Let $r\geq2$ be a fixed positive integer, and let $F$ be an infinite family of sets, each of size $r$, no two of which are disjoint. Prove that there exists a set of size $r-1$ that meets each set in $F$.

A domino is a rectangle whose length is twice its width. Any square can be divided into seven dominoes, for example as shown in the figure below. [img]https://cdn.artofproblemsolving.com/attachments/7/6/c055d8d2f6b7c24d38ded7305446721e193203.png[/img] a) Show that you can divide a square into $n$ dominoes for all $n \ge 5$. b) Show that you cannot divide a square into three or four dominoes.

Alina and Bogdan play a game on a $2\times n$ rectangular grid ($n\ge 2$) whose sides of length $2$ are glued together to form a cylinder. Alternating moves, each player cuts out a unit square of the grid. A player loses if his/her move causes the grid to lose circular connection (two unit squares that only touch at a corner are considered to be disconnected). Suppose Alina makes the first move. Which player has a winning strategy?

A floor has $2 \times 11$ dimension, and this floor covering with $1 \times 2$ rectangles. (No two rectangles overlap). How many cases we done this job?

Given is a set $A$ of $n$ elements and positive integers $k, m$ such that $4 \leq k <n$ and $m \leq \min \{k-3, \frac {n} {2}\}$. Let $A_1, A_2, \ldots, A_l$ be subsets of $A$, all with size $k$, such that $|A_i \cap A_j| \leq m$ for all $i \neq j$. Prove that there exists a subset $B$ of $A$ with at least $\sqrt[m+1]{n}+m$ elements which doesn't contain entirely any of the subsets $A_1, A_2, \ldots, A_l$.

Let $n$ be a positive integer. A regular hexagon $ABCDEF$ with side length $n$ is partitioned into $6n^2$ equilateral triangles with side length $1$. The hexagon is covered by $3n^2$ rhombuses with internal angles $60^{\circ}$ and $120^{\circ}$ such that each rhombus covers exactly two triangles and every triangle is covered by exactly one rhombus. Show that the diagonal $AD$ divides in half exactly $n$ rhombuses.

Each point of the plane is colored by one of the two colors. Show that there exists an equilateral triangle with monochromatic vertices.

Consider a $100 \times 100$ table, and identify the cell in row $a$ and column $b$, $1 \leq a, b \leq 100$, with the ordered pair $(a, b)$. Let $k$ be an integer such that $51 \leq k \leq 99$. A $k$-knight is a piece that moves one cell vertically or horizontally and $k$ cells to the other direction; that is, it moves from $(a, b)$ to $(c, d)$ such that $(|a-c|, |b - d|)$ is either $(1, k)$ or $(k, 1)$. The $k$-knight starts at cell $(1, 1)$, and performs several moves. A sequence of moves is a sequence of cells $(x_0, y_0)= (1, 1)$, $(x_1, y_1), (x_2, y_2)$, $\ldots, (x_n, y_n)$ such that, for all $i = 1, 2, \ldots, n$, $1 \leq x_i , y_i \leq 100$ and the $k$-knight can move from $(x_{i-1}, y_{i-1})$ to $(x_i, y_i)$. In this case, each cell $(x_i, y_i)$ is said to be reachable. For each $k$, find $L(k)$, the number of reachable cells.

A cube consists of $4^3$ unit cubes each containing an integer. At each move, you choose a unit cube and increase by $1$ all the integers in the neighbouring cubes having a face in common with the chosen cube. Is it possible to reach a position where all the $4^3$ integers are divisible by $3,$ no matter what the starting position is?

There are $68$ coins , each coin having a different weight than that of each other . Show how to find the heaviest and lightest coin in $100$ weighings on a balance beam. (S. Fomin, Leningrad)

The number $10^{2007}$ is written on the blackboard. Anne and Berit play a two player game in which the player in turn performs one of the following operations: 1) replace a number $x$ on the blackboard with two integers $a,b>1$ such that $ab=x$. 2) strike off one or both of two equal numbers on the blackboard. The person who cannot perform any operation loses. Who has the winning strategy if Anne starts?

In a certain country there are two tennis clubs consisting of $1000 $ and $1001$ members respectively. All the members have different playing strength, and the descending order of palying strengths in each club is known. Find a procedure which determines, within $ 11$ games, who is in the $1001$st place among the $ 2001$ players in these clubs. It is assumed that a stronger player always beats a weaker one.

Given a table with $2^n * n$ 1*1 squares ( $2^n$ rows and n column). In any square we put a number in {1, -1} such that no two rows are the same. Then we change numbers in some squares by 0. Prove that in new table we can choose some rows such that sum of all numbers in these rows equal to 0.

