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

若三个正整数 $a, b, c$ 的位数之和为 8, 且组成 $a, b, c$ 的 8 个数码能排列为2, 0,2,4,0,9,0,8, 则称 $(a, b, c)$ 为 ""幸运数组""，例如（ $9,8,202400)$ 是一个幸运数组。满足 $10<a<b<100$ 的幸运数组 $(a, b, c)$ 的个数为 $\ldots-\ldots$

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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 $2 x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}+x_{7}+x_{8}+x_{9}+x_{10}=3$ 的非负整数解共有 $\qquad$组.

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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.

在直角坐标系 xoy 中，点 $A\left(x_{1}, y_{1}\right)$ 和点 $B\left(x_{2}, y_{2}\right)$ 的坐标均为一位的正整数。 $O A$ 与 $x$ 轴正方向的夹角大于 $45^{\circ}, O B$ 与 $x$ 轴正方向的夹角小于 $45^{\circ}, B$ 在 $x$ 轴上的射影为 $B^{\prime}, A$ 在 $y$ 轴上的射影为 $A^{\prime}, \triangle O B B^{\prime}$的面积比 $\triangle O A A^{\prime}$ 的面积大 33.5, 由 $x_{1}, y_{1}, x_{2}, y_{2}$ 组成的四位数
$\overline{x_{1} x_{2} y_{2} y_{1}}=x_{1} \cdot 10^{3}+x_{2} \cdot 10^{2}+y_{2} \cdot 10+y_{1}$. 试求出所有这样的四位数, 并写出求解过程。

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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.

在直角坐标系 $x O y$ 中, 点 $A\left(x_{1}, y_{1}\right)$ 和点 $B\left(x_{2}, y_{2}\right)$ 的坐标均为一位正整数, $O A$ 与 $x$ 轴正方向的夹角大于 $45^{\circ}, O B$ 与 $x$ 轴正方向的夹角小于 $45^{\circ}, B$ 在 $x$轴上的射影为 $B^{\prime}, A$ 在 $y$ 轴上的射影为 $A^{\prime}, \triangle O B^{\prime} B$ 的面积比 $\triangle O A^{\prime} A$ 的面积大 33.5. 由 $x_{1}, y_{1}, x_{2}, y_{2}$ 组成四位数 $\overline{x_{1} x_{2} y_{2} y_{1}}=x_{1} \cdot 10^{3}+x_{2} \cdot 10^{2}+y_{2} \cdot 10+y_{1}$ 。试求出所有这样的四位数，并写出求解过程。

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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 $x 、 y 、 z$ 为非负实数，且满足方程 $4^{\sqrt{5 x+9 y+4 z}}-68 \times 2^{\sqrt{5 x+9 y+4 z}}+256=0$ ，那么 $x+y+z$ 的最大值与最小值的乘积等于 $\qquad$ ；

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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.

在底面半径为 6 的圆柱内, 有两个半径也为 6 的球面，其球心距为 13 。若作一平面与这二球面相切，且与圆柱面相交成一栯圆，则这个粨圆的长轴长与短轴长之和是 $\qquad$.

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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.

已知集合 $U=\{0,1,2, \cdots, 2021\}, S \subseteq U$, 且 $S$ 中任意两项相加不是 5 的倍数, 求 $S$ 的元素个数最大值.

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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.

已知集合 $U=\{0,1,2, \cdots, 2021\}, S \subseteq U$, 且 S 中任意两项相加不是 5 的倍数, 求 S 的元素个数最大值.

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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.

一个 $10 \times 10$ 的数表在初始状态下每个格内均填有数字1，现在对数表进行如下操作：选择某一行或某一列，并将该行（列）中所有格子内的数变为其相反数。对于非负整数 N ，如果可经过有限次操作，使得最终数表中 -1 的个数恰为 N ，则称 N 是 "好的"，则好的 N 的个数为 $\qquad$.

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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.

记所有2阶实方阵构成的线性空间为 $M_{2}(\mathbb{R})$ ，已知 $A=\left(\begin{array}{cc}3 & 7 \\ 10 & 7\end{array}\right), B=\left(\begin{array}{ll}2 & 3 \\ 4 & 5\end{array}\right)$ ，定义线性变换 $T: M_{2}(\mathbb{R}) \rightarrow M_{2}(\mathbb{R}), T(X)=A X B$ ，则线性边换 $\operatorname{trace}(T)=$ $\qquad$.

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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.

求六个元素的置换群 $S_{6}$ 中 6 阶元素的个数。

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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.

记 $I=\lim\limits_{x \rightarrow+\infty}\left(\sin \frac{1}{x^{2}}+\cos \frac{1}{x}\right)^{2 x^{2}}$, 则 $[10 I]=$ $\qquad$.

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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.

记 $I=\sum\limits_{m=1}^{+\infty} \sum\limits_{n=1}^{+\infty} \frac{1}{n!n!+m!n!}$, 则 $[10 I]=$ $\qquad$.

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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.

记 $\omega_{n}=\exp \left(\frac{2 \pi \sqrt{-1}}{n}\right)$ 。已知 n 阶傅里叶矩阵定义为
$$
\left[\begin{array}{ccccc}
1 & 1 & 1 & \cdots & 1 \\
1 & \omega_{n} & \omega_{n}^{2} & \cdots & \omega_{n}^{n-1} \\
\vdots & \vdots & \vdots & & \vdots \\
1 & \omega_{n}^{n-1} & \omega_{n}^{(n-1) \cdot 2} & \cdots & \omega_{n}^{(n-1) \cdot(n-1)}
\end{array}\right]
$$
若 A 为 2022 阶傅里叶矩阵，记其逆矩阵 $A^{-1}$ 中 $(338,2)$ 位置的元素的实部的绝对值为 I ，则 $[100000 I]=$ $\qquad$.

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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.

设 V 是所有次数不超过 100 的实系数多项式构成的线性空间。令
$$
\begin{gathered}
V_{1}=\left\{f \in V \mid f^{(2 k-1)}(0)=0, k=1,2, \cdots, 50\right\}, \\
V_{2}=\left\{f \in V \mid f^{(k)}(-1)=f^{(k)}(1)=0, k=70,71, \cdots, 98\right\},
\end{gathered}
$$
则 $\operatorname{dim}\left(V_{1}+V_{2}\right)=$ $\qquad$

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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.

现有 20 盘红灯和 10 盛绿灯。初始时，所有灯都是亮着的。我们每次从所有亮着 的灯中等可能地随机婂灭一用，直到某一种颜色的灯全部被熄灭为止。用 a 表示停止后 仍旧亮着的灯的数目的期望值，则 $[10 a]=$ $\qquad$

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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.

$f(x)=\sum_{n=0}^{\infty} 2^{-n} \cos (n x)$, 记 $A=\int_{0}^{2 \pi} f^{2}(x) \mathrm{d} x$, 则 [10A] $=$

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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 $n=5$, 令 $L_{n}=\frac{1}{n!} \frac{\mathrm{d}^{n}}{\mathrm{~d} x^{n}}\left(x^{2}-1\right)^{n}, S=\int_{-1}^{1}\left|L_{n}\right|^{2} \mathrm{~d} x$ ，则 $[|S|]=$ $\qquad$.

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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.

将方程 $(1+2+\cdots+k)-3(1+2+\cdots+t)=1$ 的全体正整数解 $(k, t)$ 按照从小到大的方式排列，得到序列 $\left\{\left(k_{n}, t_{n}\right)\right\}$ ，则 $k_{6}=$ $\qquad$。

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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.

$I=\int_{0}^{\pi / 4} \tan ^{101} x \mathrm{~d} x$ ，则 $\left[\frac{1}{3 I}\right]$

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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.

对一个复方阵, 定义 $\cos A=I+\sum_{n=1}^{\infty} \frac{(-1)^{n}}{(2 n)!} A^{2 n}$. 记矩阵 $\cos \left(\begin{array}{cc}-5 \pi & 3 \pi \\ -10 \pi & 6 \pi\end{array}\right)$ 的第二行第一列元素为 x ，则 $[|x|]=\qquad$.

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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.

在透明的球袋中有 1 个黑球和 2 个白球。接下来从袋中摸球，每次摸出一个球，然后放回袋中，并继续往袋中再放入 2 个与摸出球同色的球. 记 $S_{n}$ 为前 n 次中摸出黑球的次数. 已知第 2 次摸出的球是黑球，记 $S_{100}$ 的期望为 E ，则 $[E]=\qquad$.

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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.

5 级方阵 A 满足: 第一行元素均为 1, 第二行元素从左至右依次为 $1,-1,1,-1,1$,且 $\operatorname{trace}\left(A A^{\prime}\right)=28$. 记 $\operatorname{det} A$ 的最大值为 M, 则 $[M]=\qquad$.

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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.

令 $I=\sqrt{\frac{2}{\pi}} \int_{-\infty}^{+\infty} e^{x-10 \cosh (2 x)} \mathrm{d} x$, 则 $[|\ln I|]=$ $\qquad$.

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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.

已知圆锥面 $x^{2}+y^{2}=\frac{z^{2}}{3}$, 记沿该圆锥面从 $P(-\sqrt{3}, 3,6)$ 到 $Q(\sqrt{3}, 0,3)$ 的曲线长度的最小为值为 I, 则 $[10 I]=$ $\qquad$.

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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.

令 $A, B, C, D, E, F$ 是三阶实方阵, 且
$$
\left(\begin{array}{ccc}
I_{3} & A & C \\
& I_{3} & B \\
& & I_{3}
\end{array}\right)^{-1}=\left(\begin{array}{ccc}
I_{3} & D & F \\
& I_{3} & E \\
& & I_{3}
\end{array}\right)
$$
已知
$$
A=\left(\begin{array}{lll}
2 & & \\
& 3 & \\
& & 4
\end{array}\right), \quad B=\left(\begin{array}{lll}
1 & & 2 \\
& 2 & \\
5 & & 1
\end{array}\right)
$$
且 $C=A+B-I$ ，则 $[|\operatorname{det} F|]=$ $\qquad$.

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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.

点集 $S=\left\{(x, y) \mid x \leq 5, y \leq 4, x, y \in \mathrm{~N}^{*}\right\}$, 则由 S 中的点可组成 $\qquad$个不同的三角形。

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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.

