meetforever/GRPO_CL huggingface dataset:
user: A number is called Norwegian if it has three distinct positive divisors whose sum is equal to 2022. Determine the smallest Norwegian number.
(Note: The total number of positive divisors of a Norwegian number is allowed to be larger than 3.)
Answer: 1344
user: Find all positive integers $n>2$ such that

$$
n!\mid \prod_{\substack{p<q \leqslant n, p, q \text { primes }}}(p+q) .
$$.
Answer: 7
user: A1. Professor Oak is feeding his 100 Pokémon. Each Pokémon has a bowl whose capacity is a positive real number of kilograms. These capacities are known to Professor Oak. The total capacity of all the bowls is 100 kilograms. Professor Oak distributes 100 kilograms of food in such a way that each Pokémon receives a non-negative integer number of kilograms of food (which may be larger than the capacity of their bowl). The dissatisfaction level of a Pokémon who received $N$ kilograms of food and whose bowl has a capacity of $C$ kilograms is equal to $|N-C|$.

Find the smallest real number $D$ such that, regardless of the capacities of the bowls, Professor Oak can distribute the food in a way that the sum of the dissatisfaction levels over all the 100 Pokémon is at most $D$.
Answer: 50
user: How many ways are there to represent $10^6$ as the product of three factors? Factorizations which only differ in the order of the factors are considered to be distinct.
Answer: 784
user: Find the remainder after division of $10^{10} + 10^{10^{2}} + 10^{10^{3}}  + \cdots+ 10^{10^{10}}$  by $7$.
Answer: 5
user: Consider all positive integers written in a row:
$123456789101112131415\ldots$
Find the $206788$-th digit from the left.
Answer: 7
user: How many pairs of integers $x, y$ are there between $1$ and $1000$ such that $x^ 2 + y ^2$ is divisible by $49$?
Answer: 10153
user: How many positive integers $x$ less than $10000$ are there such that $2^x-x^2$ is divisible by $7$ ?
Answer: 2857
user: Find the number $\overline{523abc}$ divisible by $7, 8$ and $9$.
Answer: 523152
user: Given the $6$ digits: $0, 1, 2, 3, 4, 5$. Find the sum of all even four-digit numbers which can be expressed with the help of these figures (the same figure can be repeated).
Answer: 1769580
user: Suppose we have two identical cardboard polygons. We placed one polygon upon the other one and aligned. Then we pierced polygons with a pin at a point. Then we turned one of the polygons around this pin by $25^{\circ}300^{\prime}$ . It turned out that the polygons coincided (aligned again). What is the minimal possible number of sides of the polygons?
Answer: 144
user: Two seventh graders and several eightth graders take part in a chess tournament. The two seventh graders together scored eight points. The scores of eightth graders are equal. How many eightth graders took part in the tournament?
Answer: 7
user: Ninth and tenth graders participated in a chess tournament. There were ten times as many tenth graders as ninth graders. The total score of tenth graders was 4.5 times that of the ninth graders. What was the ninth graders score?
Answer: 10
user: A town has 57 bus routes. How many stops does each route have if it is known that 
1) one can reach any stop from any other stop without changing buses; 
2) for every pair of routes there is a single stop where one can change buses; 
3) each route has three or more stops?
Answer: 8
user: In a convex 1950-gon all diagonals are drawn, dividing it into smaller polygons. What is the greatest number of sides can these polygons have?
Answer: 1949
user: Dividing $x^{1951}-1$ by $P(x)=x^4+x^3+2x^2+x+1$ one gets a quotient and a remainder. Find the coefficient of $x^{14}$ in the quotient.
Answer: -1
user: Define the maximal value of the ratio of a three-digit number to the sum of its digits.
Answer: 100
user: Five men play several sets of dominoes (two against two) so that each player has each other player once as a partner and two times as an opponent. Find the number of sets and all ways to arrange the players.
Answer: 5
user: What is the least number of points that can be chosen on a circle of length 1956, so that for each of these points there is exactly one chosen point at distance 1, and exactly one chosen point at distance 2 (distances are measured along the circle)?
