 中, 已知 $\overrightarrow{A B} \cdot \overrightarrow{A C}+2 \overrightarrow{B A} \cdot \overrightarrow{B C}=3 \overrightarrow{C A} \cdot \overrightarrow{C B}$. 求 $\sin C$ 的最大值。原始答案是 \frac{\sqrt{m}}{n} 的格式，请给出 m + n 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 $P$ 为一圆锥的顶点, $A, B, C$ 是其底面圆周上的三点, 满足 $\angle A B C=90^{\circ}$, $M$ 为 $A P$ 的中点. 若 $A B=1, A C=2, A P=\sqrt{2}$, 则二面角 $M-B C-A$ 的大小为 \arctan \frac{m}{n} 的形式，请给出 m + n 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在 $\triangle A B C$ 中, 已知 $\overrightarrow{A B} \cdot \overrightarrow{A C}+2 \overrightarrow{B A} \cdot \overrightarrow{B C}=3 \overrightarrow{C A} \cdot \overrightarrow{C B}$. 求 $\sin C$ 的最大值。原始答案为 \frac{\sqrt{m}}{n} 的形式，请给出 m + n 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知正四棱锥 $V-A B C D$ 的高等于 $A B$ 长度的一半， $M$ 是侧棱 $V B$ 的中点， $N$ 是侧棱 $V D$ 上的点，满足 $D N=2 V N$ ，则异面直线 $A M, B N$ 所成角的余弦值的原始答案是 $\frac{\sqrt{m}}{n}$ 格式，请给出 m + n 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 $x_{1}, x_{2}, x_{3}$ 是非负实数, 满足 $x_{1}+x_{2}+x_{3}=1$, 求 $(x_{1}+3 x_{2}+5 x_{3})(x_{1}+\frac{x_{2}}{3}+\frac{x_{3}}{5})$的最小值和最大值。最大值的答案形式为 \frac{m}{n}，请计算最小值+m+n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在平面直角坐标系 $x O y$ 中, 椭圆 $C$ 的方程为 \frac{x^{2}}{9}+\frac{y^{2}}{10}=1, F$ 为 $C$ 的上焦点, $A$为 $C$ 的右顶点, $P$ 是 $C$ 上位于第一象限内的动点, 则四边形 $O A P F$ 的面积的最大值为 \frac{m \sqrt{n}}{k} 的形式，请给出 m + n + k 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设两个严格递增的正整数数列 $\left\{a_{n}\right\},\left\{b_{n}\right\}$ 满足: $a_{10}=b_{10}<2017$ ，对任意正整数 $n$ ，有 $a_{n+2}=a_{n+1}+a_{n}, b_{n+1}=2 b_{n}$, 请给出 $a_{1}+b_{1}$ 的所有可能值的和。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设数列 \left\{a_{n}\right\} 定义为 a_{1}=1 ，\n\na_{n+1}= \begin{cases}a_{n}+n, & a_{n} \leqslant n, \\ a_{n}-n, & a_{n}>n,\end{cases}\n求满足 a_{r}<r \leqslant 3^{2017} 的正整数 r 的个数。\n\n原始答案为 \frac{3^{m}+k}{n} 格式，请给出 m + n+k 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 $x_{1}, x_{2}, x_{3}$ 是非负实数，满足 $x_{1}+x_{2}+x_{3}=1$ ，求 $\left(x_{1}+3 x_{2}+5 x_{3}\right)\left(x_{1}+\frac{x_{2}}{3}+\frac{x_{3}}{5}\right)$的最小值和最大值。原始答案的最大值是以\frac{m}{n}的形式，请给出m+n+最小值的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在平面直角坐标系 $x O y$ 中, 粗圆 $C$ 的方程为 $\frac{x^{2}}{9}+\frac{y^{2}}{10}=1, F$ 为 $C$ 的上焦点, $A$ 为 $C$ 的右顶点, $P$ 是 $C$ 上位于第一象限内的动点, 则四边形 $O A P F$ 的面积的最大值为 \frac{m}{n}\sqrt{k} 的形式，请给出 m + n + k 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

