meetforever/GRPO_CL huggingface dataset:
user: Find the smallest positive integer $n$ such that for every integer $m$, with $0<m<1993$, there exists an integer $k$ for which
$$
\frac{m}{1993}<\frac{k}{n}<\frac{m+1}{1994}
$$
Answer: 3987
user: 设单调递增数列 $\left\{a_{n}\right\}$ 的各项均为正整数，且 $a_{7}=120, a_{n+2}=a_{n}+a_{n+1}\left(n \in \mathbf{Z}_{+}\right)$.则 $a_{8}=\qquad.$
Answer: 194
user: 若正整数 $m 、 n$ 满足 $\frac{(m+n)!}{n!}=5040$,则 $m!n$ 的值为 $\qquad$.
Answer: 144
user: 在数列 $\left\{a_{n}\right\}$ 中, $a_{4}=1, a_{11}=9$, 且任意连续三项的和都是 15, 则 $a_{2016}=\qquad$.
Answer: 5
user: 在数列 $\left\{a_{n}\right\}$ 中， $a_{4}=1, a_{11}=9$ ，且任意连续三项的和均为 15. 则 $a_{2016}=$
Answer: 5
user: 设方程 $x y=6(x+y)$ 的全部正整数解为 $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \cdots,\left(x_{n}, y_{n}\right)$, 则 $\sum_{k=1}^{n}\left(x_{k}+y_{k}\right)=\qquad$.
Answer: 290
user: 设方程 $x y=6(x+y)$ 的全部正整数解为 $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \cdots,\left(x_{n}, y_{n}\right)$. 则
$\sum_{k=1}^{n}\left(x_{k}+y_{k}\right)=\quad.$
Answer: 290
user: 设 $n$ 是正整数, 当 $n>100$ 时, $\sqrt{n^{2}+3 n+1}$ 的小数部分的前两位数是 $\qquad$.
Answer: 49
user: 已知 i 是虚数单位. 若
$z=1+\mathrm{i}+\cdots+\mathrm{i}^{2013}$,
将复数 $z$ 的共轭复数记为 $\bar{z}$, 则 $z \cdot \bar{z}=$
Answer: 2
user: 设 $(1-2 x)^{7}=\sum\limits_{k=0}^{7} a_{k} x^{k}$. 则
$2 a_{2}+3 a_{3}+4 a_{4}+5 a_{5}+6 a_{6}+7 a_{7}=$
Answer: 0
user: 设在 $5 \times 5$ 的方格表的第 $i$ 行第 $j$ 列所填的数为 $a_{i j}\left(a_{i j} \in\{0,1\}\right), a_{i j}=a_{j i}(1 \leqslant i、j \leqslant 5)$ .则表中共有五个 1 的填表方法总数为 $\qquad$ （用具体数字作答).
Answer: 326
user: 设 $a 、 b$ 为实数，对任意满足 $0 \leqslant x \leqslant 1$ 的实数 $x$ 均有 $|a x+b| \leqslant 1$.则 $|20 a+14 b|+|20 a-14 b|$的最大值为 $\qquad$.
Answer: 80
user: 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 $\qquad$.
Answer: -1
user: 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 $\overrightarrow{P B} \cdot \overrightarrow{P D}=$ $\qquad$.
Answer: 0
user: 对任意正整数 $n$ ，定义函数 $\mu(n)$ ：  
$\mu(1)=1$ ，  
且当 $n=p_{1}^{\alpha_{1}} p_{2}^{\alpha_{2}} \cdots p_{t}^{\alpha_{t}} \geqslant 2$ 时,  
$$  
\mu(n)= \begin{cases}(-1)^{t}, & \alpha_{1}=\alpha_{2}=\cdots=\alpha_{t}=1 ; \\ 0, & \text { 否则, }\end{cases}  
$$  
其中， $t \geqslant 1, p_{1}, p_{2}, \cdots, p_{t}$ 为不同的素数.  
若记 $A=\left\{x_{1}, x_{2}, \cdots, x_{k}\right\}$ 为 12 的全部不同正因数的集合，则 $\sum\limits_{i=1}^{k} \mu\left(x_{i}\right)=\qquad$.
Answer: 0
user: 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 $\qquad$.
Answer: 1
user: 设集合 $I=\{1,2, \cdots, 8\}$. 若 $I$ 的非空子集 $A 、 B$ 满足 $A \cap B=\varnothing$ ，则称有序集合对 $(A, B)$ 为 $I$ 的"隔离集合对". 那么，集合 $I$ 的隔离集合对的个数为 $\qquad$ （用具体数字作答).
Answer: 6050
user: Let $P(x)=x^{2}-1$. How many distinct real solutions does the following equation have:  
$$  
\underbrace{P(P(\ldots(P}_{2004}(x)) \ldots))=0?  
$$
Answer: 2005
user: Find the number of positive integers $x$ satisfying the following two conditions:
1. $x<10^{2006}$;
2. $x^{2}-x$ is divisible by $10^{2006}$.
Answer: 3
user: Find all prime numbers $p$ for which there exists a unique $a \in\{1,2, \ldots, p\}$ such that $a^{3}-3 a+1$ is divisible by $p$.
Answer: 3
user: Let $V$ be a 10 -dimensional real vector space and $U_{1}$ and $U_{2}$ two linear subspaces such that $U_{1} \subseteq U_{2}, \operatorname{dim}_{\mathbb{R}} U_{1}=3$ and $\operatorname{dim}_{\mathbb{R}} U_{2}=6$. Let $\mathcal{E}$ be the set of all linear maps $T: V \longrightarrow V$ which have $U_{1}$ and $U_{2}$ as invariant subspaces (i.e., $T\left(U_{1}\right) \subseteq U_{1}$ and $T\left(U_{2}\right) \subseteq U_{2}$ ). Calculate the dimension of $\mathcal{E}$ as a real vector space.