Answer: 1304
user: The distance between towns A and B is 999 km. At every kilometer of the road that connects A and B a sign shows the distances to A and B as follows:
$$\boxed{0|999}\quad\boxed{1|998}\quad\boxed{2|997}\quad\ldots\quad\boxed{998|1}\quad\boxed{999|0}$$
How many signs are there, with both distances written with the help of only two distinct digits?
Answer: 40
user: How many four-digit numbers from 0000 to 9999 (we complete a one-, two-, or three-digit number to a four-digit one by writing zeros in front of it) are there such that the sum of their first two digits is equal to the sum of their last two digits?
Answer: 670
user: On a piece of paper, write figures 0 to 9. Observe that if we turn the paper through $180^\circ$ the $0’s, 1’s$ (written as a vertical line segment, not as in the typed texts) and $8’s$ turn into themselves, the $6’s$ and $9’s$ interchange, and the other figures become meaningless. 
How many 9-digit numbers are there which turn into themselves when a piece of paper on which they are written is turned by $180^\circ$ ?
Answer: 1500
user: The sum of digits of integers k and $k +l$ is divisible by 11 and there is no number with similar properties between them. What is the greatest value of l?
Answer: 39
user: A factory produces rattles shaped in the form of a ring with 3 red and 7 blue spherical beads on it. Two rattles are said to be of the same type if one can be obtained from the other one by moving a bead along the ring or by flipping the ring over in space. How many different types of rattles can be manufactured?
Answer: 8
user: What greatest number of elements can be selected from the set of numbers $1, 2, . . . , 1963$ so that the sum of any two of the selected numbers is divisible by 26?
Answer: 76
user: What greatest number of integers can be selected from the set $1, 2, . . . , 1963$ so that no sum of any two selected numbers were divisible by their difference?
Answer: 655
user: We drew 11 horizontal and 11 vertical intersecting straight lines on a sheet of paper. We call a segment of one of the straight line drawn that connects two neighboring intersections a “link”. What least number of links must we erase in order for each intersection to be a junction of not more than 3 links?
Answer: 41
user: Let $a_1 = 1$ and for $k > 1$ define $a_k=[\sqrt{a_1+a_2+\cdots+a_{k-1}} ],$where $[x]$ denotes the integer part of x. Find $a_{1000}$.
Answer: 495
user: On an $11 × 11$ checker-board 22 squares are marked so that exactly two of the marked squares lie in each column and each row. Two arrangements of marked squares are considered equivalent if in any number of permutations of the columns and/or (independent) permutations of rows one arrangement can be obtained from the other. How many nonequivalent arrangements of marked squares are there?
Answer: 14
user: In the city of Fuchs, Ostap Bender organized a distribution of elephants to people. There were present 28 trades union members and 37 non-members. Ostap distributed the elephants so that the shares of all trades union members were equal, and the shares of non-members were also equal. It turned out that there existed only one such distribution (of all elephants). What greatest number of elephants could O. Bender have had?
Answer: 2072
user: Consider integers with the sum of their digits divisible by 7. What is the greatest difference between two consecutive such integers?
Answer: 13
user: 12 people took part in a chess tournament. After the end of the tournament every participant made 12 lists. The first list consisted of the author; the second list – of the author and of those (s)he has beaten; and so on; the 12-th list consisted of all the people on the 11-th list and those they have beaten. It is known that the 12-th list of every participant contains a person who is not on the participant’s 11-th list. How many games ended in a draw?
Answer: 54
user: Consider a paper scroll of bus tickets numbered 000 000 to 999 999. The tickets with the sum of the digits in the even places equal to the sum of the digits in the odd places are marked blue. What is the greatest difference between the numbers on two consequitive blue tickets?