正三棱锥 $P-A B C$ 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 \frac{m\sqrt{k}}{n} 的形式，请给出 m + n + k 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设两个严格递增的正整数数列 $\left\{a_{n}\right\},\left\{b_{n}\right\}$ 满足: $a_{10}=b_{10}<2017$, 对任意正整数 $n$ ，有 $a_{n+2}=a_{n+1}+a_{n}, b_{n+1}=2 b_{n}$ ，请给出 $a_{1}+b_{1}$ 的所有可能值的和。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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 $a$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 $Q$ 为有理数集, $Z$ 为整数集. 在坐标平面上，对正整数 $m$ ，定义点集
$$
A_{m}=\left\{(x, y) \mid x, y \in \mathbf{Q}, x y \neq 0, \frac{x y}{m} \in \mathbf{Z}\right\}.
$$
对线段 $M N$, 定义 $f_{m}(M N)$ 为线段 $M N$ 上属于集合 $A_{m}$ 的点的个数. 求最小的实数 $\lambda$,使得对坐标平面上的任意直线 $l$, 均存在与 $l$有关的实数 $\beta(l)$ ，满足：对直线 $l$ 上任意两点 $M 、 N$,均有
$f_{2016}(M N) \leqslant \lambda f_{2015}(M N)+\beta(l)$.
给出原始答案的形式为\frac{m}{n}，请提供m + n的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

给定整数 $n \geqslant 2$ 。设 $x_{1}, x_{2}, \cdots, x_{n}$ 为任意实数. 求
$$
2 \sum_{1 \leqslant i<j \leqslant n}\left[x_{i} x_{j}\right]-(n-1) \sum_{i=1}^{n}\left[x_{i}^{2}\right]
$$
的最大值，其中， $[x]$ 表示不超过实数 $x$ 的最大整数。请给出当 n = 5 时的最大值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

求所有的正整数对 $(x, y)$ ，满足 $x^{y}=y^{x^{-y}}$ 。请给出所有满足条件的整数对中元素的和。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The Imomi archipelago consists of $n \geqslant 2$ islands. Between each pair of distinct islands is a unique ferry line that runs in both directions, and each ferry line is operated by one of $k$ companies. It is known that if any one of the $k$ companies closes all its ferry lines, then it becomes impossible for a traveller, no matter where the traveller starts at, to visit all the islands exactly once (in particular, not returning to the island the traveller started at). Determine the maximal possible value of $k$ in terms of $n$. Please provide the value of $k$ when $n = 8$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $a_{1}, a_{2}, \\ldots, a_{n}, b_{1}, b_{2}, \\ldots, b_{n}$ be $2 n$ positive integers such that the $n+1$ products $$\begin{gathered} a_{1} a_{2} a_{3} \\cdots a_{n} \\ b_{1} a_{2} a_{3} \\cdots a_{n} \\ b_{1} b_{2} a_{3} \\cdots a_{n} \\ \vdots \\ b_{1} b_{2} b_{3} \\cdots b_{n} \end{gathered}$$ form a strictly increasing arithmetic progression in that order. Determine the smallest positive integer that could be the common difference of such an arithmetic progression.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

求实数 $a$ 的取值范围，使得对任意实数 $x$ 和任意 $\theta \in\left[0, \frac{\pi}{2}\right]$ ，恒有$$(x+3+2 \sin \theta \cos \theta)^{2}+(x+a \sin \theta+a \cos \theta)^{2} \geqslant \frac{1}{8}$$
若$a$的取值范围是$a\leq\sqrt{m}\text{或 }a\geqslant\frac{p}{q}$,求$m+p+q$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知复数 $z=1-\sin \theta+i \cos \theta\left(\frac{\pi}{2}<\theta<\pi\right)$, 求 $z$ 的共轭复数 $\bar{z}$ 的辐角主值为$\frac{a\pi}{b}-\frac{c\theta}{d}$,求$a+b+c+d$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知 $\theta=\arctan \frac{5}{12}$, 那么复数 $z=\frac{\cos 2 \theta+i \sin 2 \theta}{239+i}$ 的辐角主值是。原始答案为 $\frac{m\pi}{n}$ 的格式，请给出 $m + n$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知点 $P$ 在双曲线 $\frac{x^{2}}{16}-\frac{y^{2}}{9}=1$ 上, 并且 $P$ 到这条双曲线的右准线的距离恰是 $P$ 到这条双曲线的两个焦点的距离的等差中项，那么 $P$ 的横坐标是 $\qquad$. 原始答案为 $\frac{m}{n}$ 的形式，请给出 $m + n$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知点 $P$ 在双曲线 $\frac{x^{2}}{16}-\frac{y^{2}}{9}=1$ 上, 并且 $P$ 到这条双曲线的右准线的距离恰是 $P$ 到这条双曲线的两个焦点的距离的等差中项。那么, $P$ 的横坐标的原始答案是 $\frac{m}{n}$ 的形式，请给出 $m + n$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

