Find the digits left and right of the decimal point in the decimal form of the number \[ (\sqrt{2} + \sqrt{3})^{1980}. \]

Anna and Bob play the following game. In the beginning, Bob writes down the numbers $1, 2, ... , 2022$ on a piece of paper, such that half of the numbers are on the left and half on the right. Furthermore, we assume that the $1011$ numbers on both sides are written in some order. After Bob does this, Anna has the opportunity to swap the positions of the two numbers lying on different sides of the paper if they have different parity. Anna wins if, after finitely many moves, all odd numbers end up on the left, in increasing order, and all even ones end up on the right, in increasing order. Can Bob write down a arrangement of numbers for which Anna cannot win? For example, Bob could write down numbers in the following way: $$4, 2, 5, 7, 9, ... , 2021\,\,\,\,\,\,\,\,\,\,,\, \,\,\,\,\,\,\,\,\,\,,\, 3, 1, 6, 8, 10, ... , 2022$$ Then Anna could swap the numbers $1, 4$ and then swap $2, 3$ to win. However, if Anna swapped the pairs $3, 4$ and $1, 2$, the resulting numbers on the left and on the right would not be in increasing order, and hence Anna would not win.

How many ways are there to paint the squares of a $6 \times 6$ board black or white such that within each $2\times 2$ square on the board, the number of black squares is odd?

Find the number of $5$-digit $n$, s.t. every digit of $n$ is either $0$, $1$, $3$, or $4$, and $n$ is divisible by $15$.

The set $X$ has $1983$ members. There exists a family of subsets $\{S_1, S_2, \ldots , S_k \}$ such that: [b](i)[/b] the union of any three of these subsets is the entire set $X$, while [b](ii)[/b] the union of any two of them contains at most $1979$ members. What is the largest possible value of $k ?$

The 40 unit squares of the 9 9-table (see below) are labeled. The horizontal or vertical row of 9 unit squares is good if it has more labeled unit squares than unlabeled ones. How many good (horizontal and vertical) rows totally could have the table?

Find the number of ordered triples $(x,y,z)$ of non-negative integers satisfying (i) $x \leq y \leq z$ (ii) $x + y + z \leq 100.$

Given is a graph $G$. An [i]empty [/i] subgraph of $G$ is a subgraph of $G$ with no edges between its vertices. An edge of $G$ is called [i]important [/i] if and only if the removal of this edge will increase the size of the maximal empty subgraph. Suppose that two important edges in $G$ have a common endpoint. Prove there exists a cycle of odd length in $G$.

In a table with $88$ rows and $253$ columns, each cell is colored either purple or yellow. Suppose that for each yellow cell $c$, $$x(c)y(c)\geq184.$$ Where $x(c)$ is the number of purple cells that lie in the same row as $c$, and $y(c)$ is the number of purple cells that lie in the same column as $c$.\\ Find the least possible number of cells that are colored purple.

Players $A$ and $B$ play a game with $N \geq 2012$ coins and $2012$ boxes arranged around a circle. Initially $A$ distributes the coins among the boxes so that there is at least $1$ coin in each box. Then the two of them make moves in the order $B,A,B,A,\ldots $ by the following rules: [b](a)[/b] On every move of his $B$ passes $1$ coin from every box to an adjacent box. [b](b)[/b] On every move of hers $A$ chooses several coins that were [i]not[/i] involved in $B$'s previous move and are in different boxes. She passes every coin to an adjacent box. Player $A$'s goal is to ensure at least $1$ coin in each box after every move of hers, regardless of how $B$ plays and how many moves are made. Find the least $N$ that enables her to succeed.

One quadrant of the Cartesian coordinate system is tiled with $1\times 2$ dominoes. The dominoes don’t overlap with each other, they cover the entire quadrant and they all fit in the quadrant. Farringdon the flea is initially sitting at the origin and is allowed to jump from one corner of a domino to the opposite corner any number of times. Is it possible that the dominoes are arranged in such a way that Farringdon is unable to move to a distance greater than 2023 from the origin?

Let $n\geq 4$ be a positive integer.Out of $n$ people,each of two individuals play table tennis game(every game has a winner).Find the minimum value of $n$,such that for any possible outcome of the game,there always exist an ordered four people group $(a_{1},a_{2},a_{3},a_{4})$,such that the person $a_{i}$ wins against $a_{j}$ for any $1\leq i<j\leq 4$

In a school with $2022$ students, either a museum trip or a nature trip is organized every day during a holiday. No student participates in the same type of trip twice, and the number of students attending each trip is different. If there are no two students participating in the same two trips together, find the maximum number of trips held.