Answer: 990
user: We mark point $O_1$ on a unit circle and, using $O_1$ as the center, we mark (by means of a compass) we mark point $O_2$ on the circle (clockwise starting with $O_1$). Using point $O_2$ as a new center, we repeat the procedure in the same direction with the same radius; and so on. After we had marked point $O_{1968}$ a circle is cut through each mark so we get 1968 arcs. How many different arc lengths can we thus procure?
Answer: 3
user: A straight line cuts a square piece of paper into two parts. Another straight line cuts one of the parts into two parts. One of the three pieces of paper obtained is again cut into two parts along a line, and so on. What least number of lines must be drawn in order to obtain 100 various (perhaps, (perhaps, identical) 20-gons?
Answer: 1699
user: We multiply the number $1234567 . . . 1000$ (juxtaposed are all natural numbers 1 to 1000) by a number from 1 to 9, and strike out all 1’s in the product. We multiply the number obtained by a nonzero one-digit number once again, and strike out the 1’s, and so on, many times over. What is the least number one can obtain in this manner?
Answer: 0
user: Is there a number whose square begins with the digits $123456789$ and ends with the digits $987654321$?
Answer: 111111111
user: We cut a cardboard (perhaps, non-convex) 1000-gon along a straight line. This cut brakes it into several new polygons. What is the greatest possible number of triangles among the new polygons?
Answer: 501
user: Given a set of positive integers with the sum of any seven of them less than 15 and the sum of all the numbers in the set equal to 100, determine the least possible number of elements in this set.
Answer: 50
user: In the decimal expression of $1^1 + 2^2 + 3^3 + · · · + 999^{999} + 1000^{1000},$ what are its a) first three digits from the left? b) first four digits?
Answer: 1000
user: Find the greatest five-digit number A in which the fourth digit is greater than the fifth; the third greater than the sum of the fourth and fifth; the second greater than the sum of the third, fourth and fifth; and the first greater than the sum of the other digits.
Answer: 95210
user: Find all positive integers x satisfying the following condition: the product of the digits of x is equal to $44x − 86 868$ and their sum is equal to a cube of a positive integer.
Answer: 1989
user: The streets of a city are represented on a map as straight lines that divide a square into 25 smaller squares of side 1. (The borderline of the city is considered to be the union of 4 streets.) There is a snow plow at the bottom right corner of the bottom left square. Find the length of the shortest path for the plow to pass through all streets and come back to its starting point.
Answer: 60
user: Let $M = m(n+ 9)(m+ 2n ^2 + 3)$. What is the least number of distinct prime divisors the number M can have?
Answer: 2
user: To guard a military installation around the clock, a day shift and a night shift are required. A sentry guard may take either a day, or a night shift, or work around the clock. In these cases the guard is given a leave of absence of not less than $1, 1.5$ or 2.5 full days, respectively. What is the least number of guards necessary to ensure the security of the installation?
Answer: 4
user: A Parliament of $2000 MP$s decided to ratify the state budget of 200 expenditure items. Each $MP$ prepared a draft budget with what (s)he thinks the maximum possible allocation for each item so that the total expenditure does not exceed a given ceiling, S. For each item, the Parliament approves the maximum expenditure approved by not less than k $MP$s. What is the least value of k to ensure that the approved total does not exceed S?
Answer: 1991
user: In a $9 × 9$ square table, 9 cells are marked: those at the intersection of the 2-nd, 5-th and 8-th rows with the 2-nd, 5-th and 8-th column. In how many ways can one get from the lower left cell to the upper right one moving only upwards and to the right without entering marked cells?
Answer: 678
user: What is the least number of weights in the set that can be divided into either $4, 5$, or 6 piles of equal mass?
Answer: 11
user: Let $x_1 = 4, x_2 = 6$ and define $x_n$ for $n ≥ 3$ to be the least non-prime greater than $2x_{n−1} − x_{n−2}.$ Find $x_{1000}$.
Answer: 501500
user: To every pair of numbers $x, y$ the Rule assigns a number $x ∗ y$. Find $1993 ∗ 1935$ if it is known that
$x ∗ x = 0,$ and $x ∗ (y ∗ z) = (x ∗ y) + z$ for any $x, y, z.$
Answer: 58
user: A student did not notice a multiplication sign between two three-digit numbers and wrote one 6-digit number that happened to be 7 times greater than the product of the two three-digit numbers. Find the factors.