Let $m,n$ and $a_1,a_2,\dots,a_m$ be arbitrary positive integers. Ali and Mohammad Play the following game. At each step, Ali chooses $b_1,b_2,\dots,b_m \in \mathbb{N}$ and then Mohammad chosses a positive integers $s$ and obtains a new sequence $\{c_i=a_i+b_{i+s}\}_{i=1}^m$, where $$b_{m+1}=b_1,\ b_{m+2}=b_2, \dots,\ b_{m+s}=b_s$$ The goal of Ali is to make all the numbers divisible by $n$ in a finite number of steps. FInd all positive integers $m$ and $n$ such that Ali has a winning strategy, no matter how the initial values $a_1, a_2,\dots,a_m$ are. [hide=clarification] after we create the $c_i$ s, this sequence becomes the sequence that we continue playing on, as in it is our 'new' $a_i$[/hide] Proposed by Shayan Gholami

A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$

Place the numbers $ 1, 2, 3, 4, 5, 6, 7, 8$ at the vertices of a cuboid such that the sum of any $ 3$ numbers on a side is not less than $ 10$. Find the smallest possible sum of the 4 numbers on a side.

$(CZS 4)$ Let $K_1,\cdots , K_n$ be nonnegative integers. Prove that $K_1!K_2!\cdots K_n! \ge \left[\frac{K}{n}\right]!^n$, where $K = K_1 + \cdots + K_n$

15 of the cells of a chessboard 8x8 are chosen. We draw the segments which unite the centers of every two of the chosen squares. Prove that among these segments there are four segments which have the same length.

There are $30$ teams in NBA and every team play $82$ games in the year. Bosses of NBA want to divide all teams on Western and Eastern Conferences (not necessary equally), such that number of games between teams from different conferences is half of number of all games. Can they do it?

On a rectangular board with $m \times n$ squares ($m, n \ge 3$) there are dominoes ($2 \times 1$ or $1\times 2$ tiles), which do not overlap and do not extend beyond the board. Every domino covers exactly two squares of the board. Assume that the dominos cover the has the property that no more dominos can be added to the board and that the four corner spaces of the board are not all empty. Prove that at least $2/3$ of the squares of the board are covered with dominos.

A jury of $9$ persons should decide whether a verdict is guilty or not. Each juror votes independently with the probability $1/2$ for each of the two possibilities, and noone is allowed to be abstinent. Find the probability that a fixed juror will be a part of the majority. In the case of a jury of $n$ persons, find the values of n for which the probability of being a part of the majority is greater than, equal to, and smaller than $1/2$, respectively. (For $n = 2k$, $k +1$ votes are needed for a majority.)

A box contains $p$ white balls and $q$ black balls. Beside the box there is a pile of black balls. Two balls are taken out of the box. If they have the same color, a black ball from the pile is put into the box. If they have different colors, the white ball is put back into the box. This procedure is repeated until the last two balls are removed from the box and one last ball is put in. What is the probability that this last ball is white?

Let $n > 1$ be a fixed integer. On an infinite row of squares, there are $n$ stones on square $1$ and no stones on squares $2$, $3$, $4$, $\ldots$. Curious George plays a game in which a [i]move[/i] consists of taking two adjacent piles of sizes $a$ and $b$, where $a-b$ is a nonzero even integer, and transferring stones to equalize the piles (so that both piles have $\frac{a+b}{2}$ stones). The game ends when no more moves can be made. George wants to analyze the number of moves it takes to end the game. (a) Suppose George wants to end the game as quickly as possible. How many moves will it take him? (b) Suppose George wants to end the game as slowly as possible. Show that for all $n > 2$, the game will end after at most $\frac{3}{16}n^2$ moves. [i]Scoring note:[/i] For part (b), partial credit will be awarded for correct proofs of weaker bounds, eg. $\frac{1}{4}n^2$, $n^k$, or $k^n$ (for some $k \geq 2$).

Let $ A_1,A_2,...$ be a sequence of infinite sets such that $ |A_i \cap A_j| \leq 2$ for $ i \not\equal{}j$. Show that the sequence of indices can be divided into two disjoint sequences $ i_1<i_2<...$ and $ j_1<j_2<...$ in such a way that, for some sets $ E$ and $ F$, $ |A_{i_n} \cap E|\equal{}1$ and $ |A_{j_n} \cap F|\equal{}1$ for $ n\equal{}1,2,... .$ [i]P. Erdos[/i]

Let $n$ be a positive integer and let $z_1,z_2,\dots,z_n$ be positive integers such that for $j=1,2,\dots,n$ the inequalites $z_j \le j$ hold and $z_1+z_2+\dots+z_n$ is even. Prove that the number $0$ occurs among the values \[z_1 \pm z_2 \pm \dots \pm z_n,\] where $+$ or $-$ can be chosen independently for each operation. [i](Walther Janous)[/i]