Let $n \geq 4$ be an even integer. Consider an $n \times n$ grid. Two cells ($1 \times 1$ squares) are [i]neighbors[/i] if they share a side, are in opposite ends of a row, or are in opposite ends of a column. In this way, each cell in the grid has exactly four neighbors. An integer from 1 to 4 is written inside each square according to the following rules: [list] [*]If a cell has a 2 written on it, then at least two of its neighbors contain a 1. [*]If a cell has a 3 written on it, then at least three of its neighbors contain a 1. [*]If a cell has a 4 written on it, then all of its neighbors contain a 1.[/list] Among all arrangements satisfying these conditions, what is the maximum number that can be obtained by adding all of the numbers on the grid?

Consider a graph $G$ with $n$ vertices and at least $n^2/10$ edges. Suppose that each edge is colored in one of $c$ colors such that no two incident edges have the same color. Assume further that no cycles of size $10$ have the same set of colors. Prove that there is a constant $k$ such that $c$ is at least $kn^\frac{8}{5}$ for any $n$. [i]David Yang.[/i]

In Tlaxcala, there is a transportation system that works through buses that travel from one city to another in one direction . A set $S$ of cities is said [i]beautiful[/i] if it contains at least three different cities and from each city $A$ in $S$ at least two buses depart, each one goes directly to a different city in $S$ and none of them is $A$ (if there is a direct bus from $A$ to a city $B$ in $S$, there is not necessarily a direct bus from $B$ to $A$). Show that if there exists a beautiful set of cities $S$, then there exists a beautiful $T$ subset of $S$, such that for any two cities in $T$, you can get from one to another by taking buses that only pass through cities in $T$. Note: A bus goes directly from one city to another if it does not pass through any other city.

Find the coefficient of $x$ in the expansion of $(1 + x)(1 - 2x)(1 + 3x)(1 - 4x) ...(1 - 2008x)$.

Let $n$ be a positive integer. There is a pawn in one of the cells of an $n\times n$ table. The pawn moves from an arbitrary cell of the $k$th column, $k \in \{1,2, \cdots, n \}$, to an arbitrary cell in the $k$th row. Prove that there exists a sequence of $n^{2}$ moves such that the pawn goes through every cell of the table and finishes in the starting cell.

Anis, Banu, and Cholis are going to play a game. They are given an $n\times n$ board consisting of $n^2$ unit squares, where $n$ is an integer and $n > 5$. In the beginning of the game, the number $n$ is written on each unit square. Then Anis, Banu, and Cholis take turns playing the game, repeatedly in that order, according to the following procedure: On every turn, an arrangement of $n$ squares on the same row or column is chosen, and every number from the chosen squares is subtracted by $1$. The turn cannot be done if it results in a negative number, that is, no arrangement of $n$ unit squares on the same column or row in which all of its unit squares contain a positive number can be found. The last person to get a turn wins. Determine which player will win the game.

Let \( S \) be a set consisting of \( 2024 \) points on a plane, such that no three points in \( S \) are collinear. A line \( \ell \) passing through two points in \( S \) is called a "weakly balanced line" if it satisfies the following condition: (Condition) The line \( \ell \) divides the plane into two regions, one containing exactly \( 1010 \) points of \( S \), and the other containing exactly \( 1012 \) points of \( S \) (where each region contains no points lying on \( \ell \)). Let \( \omega(S) \) denote the number of weakly balanced lines among the lines passing through two points in \( S \). Find the smallest possible value of \( \omega(S) \).

Prove that there exist at least $100!$ ways to write $100!$ as sum of elements of set {$1!,2!,3!...99!$} (each number in sum can be two or more times)

Into each box of a $ 2012 \times 2012 $ square grid, a real number greater than or equal to $ 0 $ and less than or equal to $ 1 $ is inserted. Consider splitting the grid into $2$ non-empty rectangles consisting of boxes of the grid by drawing a line parallel either to the horizontal or the vertical side of the grid. Suppose that for at least one of the resulting rectangles the sum of the numbers in the boxes within the rectangle is less than or equal to $ 1 $, no matter how the grid is split into $2$ such rectangles. Determine the maximum possible value for the sum of all the $ 2012 \times 2012 $ numbers inserted into the boxes.