Answer: 143
user: 平面上五点 $A, B, C, D, E$ 满足 $\overrightarrow{A B}=\overrightarrow{B C}=\overrightarrow{C D}, \overrightarrow{E A} \cdot \overrightarrow{E B}=4, \overrightarrow{E B} \cdot \overrightarrow{E C}=5$, $\overrightarrow{E C} \cdot \overrightarrow{E D}=8$, 则 $\overrightarrow{E A} \cdot \overrightarrow{E D}$ 的值为 $\qquad$.
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: 有多少个不同的三边长为整数的直角三角形，其面积值为周长值的 $999$ 倍（全等的两个三角形看作相同的）？
Answer: 42
user: 对每个正整数 $n$, 记 $g(n)$ 为 $n$ 与 $2015$的最大公约数. 求满足下列条件的有序三元数组 $(a, b, c)$ 的个数:
$(1) a 、 b 、 c \in\{1,2, \cdots, 2015\} ;$
$(2) g(a) 、 g(b) 、 g(c) 、 g(a+b) 、$
$g(b+c) 、 g(c+a) 、 g(a+b+c)$
这七个数两两不同.
Answer: 138240
user: 求最大正整数 $m$ ，使得可以在 $m$ 行 $8$列的方格表的每个方格中填入 $C、G、M、O$ 这四个字母之一，并且具有如下性质：对于方格表的任意不同两行，至多存在一列，使得这两行在此列处的字母相同。
Answer: 5
user: 将 $2018$ 行 $4$ 列方格表的每个格染成红色或蓝色，使得方格表每行的红格数与蓝格数相等，每列的红格数与蓝格数也相等。记 $m$ 为满足上述要求的染色方法的总数。求 $m$除以 $2018$ 所得的余数。
Answer: 6
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
$a=m_{1}^{2}+n_{1}^{2}, a^{2}=m_{2}^{2}+n_{2}^{2},$
 且 $m_{1}-n_{1}=m_{2}-n_{2} .$
Answer: 261
user: 已知平面上有 $100$ 条直线，用 $T$ 表示由这些直线中的某三条直线围成的直角三角形的集合。求 $|T|$ 的最大值。
Answer: 62500
user: 设 $a 、 b 、 c$ 为素数，且满足 $a^{5} \mid\left(b^{2}-c\right)$ ，其中, $b+c$ 为完全平方数. 求 $a b c$ 的最小值.
Answer: 1958
user: 已知 $n$ 为正整数, 使得存在正整数 $x_{1}$, $x_{2}, \cdots, x_{n}$ 满足
$x_{1} x_{2} \cdots x_{n}\left(x_{1}+x_{2}+\cdots+x_{n}\right)=100 n$.
求 $n$ 的最大可能值.
Answer: 9702
user: Professor Oak is feeding his 100 Pokémon. Each Pokémon has a bowl whose capacity is a positive real number of kilograms. These capacities are known to Professor Oak. The total capacity of all the bowls is 100 kilograms. Professor Oak distributes 100 kilograms of food in such a way that each Pokémon receives a non-negative integer number of kilograms of food (which may be larger than the capacity of their bowl). The dissatisfaction level of a Pokémon who received $N$ kilograms of food and whose bowl has a capacity of $C$ kilograms is equal to $|N-C|$.
Find the smallest real number $D$ such that, regardless of the capacities of the bowls, Professor Oak can distribute the food in a way that the sum of the dissatisfaction levels over all the 100 Pokémon is at most $D$.