给定正整数 $n$ 和正数 $M$ 。对于满足条件 $a_{1}^{2}+a_{n+1}^{2} \leqslant M$ 的所有等差数列 $a_{1}, a_{2}, a_{3}, \cdots$, 试求 $S=a_{n+1}+a_{n+2}+\cdots+a_{2 n+1}$ 的最大值.写出$M=10,n=5$时$S$的最大值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

如图3，在直角坐标系 $x O y$中， $\\odot O: x^{2}+y^{2}=4$ 与 $x$ 轴的正半轴交于点 $A, \\odot A:(x-2)^{2}+y^{2}=r^{2}(r>0)$ 与 \\odot O 交于 $B 、 C$ 两点. (1) 求 \\overrightarrow{A B} \\cdot \\overrightarrow{A C} 的最小值; (2) 设 $P$ 为 \\odot O 上异于 $B 、 C$ 的任一点，直线 $P B 、 P C$ 与 $x$ 轴分别交于点 $M 、 N$, 求 $S_{\\triangle P O M} S_{\\triangle P O N}$ 的最大值。请提供两个部分答案之和。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

一项考试的可能得分为 $0,1,2, \cdots, 150$ ，有 100 名考生 $P_{1}, P_{2}, \cdots, P_{100}$ 考完后依顺时针围成一圈交流成绩，记 $P_{i}$ 的得分为 $a_{i}$ 。每个考生 $P_{i}$ 比较自己与相邻两人 $P_{i-1}, P_{i+1}$ (下标按模 100 理解) 的得分，定义 $P_{i}$ 的激励值 $f_{i}$ 为:
$f_{i}=\begin{cases}a_{i}(a_{i-1}+a_{i+1}-a_{i}),&a_{i}<\min\{a_{i-1},&a_{i+1}\},\\a_{i}(\max\{a_{i-1},a_{i},a_{i+1}\}-a_{i}),&a_{i}\geq\min\{a_{i-1},&a_{i+1}\}.\end{cases}$
记 $S=f_{1}+f_{2}+\cdots+f_{100}$.
(1) 求 $S$ 的最大值;
(2) 求使得 $f_{1}, f_{2}, \cdots, f_{100}$ 两两不等的 $S$ 的最大值。请给出这两个最大值的和。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

对任意的正整数 $n$, 记 $D_{n}$ 为 $n$ 的正约数全体， $f_{i}(n)(i=1,2)$ 为集合 $$ F_{i}(n)=\left\{a \in D_{n} \mid a \equiv i(\bmod 4)\right\} $$ 的元素个数. 求最小的正整数 $m$,使得 $2 f_{1}(m)-f_{2}(m)=2017$. $m$的形式满足$k \times p^{q}$,求$k+p+q$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

对正整数 $n$ ，定义 $n!=1 \times 2 \times \cdots \times n$ 。求所有的正整数 $x 、 y$, 使得 $3^{x}+x^{4}=y!+2019$ 。请给出所有解的x + y的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

求最大的正整数 $m$ ，使得存在集合 $\\{1,2, \\ldots, 100\\}$ 的 $m$ 个子集 $A_{1}, A_{2}$ ， $\\ldots, A_{m}$ 满足：对任意 $1 \\leq i<j \\leq m$ ，都有  $$  \\left|A_{i} \\bigcup A_{j}\\right| - \\left|A_{i} \\bigcap A_{j}\\right|=2.$$  请给出该最大正整数的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