Answer: 67
user: 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 $f(1)+f(2)+\cdots+f(1990)$.
Answer: 1326
user: 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 $f(1)+f(2)+\cdots+f(1990)=$ $\qquad$.
Answer: 1326
user: $1991^{2000}$ 除以 $10^{6}$ ，余数是 $\qquad$.
Answer: 880001
user: 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$\log _{3}\left|\left(z_{1} \overline{z_{2}}\right)^{2000}+\left(\overline{z_{1}} z_{2}\right)^{2000}\right|=$ $\qquad$.
Answer: 4000
user: 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 $\log _{3}\left|\left(z_{1} \overline{z_{2}}\right)^{2000}+\left(\overline{z_{1}} z_{2}\right)^{2000}\right|=$ $\qquad$.
Answer: 4000
user: 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.
Answer: 32
user: 整数 $\left[\frac{10^{93}}{10^{31}+3}\right]$ 的末两位数是 $\qquad$.
Answer: 8
user: 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.
Answer: 34
user: 整数 $\left[\frac{10^{93}}{10^{31}+3}\right]$ 的末尾两位数字是$\qquad$.（先写十位数字，后写个位数字，其中 $[x]$ 表示不超过 $x$ 的最大整数）
Answer: 8
user: 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 $\qquad$.
Answer: 7
user: 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.
Answer: 2186
user: 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.
Answer: 2186
user: 已知点集 $A=\left\{(x, y) \left\lvert\,(x-3)^{2}+(y-4)^{2} \leqslant\left(\frac{5}{2}\right)^{2}\right.\right\}$,
$B=\left\{(x, y) \left\lvert\,(x-4)^{2}+(y-5)^{2}>\left(\frac{5}{2}\right)^{2}\right.\right\}$. 则点集 $A \cap B$ 中的整点（即横、纵坐标均为整数的点）的个数是 $\qquad$.
Answer: 7
user: 直角坐标平面上, 满足不等式组 $\left\{\begin{array}{c}y \leqslant 3 x, \\ y \geqslant \frac{x}{3}, \\ x+y \leqslant 100\end{array}\right.$ 的整点个数是 $\qquad$.
Answer: 2551
user: 设 $M=\{1,2,3, \cdots, 1995\}, A$ 是 $M$ 的子集且满足条件：当 $x \in A$ 时， $15 x \notin A$ ，则 $A$ 中元素的个
数最多是 $\qquad$.
Answer: 1870
user: 求一切实数 $p$ ，使得三次方程 $5 x^{3}-5(p+1) x^{2}+(71 p-1) x+1=66 p$ 的三个根均为正整数。
Answer: 76
user: 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 $\qquad$.
Answer: 2551
user: 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。
Answer: 76
user: For what integer $a$ does $x^{2}-x+a$ divide $x^{13}+x+90$?
Answer: 2
user: 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.
Answer: 2
user: It is well known that the value of the polynomial $(x+1)(x+2) \cdots(x+n)$ is exactly divisible by $n$ for every integer $x$. Given $n$, let $k=k(n)$ be the minimal degree of any monic integral polynomial
$$
f(x)=x^{k}+a_{1} x^{k-1}+\cdots+a_{k}
$$
(with integer coefficients and leading coefficient 1) such that the value of $f(x)$ is exactly divisible by $n$ for every integer $x$.
Find the relationship between $n$ and $k=k(n)$. In particular, find the value of $k$ corresponding to $n=1000000$.
Answer: 25
user: 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 $\left.\left.g_{1}+g_{1}+g_{1}-g_{2}-g_{2}\right)\right]$
Answer: 7
user: 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?
Answer: 11
user: 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?
Answer: 11
user: 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.
Answer: 660
user: It is well known that the value of the polynomial $(x+1)(x +2) \dots (x + n)$ is exactly divisible by n for every integer x. Given n, let $k=k(n)$ be the minimal degree of any monic integral polynomial.