Answer: 50
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 $[a$, $b]=[i-m, j-\pi]$ ，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: 一副三色纸牌，共有 32 张，其中红黄蓝每种颜色的牌各 10 张，编号分别是 $1,2, \cdots, 10$ ；男有大小王牌各一张，编号均为 0 。从这副牌中任取若干张牌，然后按如下规则计算分值：每张编号为 $k$ 的牌计为 $2^{k}$ 分。若它们的分值之和为 2004 ，则称这些牌为一个"好牌组". 试求"好牌组"的个数。
Answer: 1006009
user: 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）？
Answer: 15
user: 在一个 10$\times10$ 的方格表中有一个由4$n$个 1 x 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
“毌”型(可以旋转)的图形覆盖.求正整数$n$ 的最小值.
Answer: 4
user: 设$a_{_n}=n\sqrt{5}-[n\sqrt{5}].$求数列$a_1,a_2$, $\cdots,a_{2009}$中的最大项和最小项，其中$,[x]$表示不超过实数$x$的最大整数.
Answer: 1
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
$$
\begin{array}{ll} 
& a=m_{1}^{2}+n_{1}^{2}, a^{2}=m_{2}^{2}+n_{2}^{2}, \\
\text { 且 } \quad & m_{1}-n_{1}=m_{2}-n_{2}.
\end{array}
$$
Answer: 261
user: 集合 $\{0,1, \cdots, 2012\}$ 中共有多少个元素 $k$ ，使得组合数 $\mathrm{C}_{2012}^{k}$ 是 2012 的倍数？
Answer: 1498
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: 对每个正整数 $n$, 记 $g(n)$ 为 $n$ 与 2015的最大公约数. 求满足下列条件的有序三元数组 $(a, b, c)$ 的个数:
(1) $a 、 b 、 c \in\{1,2, \cdots, 2015\} ;$
$(2) g(a) 、 g(b) 、 g(c) 、 g(a+b) 、$
$g(b+c) 、 g(c+a) 、 g(a+b+c)$
这七个数两两不同.
Answer: 138240
user: 有多少个不同的三边长为整数的直角三角形，其面积值为周长值的 999 倍（全等的两个三角形看作相同的）？
Answer: 42
user: 求最大正整数 $m$, 使得可以在 $m$ 行 8列的方格表的每个方格中填人 C、G、M、O 这四个字母之一，并且具有如下性质：对于方格表的任意不同两行，至多存在一列，使得这两行在此列处的字母相同。
Answer: 5
user: 求最小的正整数 $n$ ，使得可将 $n \times n$ 方格表中的每个格染为红、黄、蓝三种颜色之一，满足以下三个条件：
（1）每种颜色的格数目相同；
（2）若某行中有红格，则该行中必有蓝格，且无黄格；
（3）若某列中有蓝格，则该列中必有红格，且无黄格。
Answer: 45
user: 考虑所有满足以下两个条件的实数序
列 $x_{0}, x_{1}, \cdots, x_{100}$ :
(1) $x_{0}=0$;
（2）对任意整数 $i(1 \leqslant i \leqslant 100)$ 有
$1 \leqslant x_{i}-x_{i-1} \leqslant 2$.
求最大的正整数 $k \leqslant 100$ ，使得对任意这样的序列 $x_{0}, x_{1}, \cdots, x_{100}$, 均有
$$
x_{k}+x_{k+1}+\cdots+x_{100} \geqslant x_{0}+x_{1}+\cdots+x_{k-1}
$$
Answer: 67
user: 将 2018 行 4 列方格表的每个格染成红色或蓝色，使得方格表每行的红格数与蓝格数相等，每列的红格数与蓝格数也相等。记 $m$ 为满足上述要求的染色方法的总数。求 $m$除以 2018 所得的余数。
Answer: 6
user: 设正整数 $a, b, c, d$ 满足 $a>b>c>d$, 且$$a b+b c+c a+d^{2} \mid(a+b)(b+c)(c+a)$$求 $a b+b c+c a+d^{2}$ 的素因子个数的最小值. (素因子的个数计重数, 如 12 的素因子个数为 3 )。
Answer: 3
user: 设非负实数 $a_{1}, a_{2}, \cdots, a_{100}$ 满足对任意 $2 \leq i \leq 99$ ，有$$\max \left\{a_{i-1}+a_{i}, a_{i}+a_{i+1}\right\} \geq i$$求 $a_{1}+a_{2}+\cdots+a_{100}$ 的最小值.