Points $ A_1,A_2, ... ,A_n$ inside a circle and points $ B_1,B_2,...,B_n$ on its boundary are positioned so that the segments $ A_1B_1,A_2B_2, ... ,A_nB_n$ do not intersect. A bug can go from point $ A_i$ to $ A_j$ if the segment $ A_iA_j$ does not intersect any segment $ A_kB_k$, $ k \neq i, j$. Prove that the bug can go from any point $ A_p$ to any point $ A_q$ in a finite number of steps.

From a collection of $n$ persons $q$ distinct two-member teams are selected and ranked $1, \cdots, q$ (no ties). Let $m$ be the least integer larger than or equal to $2q/n$. Show that there are $m$ distinct teams that may be listed so that : [b](i)[/b] each pair of consecutive teams on the list have one member in common and [b](ii)[/b] the chain of teams on the list are in rank order. [i]Alternative formulation.[/i] Given a graph with $n$ vertices and $q$ edges numbered $1, \cdots , q$, show that there exists a chain of $m$ edges, $m \geq \frac{2q}{n}$ , each two consecutive edges having a common vertex, arranged monotonically with respect to the numbering.

Eight pieces are placed on a chessboard so that each row and each column contains exactly one piece. Prove that there are an even number of pieces on the black squares of the board.

Let $A=\{n\in\mathbb{Z}\mid 0<n<2013\}$. A subset $B\subseteq A$ is called [b]reduced[/b] if for any two numbers $x,y\in B$, we must have $x\cdot y \notin B$. For example, any subset containing the numbers $3,5,15$ cannot be reduced, and same for a subset containing $4,16$. [list=a] [*] Find the maximal size of a reduced subset of $A$. [*] How many reduced subsets are there with that maximal size? [/list]

A company has $n$ employees. It is known that each of the employees works at least one of the $7$ days of the week, with the exception of an employee who does not work any of the $7$ days. Furthermore, for any two of these $n$ employees, there are at least $3$ days of the week in which one of the two works that day and the other does not (it is not necessarily the same employee who works those days). Determine the highest possible value of $n$.

We have $ n $ kinds of puddings. There are $ a_{i} $ puddings which are $ i $-th type and those $ S = a_{1}+\cdots+a_{n} $ puddings are distinct. Now, for a given arrangement of puddings: $ p_{1}, \dots, p_{n} $. Define $ c_{i} $ to be $$ \#\{1 \le j \le S-1 \ \mid \ p_{j}, p_{j+1} \text{ are the same type.}\} $$ Show that if $ S $ is composite, then the sum of $ \prod_{i=1}^{n}{c_{i}} $ over all possible arrangements is a multiple of $ S $.

All the cells in a $8* 8$ board are colored white. Omar and Asaad play the following game: in the beginning Omar colors $n$ cells red, then Asaad chooses $4$ rows and $4$ columns and colors them black. Omar wins if there is at least one red cell. Find the least possible value for n such that Omar can always win regardless of Asaad's move.

In the following, a rhombus is one with edge length $1$ and interior angles $60^o$ and $120^o$ . Now let $n$ be a natural number and $H$ a regular hexagon with edge length $n$, which is covered with rhombuses without overlapping has been. The rhombuses then appear in three different orientations. Prove that whatever the overlap looks exactly, each of these three orientations can be viewed at the same time.

An alien with four feet wants to wear four identical socks and four identical shoes, where on each foot a sock must be put on before a shoe. How many ways are there for the alien to wear socks and shoes?

Participants to an olympiad worked on $n$ problems. Each problem was worth a [color=#FF0000]positive [/color]integer number of points, determined by the jury. A contestant gets $0$ points for a wrong answer, and all points for a correct answer to a problem. It turned out after the olympiad that the jury could impose worths of the problems, so as to obtain any (strict) final ranking of the contestants. Find the greatest possible number of contestants.

A group of mathematicians is attending a conference. We say that a mathematician is $k-$[i]content[/i] if he is in a room with at least $k$ people he admires or if he is admired by at least $k$ other people in the room. It is known that when all participants are in a same room then they are all at least $3k + 1$-content. Prove that you can assign everyone into one of $2$ rooms in a way that everyone is at least $k$-content in his room and neither room is empty. [i]Admiration is not necessarily mutual and no one admires himself.[/i] [i]Matija Bucić[/i]

Given a cube with a side of $4$. Is it possible to completely cover $3$ of its faces, which have a common vertex, with sixteen rectangular paper strips measuring $1 \times3$?