We are given a 5x5 square grid, divided to 1x1 tiles. Two tiles are called [b]linked[/b] if they lie in the same row or column, and the distance between their centers is 2 or 3. For example, in the picture the gray tiles are the ones linked to the red tile. [img]https://i.imgur.com/JVTQ9wB.png[/img] Sammy wants to mark as many tiles in the grid as possible, such that no two of them are linked. What is the maximal number of tiles he can mark?

There are $10000$ cities in country, and roads between some cities. Every city has $<100$ roads. Every cycle route with odd number of road consists of $\geq 101 $ roads. Prove that we can divide all cities in $100$ groups with $100$ cities, such that every road leads from one group to other.

Let $M$ be a set and $A,B,C$ given subsets of $M$. Find a necessary and sufficient condition for the existence of a set $X\subset M$ for which $(X\cup A)\backslash(X\cap B)=C$. Describe all such sets.

The $n$ participants of a tournament are numbered with $0$ through $n - 1$. At the end of the tournament it turned out that for every team, numbered with $s$ and having $t$ points, there are exactly $t$ teams having $s$ points each. Determine all possibilities for the final score list.

There are 100 blue lines drawn on the plane, among which there are no parallel lines and no three of which pass through one point. The intersection points of the blue lines are marked in red. Could it happen that the distance between any two red dots lying on the same blue line is equal to an integer? [i]From the folklore[/i]

Consider the following one-player game on an infinite chessboard. If two horizontally or vertically adjacent squares are occupied by a pawn each, and a square on the same line that is adjacent to one of them is empty, then it is allowed to remove the two pawns and place a pawn on the third (empty) square. Prove that if in the initial position all the pawns were forming a rectangle with the number of squares divisible by $3$, then it is not possible to end the game with only one pawn left on the board.

At a point on the real line sits a greyhound. On one of the sides a hare runs, away from the hound. The only thing known is that the (maximal) speed of the hare is strictly less than the (maximal) speed of the greyhound (but not their precise ratio). Does the greyhound have a strategy for catching the hare in a finite amount of time? ([i]Dan Schwarz[/i])

We have seven equal pails with water, filled to one half, one third, one quarter, one fifth, one eighth, one ninth, and one tenth, respectively. We are allowed to pour water from one pail into another until the first pail empties or the second one fills to the brim. Can we obtain a pail that is filled to (a) one twelfth, (b) one sixth after several such steps?

For $n$ people, if it is known that (a) there exist two people knowing each other among any three people, and (b) there exist two people not knowing each other among any four people. Find the maximum of $n$. Here, we assume that if $A$ knows $B$, then $B$ knows $A$.

Let S be a 1990-element set and P be a set of 100-ary sequences $(a_1,a_2,...,a_{100})$ ,where $a_i's$ are distinct elements of S.An ordered pair (x,y) of elements of S is said to [i]appear[/i] in $(a_1,a_2,...,a_{100})$ if $x=a_i$ and $y=a_j$ for some i,j with $1\leq i<j\leq 100$.Assume that every ordered pair (x,y) of elements of S appears in at most one member in P.Show that $|P|\leq 800$.

For a non-empty set $T$ denote by $p(T)$ the product of all elements of $T$. Does there exist a set $T$ of $2021$ elements such that for any $a\in T$ one has that $P(T)-a$ is an odd integer? Consider two cases: 1) All elements of $T$ are irrational numbers. 2) At least one element of $T$ is a rational number.

A square is partitioned in $n^2\geq 4$ rectanles using $2(n-1)$ lines,$n-1$ of which,are parallel to the one side of the square,$n-1$ are parallel to the other side.Prove that we can choose $2n$ rectangles of the partition,such that,for each two of them,we can place the one inside the other (possibly with rotation).

Let $21$ pieces, some white and some black, be placed on the squares of a $3\times 7$ rectangle. Prove that there always exist four pieces of the same color standing at the vertices of a rectangle.

Jerry writes down all binary strings of length $10$ without any two consecutive $1$s. How many $1$s does Jerry write?

A $5 \times 100$ table is divided into $500$ unit square cells, where $n$ of them are coloured black and the rest are coloured white. Two unit square cells are called [i]adjacent[/i] if they share a common side. Each of the unit square cells has at most two adjacent black unit square cells. Find the largest possible value of $n$.

A player is playing the following game. In each turn he flips a coin and guesses the outcome. If his guess is correct, he gains $ 1$ point; otherwise he loses all his points. Initially the player has no points, and plays the game until he has $ 2$ points. (a) Find the probability $ p_{n}$ that the game ends after exactly $ n$ flips. (b) What is the expected number of flips needed to finish the game?