Answer: 1683
user: 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?
Answer: 1011
user: 已知 $S=\{(i, j) \mid i 、 j=1,2, \cdots, 100\}$ 为直角坐标平面上的 $100 \times 100$ 个整点构成的集合。将 $S$ 中的每个点染为给定的四种颜色之一。求以 $S$ 中四个颜色互不相同的点为顶点且边平行于坐标轴的矩形个数的最大可能值。
Answer: 9375000
user: 设集合 $M=\{1,2, \cdots, 10\}, T$ 为 $M$ 的某些二元子集构成的集合，满足对 $T$ 中任意两个不同的元素 $\{a, b\} 、\{x, y\}$ ，均有
$11 \nmid(a x+b y)(a y+b x)$ 。
求 $T$ 的元素个数的最大值.
Answer: 25
user: 已知平面上有 100 条直线，用 $T$ 表示由这些直线中的某三条直线围成的直角三角形的集合。求 $|T|$ 的最大值。
Answer: 62500
user: 设 $a 、 b 、 c$ 为素数, 且满足 $a^{5} \mid\left(b^{2}-c\right)$,其中, $b+c$ 为完全平方数. 求 $a b c$ 的最小值.
Answer: 1958
user: 设数列 $\left\{a_{n}\right\}$ 满足
$a_{0}=\frac{1}{2}, a_{n+1}=a_{n}+\frac{a_{n}^{2}}{2012}(n=0,1, \cdots)$.
求整数 $k$, 使得 $a_{k}<1<a_{k+1}$.
Answer: 2012
user: 求最小的正整数 $m$, 使得对于任意大于 3 的质数 $p$, 都有 105 I $\left(9^{p^{2}}-29^{p}+m\right)$.
Answer: 20
user: 设集合 $M \subseteq\{1,2, \cdots, 2011\}$ 满足：在 $M$ 的任意三个元素中都可以找到两个元素 $a 、 b$ ，使得 $a \mid b$ 或 $b \mid a$ 。 求 $|M|$ 的最大值（ $|M|$表示集合 $M$ 的元素个数)。
Answer: 21
user: 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.
Answer: 911
user: 给定整数 $n \geqslant 3$ 。求最小的正整数 $k$, 使得存在一个 $k$ 元集合 $A$ 和 $n$ 个两两不同的实数 $x_{1}, x_{2}, \cdots, x_{n}$, 满足 $x_{1}+x_{2}, x_{2}+x_{3}, \cdots$, $x_{n-1}+x_{n}, x_{n}+x_{1}$ 均属于 $A$
Answer: 3
user: 已知 $T=\{1,2, \cdots, 8\}$ 。对于 $A \subseteq T, A \neq \varnothing$ ，定义 $S(A)$ 为 $A$ 中所有元素之和。问： $T$ 有多少个非空子集 $A$ ，使得 $S(A)$ 是 3 的倍数，但不是 5 的倍数？
Answer: 70
user: 求满足下述条件的最小正实数 $k$ ：对任意不小于 $k$ 的 4 个互不相同的实数 $a 、 b$ 、 $c 、 d$, 都存在 $a 、 b 、 c 、 d$ 的一个排列 $p 、 q 、 r 、 s$,使得方程 $\left(x^{2}+p x+q\right)\left(x^{2}+r x+s\right)=0$ 有 4个互不相同的实数根。
Answer: 4
user: 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。
Answer: 16
user: 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 $\boldsymbol{m}$ 。使得对 $A=\{1,2, \cdots, \boldsymbol{m}\}$ 的任意一个 14 分划 $A_{1}, A_{2}, \cdots, A_{14}$, 一定存在某个集合 $A_{1}(1 \leqslant$ $i \leqslant 14)$ ，在 $A_{i}$ 中有两个元素 $a 、 b$ 满足 $b<a \leqslant \frac{4}{3} b$ 。
Answer: 2
