Find all positive integers $n$ such that we can put $n$ equal squares on the plane that their sides are horizontal and vertical and the shape after putting the squares has at least $3$ axises.

Each of the $9$ cells in a $3\times 3$ grid is colored either blue or white with equal probability. The expected value of the area of the largest square of blue cells contained within the grid is $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

We have $2015$ prisinoers.The king gives everyone a hat coloured in one of $5$ colors.Everyone sees all hats expect his own.Now,the King orders them in a line(a prisioner can see all guys behind and in front of him).The king asks the prisinoers one by one does he know the color of his hat.If he answers [b]NO[/b],then he is killed.If he answers [b]YES[/b],then answers which color is his hat,if his answers is true,he goes to freedom,if not,he is killed.All the prisinors can hear did he answer [b]YES[/b] or [b]NO[/b],but if he answered [b]YES[/b],they don't know what did he answered(he is killed in public).They can think of a strategy before the King comes,but after that they can't comunicate.What is the largest number of prisinors we can guarentee that can survive?

The checkered plane is painted black and white, after a chessboard fashion. A polygon $\Pi$ of area $S$ and perimeter $P$ consists of some of these unit squares (i.e., its sides go along the borders of the squares). Prove the polygon $\Pi$ contains not more than $\dfrac {S} {2} + \dfrac {P} {8}$, and not less than $\dfrac {S} {2} - \dfrac {P} {8}$ squares of a same color. (Alexander Magazinov)

Let $k$ be a positive integer. Some of the $2k$-element subsets of a given set are marked. Suppose that for any subset of cardinality less than or equal to $(k+1)^2$ all the marked subsets contained in it (if any) have a common element. Show that all the marked subsets have a common element.

Sean enters a classroom in the Memorial Hall and sees a $1$ followed by $2020$ $0$'s on the blackboard. As he is early for class, he decides to go through the digits from right to left and independently erase the $n$th digit from the left with probability $\frac{n-1}{n}$. (In particular, the $1$ is never erased.) Compute the expected value of the number formed from the remaining digits when viewed as a base-$3$ number. (For example, if the remaining number on the board is $1000$, then its value is $27$.)

Let $A_1, A_2, A_3$ be nonempty sets of integers such that for $\{i, j, k\} = \{1, 2, 3\}$ holds $$(x \in A_i, y\in A_j) \Rightarrow (x + y \in A_k, x - y \in A_k).$$ Prove that at least two of the sets $A_1, A_2, A_3$ are equal. Can any of these sets be disjoint?

An $8\times 8$ chessboard is made of unit squares. We put a rectangular piece of paper with sides of length 1 and 2. We say that the paper and a single square overlap if they share an inner point. Determine the maximum number of black squares that can overlap the paper.

There are $n$ people, and given that any $2$ of them have contacted with each other at most once. In any group of $n-2$ of them, any one person of the group has contacted with other people in this group for $3^k$ times, where $k$ is a non-negative integer. Determine all the possible value of $n.$

Ten bowls are in a circle. They will go clockwise - starting somewhere filled with $1, 2, 3, ..., 9$ or $10$ marbles. You can have two choices in every move . Add a marble to neighboring shells or from two neighboring shells - if both of them are not empty - remove one marble each. Can you achieve that after finally many moves in each bowl exactly $2011$ marbles lying?

Let $a$ and $b$ be distinct positive integers. The following infinite process takes place on an initially empty board. [list=i] [*] If there is at least a pair of equal numbers on the board, we choose such a pair and increase one of its components by $a$ and the other by $b$. [*] If no such pair exists, we write two times the number $0$. [/list] Prove that, no matter how we make the choices in $(i)$, operation $(ii)$ will be performed only finitely many times. Proposed by [I]Serbia[/I].

3 countries $A, B, C$ participate in a competition where each country has 9 representatives. The rules are as follows: every round of competition is between 1 competitor each from 2 countries. The winner plays in the next round, while the loser is knocked out. The remaining country will then send a representative to take on the winner of the previous round. The competition begins with $A$ and $B$ sending a competitor each. If all competitors from one country have been knocked out, the competition continues between the remaining 2 countries until another country is knocked out. The remaining team is the champion. [b]I.[/b] At least how many games does the champion team win? [b]II.[/b] If the champion team won 11 matches, at least how many matches were played?