有 $n$ 个人，已知他们中的任意两人至多通电话一次，他们中的任意 $n-2$ 个人之间通电话的次数相等, 都是 $3 k$ 次, 其中 $k$ 是自然数, 求 $n$ 的可能值的和.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

一个球与正四面体的六条棱都相切。若正四面体的棱长为 $a$，则这个球的体积的原始答案是\frac{\sqrt{a}\pi a^b}{c}的格式，请给出a+b+c的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 $S_{n}=1+2+3+\cdots+n, n \in \mathbf{N}$. 求 $f(n)=\frac{S_{n}}{(n+32) S_{n+1}}$ 的最大值。原始答案为\frac{m}{n}的格式，请给出m + n的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若函数 $f(x)=-\frac{1}{2} x^{2}+\frac{13}{2}$ 在区间 $[a, b]$ 上的最小值为 $2 a$, 最大值为 $2 b$, 求 $[a, b]$. 标准答案格式为$[a,b]=[m,n]$ 或 $\Big[-a-\sqrt{b},\frac{c}{d}\Big]$，请提供区间 $[a, b]$ 的端点m+n+a+b+c+d和。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若复数 $z_{1}, z_{2}$ 满足 $\left|z_{1}\right|=2,\left|z_{2}\right|=3,3 z_{1}-2 z_{2}=\frac{3}{2}-i$, 则 $z_{1} \cdot z_{2}$ 的原始答案是 $-\frac{a}{b} + \frac{c}{d} i$ 的形式，请给出a+b+c+d和的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

两个半径都是 $1$ 的球 $O_{1}$ 和球 $O_{2}$ 相切，且均与直二面角 $\alpha-l-\beta$ 的两个半平面都相切，另有一个半径为 $r(r<1)$ 的小球 $O$与这个二面角的两个半平面也都相切, 同时,与球 $O_{1}$ 和球 $O_{2}$ 都外切。若原始答案的形式为 $m - \sqrt{n}$，请给出 m + n 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若实数 $x$、$y$ 满足 $x^2+y^2=1$,则$\frac{2xy}{x+y-1}$的最小值是。请给出答案中$a-\sqrt{b}$的a+b之和。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 $0<\alpha<\pi, \pi<\beta<2 \pi$. 若对任意的 $x \in \mathbf{R}$ ，都有 $$ \cos (x+\alpha)+\sin (x+\beta)+\sqrt{2} \cos x=0 $$ 恒成立，试求 \alpha, \beta 的值。给出 \alpha 和 \beta 的形式为 \frac{m\pi}{n} 和 \frac{p\pi}{q}，请给出 m+n+p+q 的值。。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

对于 $n \in \mathbf{N}_{+}$，若 $2^{n} n+1$ 是 3 的整数倍，则 $n$ 被 6 除所得余数构成的集合是_____. 请给出集合中所有元素的和。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

如图1，在第一象限内,矩形 $A B C D$ 的三个顶点 $A 、 B 、 C$分别在函数 $y=\log _{\frac{\sqrt{2}}{2}} x 、 y=x^{\frac{1}{2}} 、y=\left(\frac{\sqrt{3}}{2}\right)^{x}$ 的图像上, 且矩形的边分别平行于两坐标轴. 若点 A的纵坐标为 2, 则点 D 的横、纵坐标之和为$\frac{b}{a}$ ，求$a+b$的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

满足

$\frac{1-\sin \theta+\cos \theta}{1-\sin \theta-\cos \theta}+\frac{1-\sin \theta-\cos \theta}{1-\sin \theta+\cos \theta}=2$