$$f(x)=x^k+a_1x^{k-1}+\dots +a_k$$

(with integer coefficients and leading coefficient 1) such that the value of f(x) is exactly divisible by n for every integer x.

Find the relationship between n and $k=k(n)$. In particular, find the value of k corresponding to n=1 000 000.
Answer: 25
user: How many positive integers $n$ are there such that $n$ is an exact divisor of at least one of the numbers $10^{40}, 20^{30}$?
Answer: 2301
user: 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.
Answer: 9
user: Suppose that $G$ is a group generated by elements $A$ and $B$, that is, every element of $G$ can be written as a finite "word" $A{ }^{{ }^{1}} B^{n_{2}} A^{n_{3}} \cdots B^{n_{k}}$, 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 $B$.
Answer: 8
user: Let $V$ be the collection of all quadratic polynomials $P$ with real coefficients such that
$$\sup\{\mid P^{\prime}(0)\mid:P\in V\}.$$
Answer: 8
user: Find
$$\lim_{n\to\infty}\left[\frac{1}{n^{5}}\sum_{h=1}^{n}\sum_{k=1}^{n}(5h^{4}-18h^{2}k^{2}+5k^{4})\right].$$
Answer: -1
user: Find
$$
\lim _{n \rightarrow \infty}\left[\frac{1}{n^{5}} \sum_{h=1}^{n} \sum_{k=1}^{n}\left(5 h^{4}-18 h^{2} k^{2}+5 k^{4}\right)\right].
$$
Answer: -1
user: Consider polynomial forms $ax^2-bx+c$ with integer coefficients which have two distinct zeros in the open interval $0\lt x\lt1$. Exhibit with a proof the least positive integer value of a for which such a polynomial exists.
Answer: 5
user: 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 $g$.
Answer: 9
user: Find the largest $A$ for which there exists a polynomial
$$
P(x)=A x^{4}+B x^{3}+C x^{2}+D x+E
$$
with real coefficients, which satisfies
$$
0<P(x)<1 \text { for }-1<x<1
$$
Answer: 4
user: How many positive integers n are there such that n is an exact divisor of at least one of the numbers $10^{40}$, $20^{30}$?
Answer: 2301
user: Find the length of the longest sequence of equal nonzero digits in which an integral square can terminate (in base 10) and find the smallest square which terminates in such a sequence.
Answer: 1444
user: Consider polynomial forms $ax^2-bx+c$ with integer coefficients which have two distinct zeros in the open interval $0<x<1$. Exhibit with a proof the least positive integer value of a for which such a polynomial exists.
Answer: 5
user: What is the units (i.e., rightmost) digit of $\left[\frac{10^{20000}}{10^{100}+3}\right]$ ? Here $[x]$ is the greatest integer $\leqslant x$.
Answer: 3
user: 求同时满足下列两个条件的多项式 $f(x)=a x^{3}+b x$ 的个数:
(1) $a, b \in\{1,2, \cdots, 2013\} ;$
(2) $f(1), f(2), \cdots, f(2013)$ 中任意两数之差不是 2013 的倍数。
Answer: 7200
user: 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。
Answer: 24
user: 在一个 $10 \times 10$ 的方格表中有一个由 $4 n$ 个 $1 \times 1$ 的小方格组成的图形，它既可被 $n$个"田"型的图形覆盖，也可被 $n$ 个"田"或 "吅"型（可以旋转）的图形覆盖。求正整数 $n$ 的最小值。
Answer: 4