在平面直角坐标系内, $M \in\left\{(x, y) \left\lvert\, \frac{x^{2}}{200}+\frac{y^{2}}{8} \leq 1\right.\right\}, A(2,1)$, 若 $\Delta O M A$ 的面积不超过 3, 则满足条件的整点 $M$ 个数为 $\qquad$.

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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.

设 $M=1^{4}+2^{4}+\cdots+2013^{4}$. 则 $M$ 的个位数字为 $\qquad$.

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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.

若数列 $\left\{a_{n}\right\}$ 满足
$$
a_{n}=\left[(2+\sqrt{3})^{2^{n}}\right]\left(n \in \mathbf{Z}_{+}\right),
$$
则 $a_{2020}$ 的末尾两位数字为 $\qquad$.

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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.

已知实数 $x, y$ 满足
$3|x+1|+2|y-1| \leqslant 6$.
则 $2 x-3 y$ 的最大值是 $\qquad$.

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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.

在 $(-x y+2 x+3 y-6)^{6}$ 的展开式中， $x^{4} y^{3}$的系数是 $\qquad$ （用具体数字作答）

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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 $m>0$. 若函数
$$
f(x)=x+\sqrt{100-m x}
$$
的最大值为 $g(m)$, 求 $g(m)$ 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.

设 $(1-2 x)^{7}=\sum_{k=0}^{7} a_{k} x^{k}$. 则
$2 a_{2}+3 a_{3}+4 a_{4}+5 a_{5}+6 a_{6}+7 a_{7}=$

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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.

在矩形 $A B C D$ 中, $A B=3, A D=4, P$为矩形 $A B C D$ 所在平面上一点，满足 $P A=2$ ， $P C=\sqrt{21}$. 则 $\overrightarrow{P B} \cdot \overrightarrow{P D}=$ $\qquad$。

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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.

设直线 $y=k x+b$ 与曲线 $y=x^{3}-x$ 有三个不同的交点 $A 、 B 、 C$, 且 $|A B|=|B C|=2$ 。则 $k$ 的值为 $\qquad$.

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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.

对于集合 $\{x \mid a \leqslant x \leqslant b\}$ ，我们把 $b-a$ 称为它的长度。设集合 $A=\{x \mid a \leqslant x \leqslant$ $a+1981\}, B=\{x \mid b-1014 \leqslant x \leqslant b\}$ ，且 $A, B$ 都是集合 $U=\{x \mid 0 \leqslant x \leqslant 2012\}$的子集，则集合 $A \cap B$ 的长度的最小值是 $\qquad$ 。

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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.

给定凸 20 边形 $P$ 。用 $P$ 的 17 条在内部不相交的对角线将 $P$ 分割成 18 个三角形。所得图形称为 $P$ 的一个三角剖分图。对 $P$ 的任意一个三角剖分图 $T, P$ 的 20 条边以及添加的 17 条对角线均称为 $T$ 的边。 $T$ 的任意 10 条两两无公共端点的边的集合称为 $T$的一个完美匹配。当 $T$ 取遍 $P$ 的所有三角剖分图时, 求 $T$ 的完美匹配个数的最大值。

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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.

已知集合 $S=\{1,2,3, \cdots, N\}$ 的四个500元子集 $A_{1}, A_{2}$, $A_{3}, A_{4}$ 满足：对任意 $x, y \in S$ ，均存在某个 $i \in\{1,2,3,4\}$ ，使得 $x, y \in A_{i}$ 。求正整数 $N$ 的最大可能值。

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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.

方程 $2 x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}+x_{7}+x_{8}+x_{9}+x_{10}=3$ 的非负整数解共有 $\qquad$组.

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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.

在直角坐标系 $x O y$ 中, 点 $A\left(x_{1}, y_{1}\right)$ 和点 $B\left(x_{2}, y_{2}\right)$ 的坐标均为一位正整数, $O A$ 与 $x$ 轴正方向的夹角大于 $45^{\circ}, O B$ 与 $x$ 轴正方向的夹角小于 $45^{\circ}, B$ 在 $x$轴上的射影为 $B^{\prime}, A$ 在 $y$ 轴上的射影为 $A^{\prime}, \triangle O B^{\prime} B$ 的面积比 $\triangle O A^{\prime} A$ 的面积大 33.5. 由 $x_{1}, y_{1}, x_{2}, y_{2}$ 组成四位数 $\overline{x_{1} x_{2} y_{2} y_{1}}=x_{1} \cdot 10^{3}+x_{2} \cdot 10^{2}+y_{2} \cdot 10+y_{1}$ 。试求出所有这样的四位数，并写出求解过程。

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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 $x 、 y 、 z$ 为非负实数，且满足方程 $4^{\sqrt{5 x+9 y+4 z}}-68 \times 2^{\sqrt{5 x+9 y+4 z}}+256=0$ ，那么 $x+y+z$ 的最大值与最小值的乘积等于 $\qquad$ ；

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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.

设 $x, y, z$ 为非负实数，且满足方程 $4^{\sqrt{5 x+9 y+4 z}}-68 \times 2^{\sqrt{5 x+9 y+4 z}}+256=0$ ，那么 $x+y+z$ 的最大值与最小值的乘积等于 $\qquad$.

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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.

在底面半径为 6 的圆柱内, 有两个半径也为 6 的球面，其球心距为 13 。若作一平面与这二球面相切，且与圆柱面相交成一栯圆，则这个粨圆的长轴长与短轴长之和是 $\qquad$.

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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.

已知集合 $M=\{x, y, \lg (x y)\}$ 及 $N=\{0,|x|, y\}$ ，并且 $M=N$ ，那么 $\left(x+\frac{1}{y}\right)+\left(x^{2}+\frac{1}{y^{2}}\right)+\left(x^{3}+\frac{1}{y^{3}}\right)+\cdots+\left(x^{2001}+\frac{1}{y^{2001}}\right)$ 的值等于 $\qquad$

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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.

设集合 $A=\{1,2, m\}$ ，其中 $m$ 为实数。令 $B=\left\{a^{2} \mid a \in A\right\}, C=A \cup B$ 。若 $C$ 的所有元素之和为 6, 则 $C$ 的所有元素之积为 $\qquad$.

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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.

在 $5 \times 5$ 矩阵中, 每个元素都为 0 或 1, 且满足：五行的元素之和都相等,但五列的元素之和两两不等.这样的矩阵的个数为 $\qquad$ （答案用数值表示）.

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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.

集合 $A=\left\{n \mid n^{3}<2022<3^{n}, n \in \mathbf{Z}\right\}$ 的所有元素之和为 $\qquad$.

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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.

设 $a_{1}, a_{2}, \cdots, a_{100}$ 是非负整数，同时满足以下条件：
（1）存在正整数 $k \leq 100$ ，使得 $a_{1} \leq a_{2} \leq \cdots \leq a_{k}$ ，而当 $i>k$ 时 $a_{i}=0$ ；
（2） $a_{1}+a_{2}+a_{3}+\cdots+a_{100}=100$ ；
(3) $a_{1}+2 a_{2}+3 a_{3}+\cdots+100 a_{100}=2022$ 。
求 $a_{1}+2^{2} a_{2}+3^{2} a_{3}+\cdots+100^{2} a_{100}$ 的最小可能值.

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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 $\triangle A B C$ 的三个内角 $A, B, C$ 满足 $\cos A=\sin B=2 \tan \frac{C}{2}$, 则 $\sin A+\cos A$ $+2 \tan A$ 的值为 $\qquad$.

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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.

若正实数 $a, b$ 满足 $a^{\lg b}=2, a^{\lg a} \cdot b^{\lg b}=5$, 则 $(a b)^{\lg a b}$ 的值为 $\qquad$.

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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.

求具有下述性质的最小正整数 $k$ ：若将 $1,2, \cdots, k$ 中的每个数任意染为红色或者蓝色，则或者存在 9 个互不相同的红色的数 $x_{1}, x_{2}, \cdots, x_{9}$ 满足 $x_{1}+x_{2}+\cdots+x_{8}<x_{9}$ ，或者存在 10 个互不相同的蓝色的数 $y_{1}, y_{2}, \cdots, y_{10}$ 满足 $y_{1}+y_{2}+\cdots+y_{9}<y_{10} 。$

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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.

设实数 $a, b$ 满足: 集合 $A=\left\{x \in \mathbf{R} \mid x^{2}-10 x+a \leq 0\right\}$ 与 $B=\left\{x \in \mathbf{R} \mid b x \leq b^{3}\right\}$的交集为 $[4,9]$, 则 $a+b$ 的值为 $\qquad$.

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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.

设集合 $A=\{1,2, \cdots, 99\}, X, Y$ 均为 $A$ 的非空子集（允许 $X=Y$ ）。 $X$ 中的最大元与 $Y$ 中的最小元分别记为 $\max X, \min Y$ 。求满足 $\max X>\min Y$ 的有序集合对 $(X, Y)$ 的数目。

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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 $n$ 及实数 $x(0 \leqslant x<n)$ ，定义
$$
f(n, x)=(1-\{x\}) \cdot\binom{[x]}{n}+\{x\} \cdot\binom{[x]+1}{n}
$$
其中 $[x]$ 表示不超过实数 $x$ 的最大整数， $\{x\}=x-[x]$ 。若整数 $m, n \geqslant 2$ 满足
$$
f\left(m, \frac{1}{n}\right)+f\left(m, \frac{2}{n}\right)+\cdots+f\left(m, \frac{m n-1}{n}\right)=123,
$$
求 $f\left(n, \frac{1}{m}\right)+f\left(n, \frac{2}{m}\right)+\cdots+f\left(n, \frac{m n-1}{m}\right)$ 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.

给定整数 $n \geqslant 3$, 设 $a_{1}, a_{2}, \cdots, a_{2 n}, b_{1}, b_{2}, \cdots, b_{2 n}$ 是 $4 n$ 个非负实数, 满足
$$
a_{1}+a_{2}+\cdots+a_{2 n}=b_{1}+b_{2}+\cdots+b_{2 n}>0,
$$
且对任意 $i=1,2, \cdots, 2 n$, 有 $a_{i} a_{i+2} \geqslant b_{i}+b_{i+1}$ （这里 $a_{2 n+1}=a_{1}, a_{2 n+2}=a_{2}$, $b_{2 n+1}=b_{1}$ ), 求 $a_{1}+a_{2}+\cdots+a_{2 n}$ 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.

$\frac{1}{1-\log _{2} 20}+\frac{1}{1-\log _{5} 50}$ 的值为 $\qquad$.