Let $n$ be a nonzero even integer. We fill up all the cells of an $n\times n$ grid with $+$ and $-$ signs ensuring that the number of $+$ signs equals the number of $-$ signs. Show that there exists two rows with the same number of $+$ signs or two collumns with the same number of $+$ signs.

Robert colors each square in an empty 3 by 3 grid either red or green. Find the number of colorings such that no row or column contains more than one green square.

We have a grid of $k^2-k+1$ rows and $k^2-k+1$ columns, where $k=p+1$ and $p$ is prime. For each prime $p$, give a method to put the numbers 0 and 1, one number for each square in the grid, such that on each row there are exactly $k$ 0's, on each column there are exactly $k$ 0's, and there is no rectangle with sides parallel to the sides of the grid with 0s on each four vertices.

For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$ [i]Proposed by Shahjalal Shohag, Bangladesh[/i]

The vertices of the cube are assigned $1, 2, 3..., 8$ and then each edge we assign the product of the numbers assigned to its two extreme points. Determine the greatest possible the value of the sum of the numbers assigned to all twelve edges of the cube.

Let $$C=\{1,22,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20\}$$ and let $$S=\{4,5,9,14,23,37\}$$ Find two sets $A$ and $B$ with the properties (a) $A\cap B=\emptyset$. (b) $A\cup B=C$. (c) The sum of two distinct elements of $A$ is not in $S$. (d) The sum of two distinct elements of $B$ is not in $S$.

A circle is drawn on the board, and $2n$ points are marked on it, dividing it into $2n$ equal arcs. Petya and Vasya are playing the following game. Petya chooses a positive integer $d \leqslant n$ and announces this number to Vasya. To win the game, Vasya needs to color all marked points using $n$ colors, such that each color is assigned to exactly two points, and for each pair of same-colored points, one of the arcs between them contains exactly $(d - 1)$ marked points. Find all $n$ for which Petya will be able to prevent Vasya from winning.

What is the maximum number of coins can be arranged in cells of the table $n \times n$ (each cell is not more of the one coin) so that any coin was not simultaneously below and to the right than any other?

Let $ A$ be a subset of $ \{ 0,1,2,...,1997 \}$ containing more than $ 1000$ elements. Prove that either $ A$ contains a power of $ 2$ (that is, a number of the form $ 2^k$ with $ k\equal{}0,1,2,...)$ or there exist two distinct elements $ a,b \in A$ such that $ a\plus{}b$ is a power of $ 2$.

For a positive integer $n(\geq 5)$, there are $n$ white stones and $n$ black stones (total $2n$ stones) lined up in a row. The first $n$ stones from the left are white, and the next $n$ stones are black. $$\underbrace{\Circle \Circle \cdots \Circle}_n \underbrace{\CIRCLE \CIRCLE \cdots \CIRCLE}_n $$ You can swap the stones by repeating the following operation. [b](Operation)[/b] Choose a positive integer $k (\leq 2n - 5)$, and swap $k$-th stone and $(k+5)$-th stone from the left. Find all positive integers $n$ such that we can make first $n$ stones to be black and the next $n$ stones to be white in finite number of operations.

Given positive integers $m,n(2\le m\le n)$, let $a_1,a_2,\ldots ,a_m$ be a permutation of any $m$ pairwise distinct numbers taken from $1,2,\ldots ,n$. If there exist $k\in\{1,2,\ldots ,m\}$ such that $a_k+k$ is odd, or there exist positive integers $k,l(1\le k<l\le m)$ such that $a_k>a_l$, then call $a_1,a_2,\ldots ,a_m$ a [i]good[/i] sequence. Find the number of good sequences.

Consider $13$ weights of integer mass (in grams). It is known that any $6$ of them may be placed onto two pans of a balance achieving equilibrium. Prove that all the weights are of equal mass.

In the country of PUMACsboro, there are $n$ distinct cities labelled $1$ through $n$. There is a rail line going from city $i$ to city $j$ if and only if $i<j$; you can only take this rail line from city $i$ to city $j$. What is the smallest possible value of $n$ such that if each rail line's track is painted orange or black, you can always take the train between $2019$ cities on tracks that are all the same color? (This means there are some cities $c_1,c_2,\dots,c_{2019}$ such that there is a rail line going from city $c_i$ to $c_{i+1}$ for all $1\leq i\leq 2018$ and their rail lines' tracks are either all orange or all black.)