user: 试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  
$$  
\begin{aligned}  
& \quad a=m_{1}^{2}+n_{1}^{2}, a^{2}=m_{2}^{2}+n_{2}^{2}, \\  
& \text { 且 } \quad m_{1}-n_{1}=m_{2}-n_{2}.  
\end{aligned}  
$$
Answer: 261
user: 求同时满足下列两个条件的多项式 $f(x)=a x^{3}+b x$ 的个数:  
(1) $a, b \in\{1,2, \cdots, 2013\} ;$  
(2) $f(1), f(2), \cdots, f(2013)$ 中任意两数之差不是 2013 的倍数。
Answer: 7200
user: 求最小的正整数 $m$, 使得对于任意大于 3 的质数 $p$ ，都有 $1051\left(9^{p^{2}}-29^{p}+m\right)$ 。
Answer: 20
user: 求满足下列关系式组
$$
\left\{\begin{array}{l}
x^{2}+y^{2}=2 z^{2}, \\
z<y \leq z+50,
\end{array}\right.
$$
的正整数解组 $(x, y, z)$ 的个数.
Answer: 131
user: 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
Answer: 9800
user: 方程 $x+y+z=2010$ 满足 $x \leq y \leq z$ 的正整数解（ $\mathrm{x}, \mathrm{y}, \mathrm{z} ）$ 的个数是
Answer: 336675
user: 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 $\qquad$.
Answer: 983
user: 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
| $x^{\prime}{ }_{1,1}$ | $x^{\prime}{ }_{1,2}$ | $\ldots$ | $x^{\prime}{ }_{1,25}$ |
| :---: | :---: | :---: | :---: |
| $x^{\prime}{ }_{2,1}$ | $x^{\prime}{ }_{2,2}$ | $\ldots$ | $x^{\prime}{ }_{2,25}$ |
| $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ |
| $x^{\prime}{ }_{100,1}$ | $x^{\prime}{ }_{100,2}$ | $\ldots$ | $x^{\prime}{ }_{100,25}$ |
Answer: 97
user: 设复数 $z=\cos \theta+i \sin \theta\left(0 \leqslant \theta \leqslant 180^{\circ}\right)$ ，复数 $z ，(1+i) z, 2 \bar{z}$ 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 $S$ ，点 $S$ 到原点距离的最大值是 $\qquad$。
Answer: 3
user: 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 $\qquad$.
Answer: 3
user: 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 $\qquad$
Answer: 28
user: 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 $\qquad$.
Answer: 28
user: 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 $\_\_\_\_$ .
Answer: 49
user: 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 $\qquad$.
Answer: 5
user: 若对任意的 $x \in\left(-\frac{1}{2}, 1\right)$, 有  $\frac{x}{1+x-2 x^{2}}=\sum_{k=0}^{\infty} a_{k} x^{k}$,  则 $a_{3}+a_{4}=$ $\qquad$.
Answer: -2
user: 设$M=1^{4}+2^{4}+\cdots+2013^{4}$. 则 $M$ 的个位数字为 $\qquad$.
Answer: 1
user: 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 $d q=$.
Answer: 42
user: 若正方体 $A_{1} A_{2} A_{3} A_{4}-B_{1} B_{2} B_{3} B_{4}$ 的棱长为 1 ，则集合  
$\left\{x \mid x=\overrightarrow{A_{1} B_{1}} \cdot \overrightarrow{A_{i} B_{j}}, i, j \in\{1,2,3,4\}\right\}$  中元素的个数为 $\qquad$.
Answer: 1
user: 设正六边形 $A B C D E F$ 的边长为 1. 则 $(\overrightarrow{A B}+\overrightarrow{D C}) \cdot(\overrightarrow{A D}+\overrightarrow{B E})=$
Answer: -3
user: 用 $[x]$ 表示不超过实数 x 的最大整数, 若数列 $\left\{a_{n}\right\}$ 满足: $a_{n}=\left[(2+\sqrt{3})^{2^{n}}\right](n \in$
$\left.\mathbf{N}^{*}\right)$ ，则 $a_{2020}$ 的末尾两位数字是 $\qquad$.
Answer: 53
user: 在平面直角坐标系中$,A(1,2)$,