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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.

设 $f(x)=x^{3}+3\left(x^{2}+x+\sin \pi x\right)$, 则 $f(1-\pi)+f(\pi-3)=\qquad$.

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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.

设正整数 $a, b, c$ 中任意两个数不互素，且 $a+b+c$ 为素数. 求 $a+b+c$ 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.

小明同学任选一个各项均为非零整数的等差数列 $\left\{a_{n}\right\}$ ，随后他列出 2020 个关于 $x$ 的一元二次方程:
$$
E_{i}: a_{i} x^{2}+a_{i+1} x+a_{i+2}=0(i=1,2, \cdots, 2020),
$$
再将上述每个方程的根都写在黑板上.求此时黑板上两两不同的实根的个数的最大可能值.

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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.

对正整数 $n$ 及实数 $x(0 \leqslant x<n)$, 定义
$$
f(n, x)=(1-\{x\}) \cdot C_{n}^{[x]}+\{x\} \cdot C_{n}^{[x]+1},
$$
其中 $[x]$ 表示不超过实数 $x$ 的最大整数， $\{x\}=x-[x]$ .若整数 $m, n \geqslant 2$ 满足
$$
f\left(m, \frac{1}{n}\right)+f\left(m, \frac{2}{n}\right)+\cdots+f\left(m, \frac{m n-1}{n}\right)=123,
$$
求 $f\left(n, \frac{1}{m}\right)+f\left(n, \frac{2}{m}\right)+\cdots+f\left(n, \frac{m n-1}{m}\right)$ 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.

设 $A=\{1,2,3\}, B=\{2 x+y \mid x, y \in A, x<y\}, C=\{2 x+y \mid x, y \in A, x>y\}$ ，则 $B \cap C$ 的所有元素之和为 $\qquad$.

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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.

若正整数 $a, b, c$ 满足 $2017 \geqslant 10 a \geqslant 100 b \geqslant 1000 c$ ，则数组 $(a, b, c)$ 的个数为

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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 $\left\{a_{n}\right\}: a_{1}=7, \frac{a_{n+1}}{a_{n}}=a_{n}+2, n=1,2,3, \cdots$. 求满足 $a_{n}>4^{2018}$ 的最小正整数 $n$.

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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 $a_{i}(i=1,2,3,4)$ 均为实数, 若集合 $\left\{a_{1}, a_{2}, a_{3}, a_{4}\right\}$ 的所有非空真子集的元素之和为 28, 则 $a_{1}+a_{2}+a_{3}+a_{4}=$ $\qquad$.

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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.

有长为 $2^{n}(n=0,1, \cdots, 1009)$ 的线段各三条, 则由这 3030 条线段能构成不全等的三角形的个数为 $\qquad$ (用数字作答).

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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 $\odot O: x^{2}+y^{2}=5$ 与抛物线 $C$ : $y^{2}=2 p x(p>0)$ 交于点 $A\left(x_{0}, 2\right), A B$ 为 $\odot O$ 的直径，过点 $B$ 的直线与抛物线 $C$ 交于两点 $D$ 、 $E$ 。则 $A D$ 与 $A E$ 的斜率之积为 $\qquad$.

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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.

已知函数 $f: \mathbf{R} \rightarrow \mathbf{R}$, 若对任意实数 $x, y, z$, 都有
$\frac{1}{3} f(x y)+\frac{1}{3} f(x z)-f(x) f(y z) \geqslant \frac{1}{9}$, 求 $\sum_{i=1}^{100}[i f(i)]$.其中 [x] 代表不超过 x 的最大整数.

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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.

已知 $\odot O_{1}: x^{2}+y^{2}=2$ 和 $\odot O_{2}:(x-3)^{2}+$ $y^{2}=5$ 在第一象限内的公共点为 $A$, 过点 $A$ 的直线分别交 $\odot O_{1} 、 \odot O_{2}$ 于 $C 、 D$ 两点（ $C$ 、 $D$ 异于点 $A$ ), 且 $\overrightarrow{A C}=2 \overrightarrow{A D}$. 则直线 $C D$ 的斜率是 $\qquad$.

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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.

由 $1,2, \cdots, 9$ 这九个正整数构成的所有圆排列中, 任意相邻两数之积均不超过 60 的圆排列的个数为 $\qquad$.

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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.

设 $n \in \mathbf{Z}_{+}$. 当 $n>100$ 时， $\sqrt{n^{2}+3 n+1}$的小数部分的前两位数为 $\qquad$.

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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.

设 $n$ 是小于 100 的正整数, 且满足 $\frac{1}{3}\left(n^{2}-1\right)+\frac{1}{5} n$ 为整数. 则符合条件的所有正整数 $n$ 的和为 $\qquad$.

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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.

设 $x+\sin x \cdot \cos x-1=0$ ,$2 \cos y-2 y+\pi+4=0$ . 则 $\sin (2 x-y)$ 的值为 $\qquad$.

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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.

在矩形 $A B C D$ 中, $A B=3, A D=4, P$为矩形 $A B C D$ 所在平面上一点，满足 $P A=2$ ， $P C=\sqrt{21}$. 则 $\overrightarrow{P B} \cdot \overrightarrow{P D}=$ $\qquad$.

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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.

There are $2 n$ students in a school $(n \in \mathbb{N}, n \geq 2)$. Each week $n$ students go on a trip. After several trips the following condition was fulfilled: every two students were together on at least one trip. What is the minimum number of trips needed for this to happen?

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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.

对任意正整数 $n$, 连结原点 $O$ 与点 $A_{n}(n, n+3)$, 用 $f(n)$ 表示线段 $O A_{n}$ 上的整点个数(不计端点), 试求 $f(1)+f(2)+\cdots+f(1990)$.

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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.

在坐标平面上, 横坐标和纵坐标均为整数的点称为整点, 对任意正整数 $n$, 连接原点 $O$ 与点 $A_{n}(n, n+3)$, 用 $f(n)$ 表示线段 $O A_{n}$ 上除端点外的整点个数,则 $f(1)+f(2)+\cdots+f(1990)=$ $\qquad$.

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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.

$1991^{2000}$ 除以 $10^{6}$ ，余数是 $\qquad$.

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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.

设 复 数 $z_{1}, z_{2}$ 满 足 $\left|z_{1}\right|=\left|z_{1}+z_{2}\right|=3,\left|z_{1}-z_{2}\right|=3 \sqrt{3}$ ，则$\log _{3}\left|\left(z_{1} \overline{z_{2}}\right)^{2000}+\left(\overline{z_{1}} z_{2}\right)^{2000}\right|=$ $\qquad$.

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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.

设复数 $z_{1}, z_{2}$ 满足 $\left|z_{1}\right|=\left|z_{1}+z_{2}\right|=3,\left|z_{1}-z_{2}\right|=3 \sqrt{3}$, 则 $\log _{3}\left|\left(z_{1} \overline{z_{2}}\right)^{2000}+\left(\overline{z_{1}} z_{2}\right)^{2000}\right|=$ $\qquad$.

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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.

将正奇数集合 $\{1,3,5, \cdots\}$ 由小到大按第 $n$ 组有 $(2 n-1)$ 个奇数进行分组:
$\{1\}, \quad\{3,5,7\}, \quad\{9,11,13,15,17\}, \cdots$
（第一组）（第二组）（第三组）
则 1991 位于第 $\qquad$ 组.

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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.

整数 $\left[\frac{10^{93}}{10^{31}+3}\right]$ 的末两位数是 $\qquad$.

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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.

三位数 $(100,101, \ldots, 999)$ 共 900 个，在卡片上打印这些三位数，每张卡片上打印一个三位数，有的卡片所印的，倒过来看仍为三位数，如 198 倒过来看是 861 ；有的卡片则不然，如 531 倒过来看是,因此，有些卡片可以一卡二用，于是至多可以少打印 $\qquad$张卡片.

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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.

已知点集 $A=\left\{(x, y) \left\lvert\,(x-3)^{2}+(y-4)^{2} \leq\left(\frac{5}{2}\right)^{2}\right.\right\}, B=\left\{(x, y) \left\lvert\,(x-4)^{2}+(y-5)^{2}>\left(\frac{5}{2}\right)^{2}\right.\right\}$, 则点集 $A \cap B$ 中的整点 （即横、纵坐标均为整数的点）的个数为 $\qquad$.

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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.

将与 105 互素的所有正整数从小到大排成数列, 试求出这个数列的第 1000 项.

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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.

在直角坐标平面上, 满足不等式组 $\left\{\begin{array}{l}y \leqslant 3 x, \\ y \geqslant \frac{1}{3} x, \\ x+y \leqslant 100\end{array}\right.$ 的整点 (即横、纵坐标均为整数的点）的个数是 $\qquad$.

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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.

求一切实数 $p$ ，使得三次方程 $5 x^{3}-5(p+1) x^{2}+(71 p-1) x+1=66 p$ 的三个根均为自然数。

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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.

Let $p(x)=2+4x+3x^2+5x^3+3x^4+4x^5+2x^6$. For $k$ with $0<k<5$, deline
$$I_k=\int_{0}^{\infty}\frac{x^k}{p(x)}dx$$.
For which $k$ is $I_k$ smallest.

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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 the additive group of ordered pairs of integers ( $m, n$ ) [with addition defined componentwise: $\left.(m, n)+\left(m^{\prime}, n^{\prime}\right)=\left(m+m^{\prime}, n+n^{\prime}\right)\right]$ consider the subgroup $H$ generated by the three elements
$$
(3,8), \quad(4,-1)
$$
Then $H$ has another set of generators of the form
$$
(1, b), \quad(0, a)
$$
for some integers $a, b$ with $a>0$. Find $a$.
[Elements $g_{i}, \ldots, g_{k}$ are said to generate a subgroup $H$ if (i) each $g_{i} \in H$, and (ii) every $h \in H$ can be written as a sum $h=n_{1} g_{1}+\cdots+n_{k} g_{k}$ where the $n_{i}$ are integers (and where, for example, $3 g_{1}-2 g_{2}$ means $\left.\left.g_{1}+g_{1}+g_{1}-g_{2}-g_{2}\right)\right]$

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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.

Call a set of positive integers "conspiratorial" if no three of them are pairwise relatively prime. (A set of integers is "pairwise relatively prime" if no pair of them has a common divisor greater than 1.) What is the largest number of elements in any "conspiratorial" subset of the integers I through 16?

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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.