user: 集合 $A 、 B$ 定义如下:
$$\begin{aligned}A= & \left\{a^{3}+b^{3}+c^{3}-3 a b c \mid a 、 b 、 c \in \mathbf{N}\right\}, \\B= & \{(a+b-c)(b+c-a)(c+a-b) \mid \\& a 、 b 、 c \in \mathbf{N}\}\end{aligned}$$
设集合 $P=\{n \mid n \in A \cap B, 1 \leqslant n \leqslant 2016\}$ 。求 $P$ 的元素个数。
Answer: 980
user: 求最大的正整数 $n$ ，使得存在 $n$ 个互不相同的正整数 $x_{1}, x_{2}, \cdots, x_{n}$ ，满足
$$
x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=2017
$$
Answer: 16
user: 对每个正整数 $n$ ，定义集合$P_{n}=\left\{n^{k} \mid k=0,1, \cdots\right\}$ 。对于正整数 $a 、 b 、 c$, 若存在某个正整数 $m$ ，使得 $a-1 、 a b-12 、 a b c-2015$ 这三个数（不必两两不等）均属于集合 $P_{m}$ ，则称正整数组 $(a, b, c)$ 为"幸运的". 求所有幸运的正整数组的个数.
Answer: 25
user: 对每个正整数 $n$ ，定义集合
$P_{n}=\left\{n^{k} \mid k=0,1, \cdots\right\}$ 。
对于正整数 $a 、 b 、 c$, 若存在某个正整数 $m$ ，使得 $a-1 、 a b-12 、 a b c-2015$ 这三个数
（不必两两不等）均属于集合 $P_{m}$ ，则称正整数组 $(a, b, c)$ 为"幸运的". 求所有幸运的正整数组的个数.
Answer: 25
user: 设 $f(x)=\left[\frac{x}{1!}\right]+\left[\frac{x}{2!}\right]+\cdots+\left[\frac{x}{2013!}\right]$,其中， $[x]$ 表示不超过实数 $x$ 的最大整数。对整数 $n$, 若关于 $x$ 的方程 $f(x)=n$ 有实数解,则称 $n$ 为 "好数". 求集合 $\{1,3,5, \cdots, 2013\}$中好数的个数。
Answer: 587
user: 设集合 $M=\{1,2, \cdots, 10\}, T$ 为 $M$ 的某些二元子集构成的集合，满足对 $T$ 中任意两个不同的元素 $\{a, b\} 、\{x, y\}$ ，均有  
$11 \nmid(a x+b y)(a y+b x)$ .  
求 $T$ 的元素个数的最大值.
Answer: 25
user: 设非负实数 $a_{1}, a_{2}, \cdots, a_{100}$ 满足对任意 $2 \leq i \leq 99$ ，有
$$
\max \left\{a_{i-1}+a_{i}, a_{i}+a_{i+1}\right\} \geq i
$$
求 $a_{1}+a_{2}+\cdots+a_{100}$ 的最小值.
Answer: 1683
user: 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？
Answer: 1011
user: 一个 $100 \times 100$ 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 $n$ 次移动吃到奶酰。
Answer: 2220
user: 设正整数 $a, b, c, d$ 满足 $a>b>c>d$, 且
$$
a b+b c+c a+d^{2} \mid(a+b)(b+c)(c+a)
$$
求 $a b+b c+c a+d^{2}$ 的素因子个数的最小值。(素因子的个数计重数，如 12 的素因子个数为 3 )。
Answer: 3
user: 已知 $\alpha^{2005}+\beta^{2005}$ 可表示成以 $\alpha+\beta 、 \alpha \beta$为变元的二元多项式.求这个多项式的系数之和.
Answer: 1
user: 设 $S=\{1,2, \cdots, 2005\}$ .若 $S$ 中任意 $n$个两两互质的数组成的集合中都至少有一个质数，试求 $n$ 的最小值.