的最大负角 $\theta$ 的弧度数为 $-\frac{\pi}{a}$，求$a$的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知集合 $M=\{1,3,5,7,9\}$. 若非空集合 $A$ 满足: $A$ 中各元素都加 4 后构成 $M$ 的一个子集， $A$ 中各元素都减 4 后也构成 $M$ 的一个子集。请给出集合 $A$ 的元素之和。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知等差数列 $\left\{a_{n}\right\} 、\left\{b_{n}\right\}$ 的前 $n$ 项和分别为 $S_{n} 、 T_{n}$, 且对于一切正整数 $n$, 均有 $\frac{a_{n}}{b_{n}}=\frac{2 n-1}{3 n+1}$.则 $\frac{S_{6}}{T_{5}}$的 结果为分数形式，请给出分子与分母之和。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知函数 $f(x)=x \ln x-a x^{2}, a \in \mathbf{R}$，设函数 $F(x)=|f(x)|(x \in[1, \mathrm{e}])$ 有极小值,  $a$ 的取值范围是$\left(a, \frac{b}{\mathrm{e}}\right) \cup\left(\frac{b}{\mathrm{e}}, \frac{b}{c}\right)$，求$a+b+c$的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

用 $6$ 根等长的细铁棒焊接成一个正四面体形框架，铁棒的粗细和焊接误差不计。设此框架能容纳得下的最大球的半径为 $R_{1}$,能包容此框架的最小球的半径为 $R_{2}$ 。则 $\frac{R_{1}}{R_{2}}$等于。原始答案为 $\frac{\sqrt{m}}{n}$ 格式，请给出 $m + n$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知 $a$ 为实数, 且对任意 $k \in[-1,1]$, 当 $x \in(0,6]$ 时, $6 \ln x+x^{2}-8 x+a \leqslant k x$ 恒成立, 则 $a$的最大值是$m-m \ln m$，求$m$的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设点 $A$ 的坐标为 $(0,3)$, 点 $B, C$ 为圆 $O: x^{2}+y^{2}=25$ 上的两动点, 满足 $\angle B A C=90^{\circ}$.  $\triangle A B C$ 面积的最大值为$\frac{a+b \sqrt{c}}{d}$，求$a+b+c+d$的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知正实数 $x, y$ 满足: $\frac{1}{x+3 y}+\frac{1}{2 x+y}=1$, 则 $x+y$ 的最小值是$\frac{a}{b}(c+d \sqrt{d})$，求$a+b+c+d$的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设函数 $f(x)=x-\ln (a x+2 a+1)+2$. 若对于任意 $x \geqslant-2$, 均有 $f(x) \geqslant 0$ 恒成立，请给出实数 $a$ 的取值范围端点之和。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知实数 $x, y$ 满足 $x|x|+\frac{y|y|}{3}=1$, 则 $|\sqrt{3} x+y-4|$ 的取值范围为一个区间$[p, q)$，请给出p + q的整数部分的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若 $\triangle A B C$ 的三边 $a, b, c$ 满足 $a^{2}+b^{2}+3 c^{2}=7$, 则 $\triangle A B C$ 面积的最大值为。原始答案为\frac{\sqrt{m}}{n}的格式，请给出m + n的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

给定正整数 $n(n \geqslant 2)$. 已知 $2 n$ 个正实数 $a_{1}, a_{2}, \cdots, a_{2 n}$, 满足: $$ \sum_{k=1}^{n} a_{2 k-1} \cdot \sum_{k=1}^{n} a_{2 k}=\prod_{k=1}^{n} a_{2 k-1}+\prod_{k=1}^{n} a_{2 k}. $$ 求 $S=\sum_{k=1}^{2 n} \frac{a_{k}^{n-1}}{a_{k+1}}$ 的最小值，其中 $a_{2 n+1}=a_{1}$ 。请给出当 $n = 2$ 时的最小值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

给定正整数 $a, b(a \leqslant b)$. 数列 $\left\{f_{n}\right\}$ 满足: $$ f_{1}=a, f_{2}=b, f_{n+2}=f_{n+1}+f_{n}(n=1,2, \cdots) $$ 若对任意的正整数 $n$, 都有 $\left(\sum_{k=1}^{n} f_{k}\right)^{2} \leqslant \lambda \cdot f_{n} f_{n+1}$, 求实数 $\lambda$ 的最小值。原始答案为 $n+\sqrt{m}$，请给出 $n+m$ 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