Call a set of positive integers "conspiratorial" if no three of them are pairwise relatively prime. (A set of integers is "pairwise relatively prime" if no pair of them has a common divisor greater than 1.) What is the largest number of elements in any "conspiratorial" subset of the integers 1 through 16?

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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 positive integers $n$ and $a_1, a_2,…,a_n$ such that
$$a_1+a_2+…+a_n=1979$$
and the product $a_1a_2…a_n$ is as large as possible.

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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.

How many positive integers $n$ are there such that $n$ is an exact divisor of at least one of the numbers $10^{40}, 20^{30}$?

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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 distinct squares of the 8 by 8 chessboard C are said to be adjacent if they have a vertex or side in common. Also, g is called a C-gap if for every numbering of the squares of C with all the integers $1,2,\ldots,64$ there exist two adjacent squares whose numbers differ by at least g. Determine the largest C-gap 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.

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" $A{ }^{{ }^{1}} B^{n_{2}} A^{n_{3}} \cdots B^{n_{k}}$, where $n_{1}, \ldots, n_{k}$ are any integers, and $A^{\prime \prime}=B^{\prime \prime}=1$ as usual. Also, suppose that $A^{4}=B^{7}=A B A^{-1} B=1, A^{2} \neq 1$, and $B \neq 1$.

(a) How many elements of $G$ are of the form $C^{2}$ with $C$ in $G$?

(b) Write each such square as a word in $A$ and $B$.

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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 collection of all quadratic polynomials $P$ with real coefficients such that
$$\sup\{\mid P^{\prime}(0)\mid:P\in V\}.$$

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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.

Find
$$\lim_{n\to\infty}\left[\frac{1}{n^{5}}\sum_{h=1}^{n}\sum_{k=1}^{n}(5h^{4}-18h^{2}k^{2}+5k^{4})\right].$$

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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
$$
\lim _{n \rightarrow \infty}\left[\frac{1}{n^{5}} \sum_{h=1}^{n} \sum_{k=1}^{n}\left(5 h^{4}-18 h^{2} k^{2}+5 k^{4}\right)\right].
$$

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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.

某班有 47 个学生，所用教室有 6 排，每排有 8个座位，用 $(i, j)$ 表示位于第 $i$ 排第 $j$ 列的座位。新学期准备调整座位，设某学生原来的座位为 $(i, j)$ ，如果调整后的座位为 $(m, n)$ ，则称该生作了移动 $[a$, $b]=[i-m, j-n]$ ，并称 $a+b$ 为该生的位置数。所有学生的位置数之和记为 $S$ 。求 $S$ 的最大可能值与最小可能值之差。

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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.

求最小的正整数 $m$, 使得对于任意大于 3 的质数 $p$ ，都有 $1051\left(9^{p^{2}}-29^{p}+m\right)$ 。

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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.

双曲线 $x^{2}-y^{2}=1$ 的右半支与直线 $x=100$ 围成的区域内部（不含边界）整点（纵横坐标均为整数的点）的个数是

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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.

对于集合 $\{x \mid a \leqslant x \leqslant b\}$ ，我们把 $b-a$ 称为它的长度.设集合 $A=\{x \mid a \leqslant x \leqslant$ $a+1981\}, B=\{x \mid b-1014 \leqslant x \leqslant b\}$ ，且 $A, B$ 都是集合 $U=\{x \mid 0 \leqslant x \leqslant 2012\}$的子集，则集合 $A \cap B$ 的长度的最小值是 $\qquad$.

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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.

在 $100 \times 25$ 的长方形表格中每一格填入一个非负实数，第 $i$ 行第 $j$ 列中填入的数为 $x_{i, j}$ （ $i=1$ ， $2 ， \ldots ， 100 ; j=1,2 ， \ldots, 25 ） （$ 如表1）。然后将表1每列中的数按由小到大的次序从上到下重新排列为 $x_{1}^{\prime}$ ， $j \geq x^{\prime}{ }_{2, j} \geq \ldots \geq x^{\prime}{ }_{100, j}(j=1,2, \ldots, 25)$ 。（如表2）
求最小的自然数 $k$ ，使得只要表 1 中填入的数满足 $\sum_{j=1}^{25} x_{i}, j \leq 1 （ i=1,2, \ldots, 100 ）$ ，
则当 $i \geq k$ 时，在表2中就能保证 $\sum_{j=1}^{25} x_{i, j}^{\prime} \leq 1$ 成立。
表 1
| $x_{1,1}$ | $x_{1,2}$ | $\ldots$ | $x_{1,25}$ |
| :---: | :---: | :---: | :---: |
| $x_{2,1}$ | $x_{2,2}$ | $\ldots$ | $x_{2,25}$ |
| $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ |
| $x_{100,1}$ | $x_{100,2}$ | $\ldots$ | $x_{100,25}$ |

表 2
| $x^{\prime}{ }_{1,1}$ | $x^{\prime}{ }_{1,2}$ | $\ldots$ | $x^{\prime}{ }_{1,25}$ |
| :---: | :---: | :---: | :---: |
| $x^{\prime}{ }_{2,1}$ | $x^{\prime}{ }_{2,2}$ | $\ldots$ | $x^{\prime}{ }_{2,25}$ |
| $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ |
| $x^{\prime}{ }_{100,1}$ | $x^{\prime}{ }_{100,2}$ | $\ldots$ | $x^{\prime}{ }_{100,25}$ |

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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 $z=\cos \theta+\mathrm{i} \sin \theta\left(0^{\circ} \leqslant \theta \leqslant 180^{\circ}\right)$, 复数 $z 、(1+\mathrm{i}) z 、 2 \bar{z}$ 在复平面上对应的三个点分别是 $P 、 Q 、 R$ 。当 $P 、 Q 、 R$ 不共线时，以线段 $P Q 、 P R$ 为两边的平行四边形的第四个顶点为 $S$. 则点 $S$ 到原点距离的最大值是 $\qquad$.

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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.

如果:
(1) $a 、 b 、 c 、 d$ 都属于 $\{1,2,3,4\}$;
(2) $a \neq b, b \neq c, c \neq d, d \neq a$;
(3) $a$ 是 $a 、 b 、 c 、 d$ 中的最小值.
那么，可以组成的不同的四位数 $\overline{a b c d}$ 的个数是 $\qquad$.

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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.

设函数

$f(x)= \begin{cases}\frac{1}{p}, & x=\frac{q}{p} ; \\ 0, & x \neq \frac{q}{p},\end{cases}$

其中, $p 、 q$ 互质, 且 $p \geqslant 2$. 则满足 $x \in[0,1]$,且 $f(x)>\frac{1}{5}$ 的 x 值的个数是 $\qquad$.

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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.

已知三个互不相等的整数 $x 、 y 、 z$ 之和介于 40 与 44 之间. 若 $x, y, z$ 依次构成公差为 $d$ 的等差数列， $x+y 、 y+z 、 z+x$ 依次构成公比为 $q$ 的等比数列, 则 $d q=$.

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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.

设正六边形 $A B C D E F$ 的边长为 1. 则 $(\overrightarrow{A B}+\overrightarrow{D C}) \cdot(\overrightarrow{A D}+\overrightarrow{B E})=$

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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.

在平面直角坐标系中, $A(1,2), B(3,0)$, 点 $P$ 为圆 $(x-3)^{2}+(y-2)^{2}=1$ 上任意一点，设 $\overrightarrow{O P}=\lambda \overrightarrow{O A}+\mu \overrightarrow{O B}(\lambda, \mu \in \mathbf{R})$ ，则 $11 \lambda+9 \mu$ 的最小值是 $\qquad$.

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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.

设函数

$$f(\:x\:)=\sqrt{2x^2\:+2x\:+41}\:-\sqrt{2x^2\:+4x\:+4}\:(\:x\:\in\:\mathbf{R}\:)\:.$$
则$f(x)$的最大值为

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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 $3\times3$ 方格表中 ， 若九个格中的 每一个均被染成红 、 黄、 蓝 、 紫 四种颜色之一 ,任意 两个有公共边 的 格 的 染色均不相同 ,且该方格表 中 恰有两个格染蓝色 , 则符合条件的染色方法种数为$\qquad$ (用具体数字作答） ．

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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 $f:\{1,2, \cdots, 10\} \rightarrow\{1,2, \cdots, 5\}$ ，且对一切 $k=1,2, \cdots, 9$ ，有
$|f(k+1)-f(k)| \geqslant 3$.
则符合条件的函数 $f$ 的个数是 $\qquad$.

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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.

在 $(-x y+2 x+3 y-6)^{6}$ 的展开式中, $x^{4} y^{3}$ 的系数为 $\qquad$. (用具体数字作答)

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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 $x$ 为实数, 定义 $\lceil x\rceil$ 为不小于实数 $x$ 的最小整数 (如 $\rceil\rceil=4,\lceil-\pi\rceil=-3$ ). 则关于实数 $x$ 的方程
$\lceil 3 x+1\rceil=2 x-\frac{1}{2}$
的全部实根之和等于\_\_\_\_\_\_

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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.

$\left(x^{2}+x-\frac{1}{x}\right)^{6}$ 的展开式中的常数项是 $\qquad$ (用具体数字作答).

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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$\left\{a_n\right\}$的前$n$项和为$S_n,$满足 $S_{n}=\frac{(a_{n}+1)^{2}}{4}.$则 $S_{20}$的值为$\qquad$.

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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.

若正整数 $m 、 n$ 满足 $\frac{(m+n)!}{n!}=5040$,则 $m!n$ 的值为 $\qquad$。

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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.

Find the smallest positive integer $j$ such that for every polynomial $p(x)$ with integer coefficients and for every integer $k$, the integer

$$
p^{(j)}(k)=\left.\frac{d^j}{d x^j} p(x)\right|_{x=k}
$$

(the $j$-th derivative of $p(x)$ at $k$ ) is divisible by 2016.

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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.

How many positive integers $N$ satisfy all of the following three conditions?
(i) $N$ is divisible by 2020 .
(ii) $N$ has at most 2020 decimal digits.
(iii) The decimal digits of $N$ are a string of consecutive ones followed by a string of consecutive zeros.

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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 grasshopper starts at the origin in the coordinate plane and makes a sequence of hops. Each hop has length 5, and after each hop the grasshopper is at a point whose coordinates are both integers; thus, there are 12 possible locations for the grasshopper after the first hop. What is the smallest number of hops needed for the grasshopper to reach the point $(2021,2021)$ ?