Answer: 16
user: 设正数列 $\left\{a_{n}\right\} 、\left\{b_{n}\right\}$ 满足:
$a_{1}=b_{1}=1, b_{n}=a_{n} b_{n-1}-\frac{1}{4}(n \geqslant 2)$.
求 $4 \sqrt{b_{1} b_{2} \cdots b_{m}}+\sum_{k=1}^{m} \frac{1}{a_{1} a_{2} \cdots a_{k}}$ 的最小值,
其中 $m$ 是给定的正整数。
Answer: 5
user: 若 $x_{i}$ 是大于 1 的整数, 记 $f\left(x_{i}\right)$ 是 $x_{i}$ 的最大素因数。令 $x_{i+1}=x_{i}-f\left(x_{i}\right)(i \in \mathbf{N})$ 。
令 $V\left(x_{0}\right)$ 是 $f\left(x_{0}\right), f\left(x_{1}\right), \cdots, f\left(x_{k\left(x_{0}\right)}\right)$ 中不同数的个数. 求 $V(2), V(3), \cdots, V(781)$ 中的最大数，并说明理由。
Answer: 5
user: 设 $n$ 为正整数. 若平面中存在两点 $A, B$ 及 2024 个不同的点 $P_{1}, P_{2}, \cdots, P_{2024}$ ，满足：线段 $A B$ 及各条线段 $A P_{i}, B P_{i}(i=1,2, \cdots, 2024)$ 的长度均为不超过 $n$ 的正整数，求 $n$ 的最小值。
Answer: 39
user: $(1)$求不定方程
$m n+n r+m r=2(m+n+r)$
的正整数解 $(m, n, r)$ 的组数;
Answer: 7
user: 试求实数 $a$ 的个数，使得对于每个 $a$ ，关于 $x$ 的三次方程 $x^{3}=a x+a+1$ 都有满足 $|x|<1000$ 的偶数根.
Answer: 999
user: 试求最小的正整数 $n$ ，使得对于满足条件 $\sum\limits_{i=1}^{n} a_{i}=2007$ 的任一具有 $n$ 项的正整数数列 $a_{1}, a_{2}, \cdots, a_{n}$, 其中必有连续的若干项之和等于 $30$ .
Answer: 1018
user: 在一个圆周上给定 $8$ 个点 $A_{1}, A_{2}, \cdots$, $A_{8}$. 求最小的正整数 $n$, 使得以这 $8$ 个点为顶点的任意 $n$ 个三角形中，必存在两个有公共边的三角形。
Answer: 9
user: 将时钟盘面上标有数字 $1,2, \cdots, 12$ 的十二个点分别染红、黄、蓝、绿四种颜色，每色三个点，以这些点为顶点作 $n$ 个凸四边形，使其满足：  
$(1)$每个四边形的四个顶点四色都有；  
$(2)$对于其中任何三个四边形，都存在某一色，染有该色的三个顶点所标数字互不相同。  
求 $n$ 的最大值.
Answer: 9
user: 设 $m$ 为大于 $1$ 的整数，数列 $\left\{a_{n}\right\}$ 定义如下：  
$$  
\begin{aligned}  
& a_{0}=m, a_{1}=\varphi(m) ,\\  
& a_{2}=\varphi^{(2)}(m)=\varphi(\varphi(m)), \cdots, \\  
& a_{n}=\varphi^{(n)}(m)=\varphi\left(\varphi^{(n-1)}(m)\right)  ,
\end{aligned}  
$$  
其中， $\varphi(m)$ 为欧拉函数.
若对于任意非负整数 $k$ ，均有 $a_{k+1} \mid a_{k}$ ，求不超过 $2016$ 的最大正整数 $m$ 的值.
Answer: 1944
user: 集合 $A=\{1,2, \cdots, n\}$.若存在非空集合 $B C$, 使得 $B \cap C=\varnothing, B \cup C=A$, 且 $B$ 中元素的平方和为 $M, C$ 中元素的平方和为 $N$ ，满足 $M-N=2016$ ，求 $n$ 的最小值.
Answer: 19