如图1，已知正方形 ABCD 的边长为 3 $E$为$DC$ 的中点$,AE$ 与 $BD$交于点 $F.$ 则$
\overrightarrow{FD}\cdot\overrightarrow{DE}$的原始答案为$-\frac{m}{n}$格式，请给出m + n的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设等比数列 ${an}$的前$n$项和 为$S_n$ ，且$a_{n+1}=2S_{n}+\frac{1}{2}(n\in\mathbf{Z}_{+}).$\求数列 ${an}$的通项公式。已知原始答案为$\frac{a}{b}\times c^{n-1}$的形式，请给出$a + b + c$的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知直线 $l: y=\sqrt{3} x+4$, 动圆 $O: x^{2}+y^{2}=r^{2}(1<r<2)$, 菱形 $A B C D$ 的一个内角为 $60^{\circ}$, 顶点 $A, B$ 在直线 $l$ 上, 顶点 $C, D$ 在圆 $O$ 上, 当 $r$ 变化时, 求菱形 $A B C D$ 的面积 $S$ 的取值范围. 计算该取值范围的端点之和，并将结果四舍五入为最接近的整数.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

某人抛郑一枚硬币，出现正面向上和反面向上的概率均为 $\frac{1}{2}$. 构造数列 $\left\{a_{n}\right\}$ ，使\na_{n}= \begin{cases}1, & \text { 第 } n \text { 次正面向上; } \\ -1, & \text { 第 } n \text { 次反面向上. }\end{cases}\n记 $S_{n}=a_{1}+a_{2}+\cdots+a_{n}$. 则 $S_{2} \neq 0$, 且 $S_{8}=2$的概率为\n原始答案为 \(\frac{m}{n}\) 的格式，请给出 m + n 的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let P be a polynomial, with real coefficients, in three variables and F be a function of two variables such that

$P(ux,uy, uz)=u^{2}F(y-x,z-x,z-x)$ 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 $|C-A|$. The original answer is in the format $(m/k)\sqrt{n}$, please give the value of m + n + k.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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 endpoints of the interval for which the series converges.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $k$ be a nonnegative integer. Evaluate $$\sum_{j=0}^k2^{k-j}\binom{k+j}j.$$ Please provide the value of the expression when $k = 1$. The original answer is in $4^{k}$ format, please give the value of $m + n$

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $n$ be an integer with $n\geq2.$ Over all real polynomials $p(x)$ of degree $n$, what is the largest possible number of negative coefficients of $p(x)^2?$ Provide the value when $n=2$. directly give the value of the answer when n=4

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $n$ be an even positive integer. Let $p$ be a monic, real polynomial of degree $2n;$ that is to say, $p(x)=x^{2n}+a_{2n-1}x^{2n-1}+\cdots+a_{1}x+a_{0}$ for some real coefficients $a_{0},\ldots,a_{2n-1}.$ Suppose that $p(1/k)=k^{2}$ for all integers $k$ such that $1\leq|k|\leq n.$ Find all other real numbers $x$ for which $p(1/x)=x^2.$ The original answer is in the form of $\pm m/n!$, please give the value of m+ n