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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.

$\operatorname{Let}\left(x_{1},y_{1}\right)=\left(0.8,0.6\right)$and let $x_{n+1}=x_{n}\cos y_{n}-y_{n}\sin y_{n}$ and $y_{n+1}=x_{n}\sin y_{n}+$ $y_{n}$cos$y_n$ for $n=1,2,3,\ldots.$ For each of $\lim_{n\to\infty} x_n$ and $\lim_{n\to\infty }y_n$, prove that the limit exists and find it or prove that the limit does not exist.

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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 $p(x)$ be a nonzero polynomial of degree less than 1992 having no noncon-
stant factor in common with $x^3-x.$ Let

$$\frac{d^{1992}}{dx^{1992}}\left(\frac{p(x)}{x^3-x}\right)=\frac{f(x)}{g(x)}$$

for polynomials $f(x)$ and $g(x).$ Find the smallest possible degree of $f(x)$.

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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 positive integer $n$, define $d(n)$ to be the sum of the digits of $n$ when written in binary (for example, $d(13)=1+1+0+1=3).$Let
$$S=\sum_{k=1}^{2020}(-1)^{d(k)}k^3.$$
Determine $S$ modulo 2020.

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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 $k(n)$ be the number of ones in the binary representation of 2023·n. What is the minimum value of $k(n)?$

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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 $P_n$ be the number of permutations $\pi$ of $\{1,2,\ldots,n\}$
such that
$|i-j|=1$  implies  $|\pi(i)-\pi(j)|\leq2$
for all $i,j$ in $\{1,2,\ldots,n\}.$ Show that for $n\geq2$,the quantity
$$P_{n+5}-P_{n+4}-P_{n+3}+P_n$$
does not depend on $n$, and find its value.

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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.

Define $C(\alpha)$ to be the coefficient of $x^{1992}$ in the power series expansion about $x=0$ of $(1+x)^{\alpha}$. Evaluate
$$
\int_{0}^{1} C(-y-1)\left(\frac{1}{y+1}+\frac{1}{y+2}+\frac{1}{y+3}+\cdots+\frac{1}{y+1992}\right) d y.
$$

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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.

我们称 $A_{1}, A_{2}, \cdots, A_{n}$ 为集合 $A$ 的一个 $n$ 分划，如果
(1) $A_{1} \cup A_{2} \cup \cdots \cup A_{n}=A$;
(2) $A_{i} \cap A_{j} \neq \varnothing, 1 \leqslant i<j \leqslant n$.
求最小正整数 $m$ ，使得对 $A=\{1,2, \cdots, m\}$ 的任意一个 14 分划 $A_{1}, A_{2}, \cdots, A_{14}$ ，一定存在某个集合 $A_{i}(1 \leqslant$ $i \leqslant 14)$ ，在 $A_{1}$ 中有两个元素 $a 、 b$ 满足 $b<a \leqslant \frac{4}{3} b$ 。

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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.

一个$100\times100$方格表的左上角小方格中有一只老鼠，右下角小方格中有一块奶酪。老鼠希望移动到右下角小方格中吃奶酪，每次可以从一个小方格移动到相邻的小方格（两个小方格相邻指它们有公共边）。现在在一些小方格的边上放置隔板，老鼠在移动时不能越过隔板。称一种放置隔板的方式是“仁慈的”，如果放置隔板后老鼠仍能吃到奶酪。求最小的正整数$n$，使得对任意一种“仁慈的”放置2023个隔板的方式，老鼠都能通过不超过$n$次移动吃到奶酪。

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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.

某个国家有2023个岛和2022座桥，任意一座桥连接两个不同的岛，任意两个岛之间至多有一座桥相连，且可以从任意一个岛通过若干座桥到达其他任何岛。若三个岛中的某个岛与另两个岛都有桥连接，则称这三个岛组成“岛群”.已知任两个“岛群”中都有相同的岛，那么恰有一座桥的岛最少有多少个？

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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 $1,2,3,4,5,6,7,8$ 分别放在正方体的八个顶点上，使得每一个面上的任意三个数之和均不小于 10 。求每一个面上四个数之和的最小值。

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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.

求所有的整数 $n$ ，使得  $n^{4}+6 n^{3}+11 n^{2}+3 n+31$  是完全平方数.

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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.

有 $n(n>12)$ 个人参加某次数学邀请赛，试卷由十五道填空题组成，每答对一题得 1 分，不答或答错得 0 分。分析每一种可能的得分情况发现：只要其中任意 12 个人得分之和不少于 36 分，则这 $n$ 个人中至少有 3 个人答对了至少三道同样的题. 求 $n$ 的最小可能值。

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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.

集合 $I=\{1,2, \cdots, 2020\}$. 称
$$
\begin{aligned}
& W=\{w(a, b)=(a+b)+a b \mid a, b \in I\} \cap I, \\
& Y=\{y(a, b)=(a+b) \cdot a b \mid a 、 b \in I\} \cap I, \\
& X=W \cap Y,
\end{aligned}
$$
分别为"吴"集合、"越"集合、"西子"集合. 试求西子集合中最大数与最小数之和。

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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.

今有 $n$ 个人排成一个横队, 从左到右按 $1,2, \cdots, n$ 报数，凡报奇数者退出队伍；剩下的人再从右到左按 $1,2, \cdots$ 重新报数，凡报奇数者退出队伍；然后剩下的人又从左到右重新报数。如此进行，使得最后队伍中只剩下一人. $f(n)$ 表示最后剩下的这个人在第一次报数时的号码。求 $f(n)$ 的表达式，并给出 $f(2022)$ 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.

小陶同学玩如下游戏：取定大于 1 的常数 $v$ ；对正整数 $m$ ，游戏的第 $m$ 轮与第 $m+1$ 轮间隔为 $2^{-m}$ 单位时长；第 $m$ 轮在平面上有策略地取一个半径为 $2^{-m+1}$ 的圆形安全区域（含边界，取圆时间忽略不计）；每轮取定的圆形安全区域将在整个游戏剩余时间内保持圆心不动，半径以速率 $v$ 匀速减小（半径减小到零时，去掉该圆形安全区域）；若可在第 100 轮之前（含第 100 轮）的某轮将圆形安全区域取在剩余的安全区域的并集内，则取得游戏胜利。若小陶同学有必胜策略，求 $\left[\frac{1}{v-1}\right]$ 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.

设 $1,2, \cdots, 9$ 的所有排列 $X=(x_{1}, x_{2}, \cdots, x_{9})$ 的集合为 $A$ 。对任意的 $X \in A$ ，记
$$
\begin{aligned}
& f(X)=x_{1}+2 x_{2}+\cdots+9 x_{9}, \\
& M=\{f(X) \mid X \in A\}.
\end{aligned}
$$
求 $|M|$ ( $|M|$ 表示集合 $M$ 的元素个数)。

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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.

求正整数 $n$ 的最小值, 使得
$$
\begin{aligned}
& \sqrt{\frac{n-2011}{2012}}-\sqrt{\frac{n-2012}{2011}} \\
& <\sqrt[3]{\frac{n-2013}{2011}}-\sqrt[3]{\frac{n-2011}{2013}}.
\end{aligned}
$$

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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.

已知奇素数 $x 、 y 、 z$ 满足
$$
x\left|\left(y^{5}+1\right), y\right|\left(z^{5}+1\right), z \mid\left(x^{5}+1\right)
$$
求乘积 $x y z$ 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.

青青草原上生活着编号为 $1,2, \cdots, 7$的 $7$ 只羊和编号为 $1,2, \cdots, 2017$ 的 $2017$ 匹狼。在该草原上有如下奇怪的规则：
（1）定义 $P(n)$ 为小于 $n$ 的素数个数，仅当 $P(i) \equiv j(\bmod 7)$ 时，编号为 $i$ 的狼可以吃掉编号为 $j$ 的羊（也可以不吃）；
（2）若编号为 $i$ 的狼吃了编号为 $j$ 的羊，则它会立刻变成编号为 $j$ 的羊；
（3）每匹狼在确保不会被吃的前提下都非常想体验作为一只羊的生活。
假设每匹狼都很聪明，求最后草原上会剩下多少匹狼？

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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.

求最大的实数$C$，使得不等式
$$\sum_{k=1}^{100}\frac{1}{x_{k}}\cdot\sum_{k=1}^{100}\left(x_{k}^{3}+2x_{k}\right)-\left(\sum_{k=1}^{100}\sqrt{x_{k}^{2}+1}\right)^{2}\geq C$$
对任意100个两两不同的正整数$x_{1}, x_{2}, \ldots, x_{100}$成立。

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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 $n$ 次实系数多项式 $f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}$ 满足:对任意 $x \in[0,1] \cup[99,100]$ ，都有 $-1 \leq f(x) \leq 1$ 。
(1) 当 $n=2$ 时, 求 $f(50)$ 的最大可能值;
(2) 当 $n=100$ 时, 求 $f(50)$ 的最大可能值.

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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.

若集合 $X \subseteq\{1,2, \cdots, n\}$ 满足: 在 $X$ 的任意五元子集 $A$ 中都可以找到两个元素 $a 、 b(a<b)$ 使得 $a \mid b$, 则记 $|X|$ 的最大可能值是 $f(n)$ （ $|X|$ 表示集合 $X$ 中的元素个数）。求$f(2023)+f(3000)+f(10000)$.

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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.

某国足协规定：在联赛中比赛双方胜一场得$a$分，平一场得$b$分，负一场得0分，其中，实数$a>b>0$。若一支球队经过$n$场比赛后总得分恰有2015种可能，求$n$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.

已知数列 $\left\{x_{n}\right\}$ 满足:
$x_{1}=a, x_{2}=b, x_{n+2}=3 x_{n+1}+2 x_{n}$.
若存在整数 $k \geqslant 3$, 使得 $x_{k}=2019$, 则这样的有序正整数对 $(a, b)$ 的个数为 $\_\_\_\_$.

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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.

长春嘉年华 $G$ 共有 101 个景点，任意两个景点之间总可以通过一些栈道相连，且每条栈道恰连接两个景点，每两个景点之间至多有一条栈道. 若从某个景点出发，沿一系列不同栈道经过不同景点之后回到了该景点, 则把这样的一系列栈道称为 "环线", 并将环线中所含的栈道条数定义为环线的"长度". 若该嘉年华 $G$ 的每条栈道都落在一个长度至多为 $c(G)$ 的环线上, 且栈道总数为 $e(G)$, 求 $c(G)+e(G)$ 的最小可能值.