$B(3,0),P$ 为圆$(x-3)^2+(y-2)^2=1$上任

意一点.设

$$\overrightarrow{OP}=\lambda\overrightarrow{OA}+\mu\overrightarrow{OB}(\lambda,\mu\in\mathbf{R}).$$
则 11$\lambda+9\mu$的最小值为
Answer: 12
user: 在平面直角坐标系中, $A(1,2)$, $B(3,0), P$ 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 $11 \lambda+9 \mu$ 的最小值为 $\qquad$.
Answer: 12
user: 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
Answer: 5
user: 设函数
$f(x)=\sqrt{2 x^{2}+2 x+41}-\sqrt{2 x^{2}+4 x+4}(x \in \mathbf{R})$.则 $f(x)$ 的最大值为 $\qquad$.
Answer: 5
user: 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\_\_\_\_\_\_
Answer: 4
user: 集合的容量是指集合中元素的和。则满足条件 “$A \subseteq \{1, 2, \ldots, 7\}$ 且若 $a \in A$ 时，必有 $8-a \in A$” 的所有非空集合 $A$ 的容量的总和是\_\_\_\_\_\_（用具体数字作答）。
Answer: 224
user: 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） ．
Answer: 3816
user: 在凸五边形 $A B C D E$ 中, $D E / / A C$, $\angle A D C=90^{\circ}, A C=12, C D=6, A C$ 平分 $\angle D A B, \angle B C E=60^{\circ}$. 则 $\triangle B C E$ 面积的最小值为 $\qquad$.
Answer: 27
user: 已知函数 $f(x)$ 在区间 $(0,+\infty)$ 上严格单调递减，对任意的 $x \in(0,+\infty)$ ，均有
$$
f(x) f\left(f(x)+\frac{2}{x}\right)=\frac{1}{3}.
$$
记 $g(x)=f(x)+4 x^{2}(x \in(0,+\infty))$. 则函数 $g(x)$ 的最小值是 $\qquad$.
Answer: 3
user: 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 $\qquad$.
Answer: 288
user: 已知数列 $\left\{a_{n}\right\}$ 满足:
$$
\begin{aligned}
& a_{1}=1, a_{2}=2, a_{2 k+1}=\frac{a_{2 k}^{2}}{a_{2 k-1}} \\
& a_{2 k+2}=2 a_{2 k+1}-a_{2 k}\left(k \in \mathbf{Z}_{+}\right)
\end{aligned}
$$
则 $a_{2022}$ 的末两位数字是 $\qquad$.
Answer: 32
user: 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\_\_\_\_\_\_
Answer: -4
user: 已知 $m > 0$。若函数

\[
f(x) = x + \sqrt{100 - mx}
\]

的最大值为 $g(m)$，求 $g(m)$ 的最小值。
Answer: 10
user: $\left(x^{2}+x-\frac{1}{x}\right)^{6}$ 的展开式中的常数项是 $\qquad$ (用具体数字作答).
Answer: -5
user: 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$\qquad$.
Answer: 0
user: 设 $x \in \mathbf{R}$. 则函数
$$
f(x)=|2 x-1|+|3 x-2|+|4 x-3|+|5 x-4|
$$
的最小值为 $\qquad$.
Answer: 1
user: 设方程 $x y=6(x+y)$ 的全部正整数解为 $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \cdots,\left(x_{n}, y_{n}\right)$, 则 $\sum_{k=1}^{n}\left(x_{k}+y_{k}\right)=$
$\qquad$ $\cdot$
Answer: 290
user: 设方程 $x y=6(x+y)$ 的全部正整数解为 $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \cdots,\left(x_{n}, y_{n}\right)$ . 则
$\sum_{k=1}^{n}\left(x_{k}+y_{k}\right)=\qquad$.
Answer: 290
user: 设方程 $x y=6(x+y)$ 的全部正整数解为 $\left(x_1, y_1\right),\left(x_2, y_2\right), \cdots,\left(x_n, y_n\right)$. 则