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $a_0=5/2$ and $a_k=a^2_{k-1}-2$ for $k\geq1.$ Compute $$\prod\limits_{k=0}^{\infty}\left(1-\frac{1}{a_k}\right)$$ in closed form. The original answer is in \frac{m}{n} format, please give the value of m + n.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Suppose that $f$ is a function from $\mathbb{R}$ to $\mathbb{R}$ such that $$f(x)+f\left(1-\dfrac{1}{x}\right)=\arctan x$$ for all real $x\neq0.$ (As usual, y=arctan $x$ means $-\pi/2<$ $y<\pi/2$ and tan$y=x.)$ Find $$\int_0^1f(x)\:dx.$$ The original answer is in the format \(\frac{m \pi}{n}\). Please give the value of \(m + n\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k\leqslant n.$ The rules of the game are: $(i)$ $A$ player cannot choose a number that has been chosen by either player on any previous turn. $(ii)$ $A$ player cannot choose a number consecutive to any of those the player has already chosen on any previous turn. $(iii)$ The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game. The player $A$ takes the first turn. Determine the sum of all $n$ for which the game ends in a draw, assuming that both players use optimal strategies.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f:\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ is called $k$-good if $\gcd(f(m)+n,f(n)+m)\leqslant k$ for all $m\neq n.$ Find the smallest positive integer $k$ such that there exists a $k$-good function.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $q$ be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard: $\bullet$ In the first line, Gugu writes down every number of the form $a-b$, where $a$ and $b$ are two (not necessarily distinct) numbers on his napkin. $\bullet$ In the second line, Gugu writes down every number of the form $qab$, where $a$ and $b$ are two (not necessarily distinct) numbers from the $frst\textit{line. }$ $\bullet$ In the third line, Gugu writes down every number of the form $a^2+b^2-c^2-d^2$, where $a,b,c,d$ are four (not necessarily distinct) numbers from the $frst\textit{line. }$ Determine all values of $q$ such that, regardless of the numbers on Gugu's napkin, every number in the second line is also a number in the third line. Please provide the sum of all possible values of $q$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

For each permutation $y_1,y_2,\cdots,y_n$ of these numbers we have $\sum\limits_{i=1}^{n-1}y_iy_{i+1}=y_1y_2+y_2y_3+y_3y_4+\cdots+y_{n-1}y_n\geqslant-1$. Find the largest constant $K=K(n)$ such that $\sum\limits_{1\leqslant i<j\leqslant n}x_ix_j\geqslant K$ holds for every Shiny$n$-tuple $(x_1,x_2,\cdots,x_n)$. Please provide the value of $K$ when $n=3$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Sir Alex plays the following game on a row of $9$ cells. Initially, all cells are empty. In each move, Sir Alex is allowed to perform exactly one of the following two operations: (1) Choose any number of the form $2^j$, where $j$ is a non-negative integer, and put it into an empty cell. (2) Choose two (not necessarily adjacent) cells with the same number in them; denote that number by $2^j.$ Replace the number in one of the cells with $2^j+1$ and erase the number in the other cell. At the end of the game, one cell contains the number $2^n$, where $n$ is a given positive integer, while the other cells are empty. Determine the maximum number of moves that Sir Alex could have made, in terms of $n$. Please provide the value of the maximum number of moves when $n = 10$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $n>1$ be an integer. An $n\times n\times n$ cube is composed of $n^3$ unit cubes. Each unit cube is painted with one color. For each $n\times n\times1$ box consisting of $n^2$ unit cubes (of any of the three possible orientations), we consider the set of the colors present in that box (each color is listed only once). This way, we get $3n$ sets of colors, split into three groups according to the orientation. It happens that for every set in any group, the same set appears in both of the other groups. Determine, in terms of $n$, the maximal possible number of colors that are present. Please provide the maximum number of colors when n = 6.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $n$ be a given positive integer. In the Cartesian plane, each lattice point with nonnegative coordinates initially contains a butterfly, and there are no other butterflies. The neighborhood of a lattice point $c$ consists of all lattice points within the axis-aligned $(2n+1)\times (2n+1)$ square centered at $c$, apart from $c$ itself. We call a butterfly $lonely, crowded$, or $comfortable$, depending on whether the number of butterflies in its neighborhood $N$ is respectively less than, greater than, or equal to half of the number of lattice points in $N.$ Every minute, all lonely butterflies fly away simultaneously. This process goes on for as long as there are any lonely butterflies. Assuming that the process eventually stops, determine the number of comfortable butterflies at the final state. Please provide the number of comfortable butterflies when $n = 3$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $a_0, a_1, a_2, \ldots$ be a sequence of real numbers such that $a_0=0, a_1=1$, and for every $n \geqslant 2$ there exists $1 \leqslant k \leqslant n$ satisfying
$$
a_n=\frac{a_{n-1}+\cdots+a_{n-k}}{k}
$$
Find the maximal possible value of $a_{2018}-a_{2017}$. The original answer is in the form \frac{m}{n^2}, please give the value of $m + n$.

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