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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 $A=\{1,2, \cdots, 10\}$. 定义函数 $f: A \rightarrow A$ 满足:
(1) 对于任意的 $x 、 y \in A$, 若 $x \leqslant y$, 则
$f(x) \leqslant f(y)$;
(2) $f(3)=4$.
这样的函数 $f$ 的个数为  $\qquad$.

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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.

定义运算 $a * b=a b-5\left[\frac{a b}{5}\right]$ ，其中， [ $x$ ] 表示不超过实数 $x$ 的最大整数, 集合 $A=$ $\{0,1,2,3,4\}$ ，双射 $f: A \rightarrow A$ 满足 $f(a * b)=$ $f(a) * f(b)$. 则满足条件的 $f$ 共有 $\qquad$个.

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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
$$
\begin{aligned}
f(x)= & a_{2020} x^{2020}+a_{2019} x^{2019}+\cdots+ \\
& a_{2018} x^{2018}+a_{1} x+a_{0},
\end{aligned}
$$
其中， $a_{i} \in \mathbf{Z}, i=0,1, \cdots, 2020$. 求有序数对 $(n, p)$ ( $p$ 为素数) 的个数，满足 $p^{2}<n<2020$ ，且对于任意的 $i=0,1, \cdots, n, \mathrm{C}_{n}^{i}+f(i)$ 模 $p$ 同余。

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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.

设方程 $x y z=900$ 的全部正整数解为 $\left(x_{i}, y_{i}, z_{i}\right)(1 \leqslant i \leqslant n)$. 则 $\sum_{k=1}^{n}\left(x_{k c}+y_{k}+z_{k}\right)=\qquad$.

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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.

有 2023 个顶点的完全图
(即任意两个点之间均有连线), 现将每条边进行染色，有公共点的任意两条边颜色不同，定义每个点所连出的边的颜色为一个颜色集。求不同颜色集个数的最小值。

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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.

无锡市有100个景点，各景点每接待1名游客则景点在旅游局积1分。现在共有100名游客在100天的旅游季内参观这些景点。每天若干人出门参观互不相同的景点，且每名游客自行决定参观一个景点或休息不参观。在旅游季结束时，旅游局发现任何一名游客未曾在不同的两天里参观同一个景点，且任何两名曾在同一天出门的游客均未曾参观过同一个景点。求各景点该旅游季在旅游局获得的最大可能积分总和。

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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$\boldsymbol{u}=\left(u_{1},u_{2},u_{3}\right)$和$\boldsymbol{v}=$$\left(v_{1},v_{2},v_{3}\right)$是空间向量，满足$u_{i}、v_{i}(i=1,2,3)$均为整数，且
$0.9999<\cos\langle\boldsymbol{u},\boldsymbol{v}\rangle<1$。
记$S=\left|u_{1}\right|+\left|u_{2}\right|+\left|u_{3}\right|+\left|v_{1}\right|+\left|v_{2}\right|+\left|v_{3}\right|$。求$[\sqrt{S}]$的最小可能值，其中，$[x]$表示不超过实数$x$的最大整数。

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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.

小明在两个 $3 \times 3$ 方格表 $A 、 B$ 中随机填入 $1,2, \cdots, 9$ ，使得每个数字恰在每个方格表中出现一次。对于 $1 \leqslant i<j \leqslant 9$ ，若 $i 、 j$ 在 $A$中位于同一行或同一列，且在 $B$ 中位于同一行或同一列，则称 $(i, j)$ 为"紧密数对"。紧密数对个数的数学期望是$\qquad.$

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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.

全集 $U=\{1,2, \cdots, 50\}, A=\{a, b\}$ 。若集合 $C_{U} A$ 中所有元素之和等于 $a b$, 则 $a b=$
$\qquad$.

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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.

将两个相同的白球、三个相同的红球、四个相同的黑球全部放入三个不同的袋子
中. 则没有空袋的放法数为 $\qquad$.

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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.

已知 $S_{n}$ 为数列 $\left\{a_{n}\right\}$ 的前 $n$ 项和，规定 $S_{0}=0$ 。若对任意的 $n \in \mathbf{Z}_{+}$，均有
$$
\begin{aligned}
& \frac{a_{n}}{2017}=-\frac{2017+S_{n-1}}{n}, \\
& \text { 则 } \sum_{n=1}^{2017} 2^{n} a_{n}=
\end{aligned}
$$

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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.

$\cos18^{\circ}+2\cos36^{\circ}+\cdots+20\cos360^{\circ}$
=_______。

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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$P(4,2)$，过点$P$的直线$l$与$x$轴、$y$轴的正半轴分别交于$A、B$两点，$O$为坐标原点。则$\triangle AOB$周长的最小值为_______。

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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$a_{1},a_{2},\cdots,a_{k}\left(k\in\mathbf{Z}_{+}\right)$均为大于1的整数，且满足$$\left(a_{1}!\right)\left(a_{2}!\right)\cdots\left(a_{k}!\right)\mid2017!$$。当$k$变化时，求$\sum\limits_{i=1}^{k}a_{i}$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.

已知$f:\mathbf{Z}\rightarrow\mathbf{Z}$为偶函数，且$f(1)=1,f(2017)\neq1$。若对任意的$x、y\in\mathbf{Z}$均满足
$$2f(x+y)-f(x)-f(y)\leqslant|f(x)-f(y)|$$，求$f(2018)$的所有可能值。

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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$\left\{a_{n}\right\}$的前$n$项和为$S_{n}$，且
$a_{1}=3,S_{n}=2a_{n}+\frac{3}{2}\left((-1)^{n}-1\right)$。
若$\left\{a_{n}\right\}$中存在三项$a_{1}、a_{p}、a_{q}(p、q\in\left.\mathbf{Z}_{+},1<p<q\right)$成等差数列，则$q-p$=______。

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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.

已知不等式 $\left|a x^{2}+b x+a\right| \leqslant x$ 对 $x \in$ $[1,2]$ 恒成立. 则 $3 a+b$ 的最大值为 _____.

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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 $a_{n}$ 是与 $\sqrt{\frac{n}{2}}$ 的差最小的整数， $b_{n}$ 是与 $\sqrt{2 n}$ 的差最小的整数，记 $\left\{\frac{1}{a_{n}}\right\}$ 的前 $n$ 项和为 $S_{n},\left\{\frac{1}{b_{n}}\right\}$的前 $n$ 项和为 $T_{n}$ ，则 $2 T_{100}-S_{100}$ 。

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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.

已知 $f(x)=x^{2}-53 x+196+\left|x^{2}-53 x+196\right|$, 求 $f(1)+f(2)+\cdots+$ $f(50)$ 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 $k$ 的最小值，使得 $7 \times 7$ 方格表中挖去 $k$ 个格后，不存在 $T$ 字型 $(T$ 字型指一个方格及三个与其有公共边的方格构成的图形)。

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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 positive integers $n$ and $a_{1}, a_{2}, \ldots, a_{n}$ such that
$$
a_{1}+a_{2}+\cdots+a_{n}=1979
$$
and the product $a_{1} a_{2} \cdots a_{n}$ is as large as possible.

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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 $f: \mathbf{R} \rightarrow \mathbf{R}$, 若对任意实数 $x 、 y 、 z$都有

$$
\begin{aligned}
& \frac{1}{3} f(x y)+\frac{1}{3} f(x z)-f(x) f(y z) \geqslant \frac{1}{9} ,
\end{aligned}
$$

求 $\sum_{i=1}^{100}[i f(i)]$.其中, $[x]$ 表示不超过实数 $x$ 的最大整数。

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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.

设 $f(x)$ 是二次函数, 且 $2 x^{2}-4 x+3 \leqslant f(x) \leqslant 3 x^{2}-6 x+4, f(3)=11$, 则 $f(5)=$?.

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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.

若实数 $x, y$ 满足 $4 x^{2}-4 x y+2 y^{2}=1$ ，则 $3 x^{2}+x y+y^{2}$ 的最大值与最小值的和为？

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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.

已知抛物线 $\Omega: y=x^{2}$, 动线段 A B 在直线 $y=\sqrt{3} x-3$ 上 ( B 在 A 右侧), 且 $|A B|=2 \sqrt{3}$. 过 A 作 $\Omega$的切线, 取左边的切点为 M. 过 B 作 $\Omega$ 的切线, 取右边的切点为 N. 当 $M N / / A B$ 时, 求点 A 的横坐标.

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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.

Alice drew a regular 2021-gon in the plane. Bob then labelled each vertex of the 2021-gon with a real number, in such a way that the labels of consecutive vertices differ by at most 1. Then, for every pair of non-consecutive vertices whose labels differ by at most 1, Alice drew a diagonal connecting them. Let $d$ be the number of diagonals Alice drew. Find the least possible value that $d$ can obtain.

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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.

Every positive integer is marked with a number from the set $\{0,1,2\}$, according to the following rule:
if a positive integer $k$ is marked with $j$, then the integer $k+j$ is marked with 0.
Let $S$ denote the sum of marks of the first 2019 positive integers. Determine the maximum possible value of $S$.

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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 $a_{1} 、 a_{2} 、 a_{3} 、 a_{4}$ 均为实数。若集合 $\left\{a_{1}, a_{2}, a_{3}, a_{4}\right\}$ 的所有非空真子集的元素之和为 28, 则 $a_{1}+a_{2}+a_{3}+a_{4}=$   .

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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.

$x \&(y \& z)=x \& y+z, x \& x=0$, 求 $2000 \& 2022$

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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.

若三个正整数$a,b,c$使得$\sqrt{a+b}$，$\sqrt{b+c}$，$\sqrt{c+a}$是三个连续正整数，则$a^{2}+b^{2}+c^{2}$的最小值是 ？

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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.