$$
\sum_{k=1}^n\left(x_k+y_k\right)=
$$
Answer: 290
user: 若正整数 $m 、 n$ 满足 $\frac{(m+n)!}{n!}=5040$,则 $m!n$ 的值为 $\qquad$。
Answer: 144
user: 在数列 $\left\{a_{n}\right\}$ 中, $a_{4}=1, a_{11}=9$, 且任意连续三项的和都是 15, 则 $a_{2016}=$ $\qquad$.
Answer: 5
user: The sequence of digits $1~2~3~4~5~6~7~8~9~1~0~1~1~1~2~1~3~1~4~1~5~1~6~1~7~1~8~1~9~2~0~2~1 \ldots$  
is obtained by writing the positive integers in order. If the $10^{n}$ th digit in this sequence occurs in the part of the sequence in which the $m$-digit numbers are placed, define $f(n)$ to be $m$. For example, $f(2)=2$ because the 100 th digit enters the sequence in the placement of the two-digit integer 55. Find, with proof, $f(1987)$.
Answer: 1984
user: 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.
Answer: 8
user: 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.
Answer: 508536
user: 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 $(2021,2021)$ ?
Answer: 578
user: Let $a_0=1, a_1=2$, and $a_n=4 a_{n-1}-a_{n-2}$ for $n \geq 2$. Find an odd prime factor of $a_{2015}$.
Answer: 181
user: Compute

$$\log_2\left(\prod\limits_{a=1}^{2015}\prod\limits_{b=1}^{2015}(1+e^{2\pi iab/2015})\right)$$

Here $i$ is the imaginary unit (that is, $i^2=-1)$
Answer: 13725
user: Find the smallest positive integer $n$ such that for every integer $m$,
with $0<m<1993$, there exists an integer $k$ for which
$$\frac{m}{1993}<\frac{k}{n}<\frac{m+1}{1994}.$$
Answer: 3987
user: $\operatorname{Let}\left(x_{1},y_{1}\right)=\left(0.8,0.6\right)$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.
Answer: -1
user: 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 $f(x)$.
Answer: 3984
user: 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.
Answer: 1990
user: Suppose that a positive integer $N$ can be expressed as
the sum of $k$ consecutive positive integers
$$N=a+(a+1)+(a+2)+\cdots+(a+k-1)$$
for $k=2017$ but for no other values of $k>1.$ Considering all positive integers $N$ with this property, what is the smallest positive integer $a$ that occurs in any of these expressions?
Answer: 16
user: 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 $k(n)?$
Answer: 3
user: For a nonnegative integer $n$ and a strictly increasing sequence of real numbers $t_0,t_1,\ldots,t_n$, let $f(t)$ be the corresponding real-valued function defined for $t\geq t_0$ by the
following properties:
(a) $f(t)$ is continuous for $t\geq t_0$,and is twice differentiable for all $t>t_0$ other than $t_1,\ldots,t_n;$
(b)$f( t_0) = 1/ 2;$
(c) $\lim _{r\to t_{k}^{+ }}f^{\prime }( t) = 0$ for $0\leq k\leq n;$
(d) For $0\leq k\leq n-1$, we have $f^{\prime\prime}(t)=k+1$ when
$t_k<t<t_{k+1}$, and $f^{\prime\prime}(t)=n+1$ when $t>t_n.$

Considering all choices of $n$ and $t_0,t_1,\ldots,t_n$ such that $t_k\geq t_{k-1}+1$ for l$\leq k\leq n$,what is the least possible value of $T$ for which $f(t_0+T)=2023?$
Answer: 29
user: 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.
Answer: 4
user: How many primes among the positive integers, written as usual in base 10, are such that their digits are alternating 1's and 0's, beginning and ending with 1?
Answer: 0