若定义在 $\mathbb{R}$ 上的光滑函数 $h(x)$ 在 $x=0$ 处的泰勒展开为 $c_{k} x^{k}+c_{k+1} x^{k+1}+\cdots$,其中 $c_{k} \neq 0$ ，则定义 $\operatorname{ord}(h)=k$ ，若 $h(x)$ 在 $x=0$ 处泰勒展开系数均为 0 ，则定义 $\operatorname{ord}(h)=\infty$ ，令 $S=\left\{f(x) \mid f(x)\right.$ 为定义在 $\mathbb{R}$ 上的光滑奇函数，满足 $\left.f^{\prime}(0)=1\right\}$ ，设
$$
m=\min \{\operatorname{ord}(f(g(x))-g(f(x))) \mid f(x) \in S, g(x) \in S\}
$$
1. 求 $m$;
2. 对于 $f(x), g(x) \in S$ ，设 $f^{(i)}(0)=a_{i}, g^{(i)}(0)=b_{i}, i=2,3, \ldots$ ，求 $f(g(x))-g(f(x))$在 $x=0$ 处泰勒展开式中 $x^{m}$ 的系数关于 $a_{i}, b_{i}$ 的表达式。

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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.

对非负实数 $x$ 令 $\pi(x)$ 为不超过 $x$ 的素数个数，如 $\pi(1)=0, \pi(2)=1$ ，再令 $f(x)$为第 $\left[\frac{x}{4}\right]+1$ 个素数, 求 $\int_{0}^{100}\left(\pi(x)+\frac{1}{4} f(x)\right) \mathrm{d} x$ 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.

$I=\int_{0}^{\frac{\pi}{2}} \frac{1}{1+(\cos t)^{2}} d t$, 则 $[10 I]=$？

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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.

考虑如下 $2021 \times 2021$ 的方阵 $M=\left(\begin{array}{cccccc}0 & 2 & 3 & 4 & \cdots & 2021 \\ 1 & 0 & 3 & 4 & \cdots & 2021 \\ 1 & 2 & 0 & 4 & \cdots & 2021 \\ 1 & 2 & 3 & 0 & \cdots & 2021 \\ \vdots & \vdots & \vdots & \vdots & \ddots & 2021 \\ 1 & 2 & 3 & 4 & \cdots & 2021\end{array}\right)$, 这里 $M$ 的行通过依次地将行向量 $(1,2,3, \cdots, 2020,2021)$ 的某个位置替换为 0 得到. 则 $\left[\dfrac{1}{2020!} \operatorname{det} M\right]=$ ？

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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.

令 $A=\left(\begin{array}{rrr}-\frac{2}{3} & 1 & 0 \\ -1 & \frac{4}{3} & 0 \\ 2 & -\frac{4}{3} & \frac{2}{3}\end{array}\right)$. 令 $M=A-A^{2}+A^{3}-A^{4}+\cdots+(-1)^{n-1} A^{n}+\cdots$, 则
$[2021 \operatorname{det} M]=$ ？

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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.

如果实数 $k$ 使得关于 $x, y, z$ 的线性方程组 $\begin{cases}k x+y+2 z=1 \\ 3 x+k y+4 z=1 \\ -x+2 y+z=2\end{cases}$ 无解, 则称其是一个“好实数”. 设所有好实数之和为 $I$, 则 $[I]=$ ？

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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.

已知下列结论成立：在复平面上的多项式 $f(z), g(z)$ 和实数 $r>0$ ，若对 $|z|=r$ ，都有 $|g(z)|<|f(z)|$ ，则在 $\{z||z|<r\}$ 中， $f(z)$ 与 $f(z)+g(z)$ 的零点数相等 (计算重数)。现已知多项式 $z^{9}+2 z^{5}-8 z^{3}+3 z+1$, 求其在 $\{z|1<|z|<2\}$ 上的零点个数（计算重数）。

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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.

已知二元函数
$f(x, y)=\sqrt{4 y^{2}-12 y+10}+\sqrt{18 x^{2}-18 x+5}+\sqrt{18 x^{2}+4 y^{2}-12 x y+6 x-4 y+1}$
的最小值为 $a$ ，求 $\left[8 a^{2}\right]$ 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.

A rectangle $ABCD$ is made up of five small congruent rectangles as shown in the given figure. Find the perimeter, in cm, of $ABCD$ if its area is $6750 cm^2$.

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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.

Evaluate

$$\begin{aligned}
&\begin{aligned}&~~~~\quad1\left(\frac11+\frac12+\frac13+\frac14+\frac15+\frac16+\frac17+\frac18+\frac19+\frac1{10}\right)\end{aligned} \\
&+\quad3\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}\right) \\
&+\quad5\left(\frac13+\frac14+\frac15+\frac16+\frac17+\frac18+\frac19+\frac1{10}\right) \\
&+\quad7\left(\frac14+\frac15+\frac16+\frac17+\frac18+\frac19+\frac1{10}\right) \\
&\begin{array}{cc}+&9\left(\frac15+\frac16+\frac17+\frac18+\frac19+\frac1{10}\right)+11\left(\frac16+\frac17+\frac18+\frac19+\frac1{10}\right)\end{array} \\
&\begin{array}{cc}+&13\left(\frac17+\frac18+\frac19+\frac1{10}\right)+15\left(\frac18+\frac19+\frac1{10}\right)\end{array} \\
&+\quad17\left(\frac19+\frac1{10}\right)+19\left(\frac1{10}\right)
\end{aligned}$$

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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 that $A^4=75600\times B$. If $A$ and $B$ are positive integers, find the smallest value of $B$.

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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 $x$ be a fraction between $\frac{35}{36}$ and $\frac{91}{183}$. If the denominator of $x$ is $455$ and the numerator and denominator have no common factor except $1$, how many possible values are there for $x$?

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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.

An isosceles right triangle is removed from each corner of a square piece of paper so that a rectangle of unequal sides remains. If the sum of the areas of the cut-off pieces is $200$ and the lengths of the legs of the triangles cut off are integers, find the area of the rectangle.

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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.

$P(x)$ is a polynomial of degree $3n$ such that
$$\begin{array}{rcl}P(0)=P(3)=\cdots&=&P(3n)=2,\\P(1)=P(4)=\cdots&=&P(3n-2)=1,\\P(2)=P(5)=\cdots&=&P(3n-1)=0,\quad and\\&&P(3n+1)=730.\end{array}$$
Determine $n$.

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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.

What is the smallest integer $n$, greater than one, for which the root-mean-square of the first $n$ positive integers is an integer?
Note. The root-mean-square of  $n$  numbers $a_1, a_2, \cdots, a_n$ is defined to be
$\left[\frac{a_1^2 + a_2^2 + \cdots + a_n^2}n\right]^{1/2}$

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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.

设函数 f(x)=|2-\log_3x|，正实数 a,b,c 满足 a<b<c，且 f(a)=2f(b)=2f(c). 求 \df{ac}{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.

定义在 \bR 上的奇函数 f(x) 同时是以 2 为周期的周期函数，并且在 x\in(0,1) 时，f(x)=\ln(x^2-2x+\frac{3}{2}). 则函数 f(x) 在 [0,10] 上的零点个数为__________.（李纪琛供题）

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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.

若存在实数 a 及正整数 n 使得 f(x)=\cos2x-a\sin x 在 (0,n\pi) 内恰有 2022 个零点，则满足条件的正整数 n 的值有__________个.

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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:\bR\rightarrow\bR，若对任意实数 x,y,z 都有\df{1}{3}f(xy)+\df{1}{3}f(xz)-f(x)f(yz)\geq\df{1}{9},求 \ds{i=1}{100}[if(i)]，其中 [x] 代表不超过 x 的最大整数.

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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.

设 A 是一个 11 元实数集，令集合 B=\{uv\ |\ u,v\in A,u\neq v\}. 求 B 的元素个数的最小值.

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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.

对于一个由至少两个正整数构成的集合，若每个元素均与其他至少一个元素不互质，则称该集合为"芳香集".设 P(n)=n^2+n+1. 求最小的正整数 b，使得存在非负整数 a，满足集合 \{P(a+1),P(a+2),\cdots,P(a+b)\} 是芳香集.

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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.

求最小的正整数 m，使得对任意大于 3 的质数 p，都有105\mid 9^{p^2}-29^p+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.

设数列 \{a_n\} 满足 a_0=\df{1}{2}, a_{n+1}=a_n+\df{a_n^2}{2012}, n\geq0. 求所有的整数 k，使得 a_k<1<a_{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.

设整数 n\geq 4，A=\{a_1,a_2,\cdots,a_n\} 是整数集，满足 2023\in A，且 A 中任意两个元素的差的绝对值是素数. 则这样的集合 A 共有_________个.

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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.

正整数 n 的分拆 P=(x_1,x_2,\cdots,x_n) 是一个非负整数组，满足 \ds{k=1}{n}kx_k=n. 正整数 m(m\leq n) 的分拆 Q=(y_1,y_2,\cdots,y_m) 称为与 P 相容，如果 y_i\leq x_i, i=1,2,\cdots,m.记 S(n) 为 n 的满足下述要求的分拆 P 的个数：对每个小于 n 的正奇数 m，m 恰有一个与 P 相容的分拆；对每个小于 n 的正偶数 m，m 恰有两个与 P 相容的分拆. 求 S(2010) 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,a_2,\cdots,a_{100} 是非负整数，同时满足以下条件：(1) 存在正整数 k\leq100，使得 a_1\leq a_2\leq\cdots\leq a_k，而当 i>k 时 a_i=0；(2) a_1+a_2+a_3+\cdots+a_{100}=100；(3) a_1+2a_2+3a_3+\cdots+100a_{100}=2022.求 a_1+2^2a_2+3^2a_3+\cdots+100^2a_{100} 的最小可能值.

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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 n 为正整数，且 \ds{k=0}{n}\df{(-1)^k\text{C}_n^k}{k^3+9k^2+26k+24}=\df{1}{312}，则 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 9 元集合 A=\{a+bi\ |\ a,b\in\{1,2,3\}\}，其中 i 是虚数单位. \alpha=(z_1,z_2,\cdots,z_9) 是 A 中所有元素的一个排列，满足 |z_1|\leq|z_2|\leq\cdots\leq|z_9|，则这样的排列 \alpha 的个数为__________.

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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.

已知首项系数为 1 的五次多项式 f(x) 满足：f(n)=8n, n=1,2,\cdots,5，则 f(x) 的一次项系数为__________.

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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.

在奥林匹克高级中学有 2010 名学生和 100 间教室. 开始时，每名学生任意进入到一间教室. 只要所有学生不都在同一间教室，且存在教室 Y 中的学生数目不少于教室 X 中的学生数目，则教室 X 中的一名学生走到教室 Y 中. 若每分钟只移动一名学生，且此过程在 M 分钟后结束，求 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.

设正整数 n 的正因数为 1=d_1<d_2<\cdots<d_k=n，其中 k\geq6. 求所有的 n 使得 n=d_5^2+d_6^2.

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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.

设 a_1,a_2,\cdots,a_{20}\in\{1,2,\cdots,5\}，b_1, b_2, \cdots, b_{20} \in\{1,2,\cdots,10\}. 集合 X=\{(i,j)\ |\ 1\leq i<j\leq 20,(a_i-a_j)(b_i-b_j)<0\}, 求 X 的元素个数的最大值.

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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) 的定义域为 (0,+\infty)，且满足f(x)-2xf\left(\df{1}{x}\right)+x^2=0,则 f(x) 的最小值为__________.

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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.

将一个 3\times 3 的正方形的四个角上各去掉一个单位正方形所得到的图形称为" 十字形". 在一个 10\times 11 的棋盘上，最多可以放置多少个互不重叠的十字形？（每个十字形恰好盖住棋盘上的 5 个小方格）

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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.

定义运算 a\ast b=ab-5\left[\df{ab}{5}\right]. 集合 A=\{0,1,2,3,4\}，双射 f:A\rightarrow A 满足 f(a\ast b)=f(a)\ast f(b)，则满足条件的 f 共有__________个.

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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 \{a_n\} 满足 a_{n-1}=a_n+a_{n-2}, n\geq3. 记前 n 项和为 S_n，若 S_{2018}=2017, S_{2019}=2018，则 S_{2020}=\underline{\qquad\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.

设方程 z^3-2z-4=0 的三个复根为 z_1, z_2, z_3，则它们在复平面上对应的点构成三角形的面积为________.

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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.

若能从 1,2,\cdots,100 中取出 k 个两两互质的数，则 k 的最大值为________.

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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.

给定实数集合 A,B，定义运算A\otimes B=\{x\ |\ x=ab+a+b,a\in A,b\in B\}.设 A=\{0,2,4,\cdots,18\},B=\{98,99,100\}，则 A\otimes B 中的所有元素之和为__________.

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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 a_{n+1}=\left[\df{a_n}{2}\right]+\left[\df{a_n}{3}\right], n=1,2,\cdots,7，这里 [x] 表示不超过 x 的最大整数. 若 a_8=8，则正整数 a_1 有__________种可能的取值情况.

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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.

问：至多能从集合 \{1,2,\cdots,100\} 中取出多少个数，使得其中任意不同两数的和与积均不为 100 的倍数？

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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.

在一个社交网站上有 2022 名用户，其中一些是朋友，朋友关系是相互且永久的. 从现在开始，两名陌生用户只有至少有两个公共朋友，才允许他们成为朋友. 问：初始时至少需要有多少对朋友，才能使得最终任两名用户都成为朋友？

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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.

对于正整数 n，将其各位数字之和记为 s(n)，各位数字之积记为 p(n). 若 s(n)+p(n)=n 成立，就称 n 为巧合数，则所有巧合数的和为__________.

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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.

称一个九位数是平衡的，如果数字 1\sim9 都出现. 将所有平衡的数按递增顺序依次写下，设 S 是所得的数字序列. 求最小的正整数 k，使得 S 的任意两个由 k 个连续数字构成的子列互不相同.

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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.

设平面向量 \textbf{a},\textbf{b} 满足：|\textbf{a}|=1, |\textbf{b}|=2, \textbf{a}\bot\textbf{b}. 点 O, A, B 为平面上的三点，满足：\overrightarrow{OA}=2\textbf{a}+\textbf{b}，\overrightarrow{OB}=-3\textbf{a}+2\textbf{b}，则 \triangle AOB 的面积为__________.

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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.

已知抛物线 \Omega:y=x^2，动线段 AB 在直线 y=\sqrt{3}x-3 上（B 在 A 右侧），且 |AB|=2\sqrt{3}. 过 A 作 \Omega 的切线，取左边的切点为 M. 过 B 作 \Omega 的切线，取右边的切点为 N. 当 MN\parallel AB  时，求点 A 的横坐标.

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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 x^3-9x^2+8x+2=0 有 3 个实根 p,q,r，则 \df{1}{p^2}+\df{1}{q^2}+\df{1}{r^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.

老师为学生购买纪念品，商店中有三种不同类型的纪念品（同类型的纪念品完全相同），价格分别为 1 元、2 元、4 元. 李老师计划用 101 元，且每种纪念品至少购买一件，则共有__________种不同的购买方案.

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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.

设集合 A=\{1,2,\cdots,100\}，函数 f:A\rightarrow A 满足：(1) 对任意 1\leq i\leq 99，|f(i)-f(i+1)|\leq 1；(2) 对任意 1\leq i\leq 100，f(f(i))=100.求 \ds{i=1}{100}f(i) 的最小可能值.

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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 k，使得对任意满足 k(ab+bc+ca)>5(a^2+b^2+c^2) 的正实数 a,b,c，都存在三边长分别为 a,b,c 的三角形.

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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.

求最大的正实数 k，使得对任意满足 a+b+c=1 的非负实数 a,b,c，均有\ds{}{}\df{a}{1+9bc+k(b-c)^2}\geq\df{1}{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.

求最大的正整数 n，使得存在 n 个正整数 x_1<x_2<\cdots<x_n，满足x_1+x_1x_2+\cdots+x_1x_2\cdots x_n=2021.

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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.

在正整数 1,2,\cdots,20210418 中，有多少个数的数码和是 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.

使得 3n+1 与 5n+1 同时成为完全平方数的最小正整数 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.

方程 \df{1}{x}+\df{1}{y}=\df{1}{2021} 的正整数解的组数为__________.

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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)=x^2-2ax-2a|x-a|+1 有且仅有三个零点，则实数 a 的值为__________.

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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 \{a_n\} 的前 n 项和为 S_n，且 a_1=3, S_n=2a_n+\df{3}{2}((-1)^n-1). 若 \{a_n\} 中存在三项 a_1, a_p, a_q (p,q\in\bN_+,1<p<q) 成等差数列，则 q-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.

设 X=\{1,2,\cdots,10\}，f:X\rightarrow X 的映射满足① f\circ f=I_X，其中 f\circ f 是复合映射，I_X 为 X 上的恒等映射；② |f(i)-i|\leq 2，对任意 i\in X.求映射 f 的数目.

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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.

S 是集合 \{1,2,\cdots,2023\} 的子集，满足任意两个元素的平方和不是 9 的倍数，则 |S| 的最大值是__________.（这里 |S| 表示 S 的元素个数）

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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.

已知函数 f(x)=\sin\omega x+\sin 2x，其中 \omega\in\bN_+, \omega\leq2023. 若 f(x)<2 恒成立，则满足题设的常数 \omega 的个数为__________.

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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.

使得 n^2+2023n 为平方数的正整数 n 的最小值是__________.

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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.

现有 11 位同学报名博物馆的志愿讲解活动，活动从上午 9 点开始到下午 5 点结束，每小时安排一场公益小讲堂，每场需要 1 位同学为参观的游客提供讲解服务. 为避免同学们劳累，馆方在排班时不会让同一人连续讲解 2 场，并且第一场与最后一场需要两位不同的同学负责. 则馆方共有__________种排班方式.

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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,a_2,\cdots,a_6;b_1,b_2,\cdots,b_6;c_1,c_2,\cdots,c_6 都是 1, 2, \cdots, 6 的排列. 求 \ds{i=1}{6}a_ib_ic_i 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.

某国有 100 个城市，分别编号 1,2,\cdots,100. 在一些城市之间有道路相连，两个城市之间至多只有一条道路. 称有道路相连的两个城市是相邻的，一位外国游客计划来该国旅行 100 次. 对 1\leq i\leq 100，在第 i 次旅行中他从编号为 i 的城市开始，之后每天前往与当前所在城市相邻且编号最小的城市，如果该城市在本次旅行中已经游览过，则不再前往且本次旅行结束.已知 100 次旅行结束后，该旅客游览每个城市的次数恰好相同. 求该国道路个数的最大可能值.

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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=9+10i（i 为虚数单位），若正整数 n 满足 |z^n|\leq 2023，则 n 的最大值为__________.

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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.

设集合 X=\{1,2,\cdots,2001\}. 求最小的正整数 m，使得对 X 的任一 m 元子集 W，都存在 u,v\in W（u,v 可以相同），满足 u+v 是 2 的幂.

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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 n，使得存在 k 及大于 3 的整数 n_1, n_2, \cdots, n_k，满足n=n_1n_2\cdots n_k=2^{\frac{1}{2^k}(n_1-1)(n_2-1)\cdots(n_k-1)}-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.

求最大的实数 C，使得不等式\left(\ds{k=1}{100}\df{1}{x_k}\right)\left(\ds{k=1}{100}(x_k^3+2x_k)\right)-\left(\ds{k=1}{100}\sqrt{x_k^2+1}\right)^2\geq C对任意两两不同的正整数 x_1, x_2, \cdots, x_{100} 成立.

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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.

设正整数 a,b 满足 a<b\leq100. 若存在正整数 k，使得 ab\mid a^k+b^k，则称数对 (a,b) 是"好的". 求所有好数对的个数.

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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.

设复数列 \{z_n\} 满足 |z_1|=1,|z_n+z_{n-1}|=n, n\geq2. 记 |z_{20}-z_{23}| 和 |z_{20}+z_{23}| 的最大值分别为 m 和 n，则 m-n 的值为__________.

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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 及实数 x(0\leq x<n)，定义f(n,x)=(1-\{x\})\cdot\text{C}_n^{[x]}+\{x\}\cdot\text{C}_n^{[x]+1},其中 [x] 表示不超过实数 x 的最大整数，\{x\}=x-[x]. 若整数 m,n\geq2 满足f\left(m,\df{1}{n}\right)+f\left(m,\df{2}{n}\right)+\cdots+f\left(m,\df{mn-1}{n}\right)=123,求 f\left(n,\df{1}{m}\right)+f\left(n,\df{2}{m}\right)+\cdots+f\left(n,\df{mn-1}{m}\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.

求满足下列条件的大于 5 的最小奇数 a：存在正整数 m_1, n_1, m_2, n_2，使得 a=m_1^2+n_1^2，a^2=m_2^2+n_2^2，且 m_1-n_1=m_2-n_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.

平面上有 2022 个点，满足任意三点不共线. 每个点被染为红色或蓝色，使得每三个（不同的）红点构成的三角形都至少包含一个蓝点. 求红点个数的最大可能值.

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