Let \(n\) be a positive integer. On a \(2 \times 3 n\) board, we mark some squares, so that any square (marked or not) is adjacent to at most two other distinct marked squares (two squares are adjacent when they are distinct and have at least one vertex in common, i.e. they are horizontal, vertical or diagonal neighbors; a square is not adjacent to itself). (a) What is the greatest possible number of marked square? (b) For this maximum number, in how many ways can we mark the squares? configurations that can be achieved through rotation or reflection are considered distinct.

Let $n$ be a natural number. At first the cells of a table $2n$ x $2n$ are colored in white. Two players $A$ and $B$ play the following game. First is $A$ who has to color $m$ arbitrary cells in red and after that $B$ chooses $n$ rows and $n$ columns and color their cells in black. Player $A$ wins, if there is at least one red cell on the board. Find the least value of $m$ for which $A$ wins no matter how $B$ plays.

Let $n$ be a positive integer. We are given a $3n \times 3n$ board whose unit squares are colored in black and white in such way that starting with the top left square, every third diagonal is colored in black and the rest of the board is in white. In one move, one can take a $2 \times 2$ square and change the color of all its squares in such way that white squares become orange, orange ones become black and black ones become white. Find all $n$ for which, using a finite number of moves, we can make all the squares which were initially black white, and all squares which were initially white black. Proposed by [i]Boris Stanković and Marko Dimitrić, Bosnia and Herzegovina[/i]

In each cell of a $2025 \times 2025$ board, a nonnegative real number is written in such a way that the sum of the numbers in each row is equal to $1$, and the sum of the numbers in each column is equal to $1$. Define $r_i$ to be the largest value in row $i$, and let $R = r_1 + r_2 + ... + r_{2025}$. Similarly, define $c_i$ to be the largest value in column $i$, and let $C = c_1 + c_2 + ... + c_{2025}$. What is the largest possible value of $\frac{R}{C}$? [i]Proposed by Paulius Aleknavičius, Lithuania, and Anghel David Andrei, Romania[/i]

If $46$ squares are colored red in a $9\times 9$ board, show that there is a $2\times 2$ block on the board in which at least $3$ of the squares are colored red.

Let $n \ge 2$ be a positive integer. Each square of an $n\times n$ board is coloured red or blue. We put dominoes on the board, each covering two squares of the board. A domino is called [i]even [/i] if it lies on two red or two blue squares and [i]colourful [/i] if it lies on a red and a blue square. Find the largest positive integer $k$ having the following property: regardless of how the red/blue-colouring of the board is done, it is always possible to put $k$ non-overlapping dominoes on the board that are either all [i]even [/i] or all [i]colourful[/i].

All cells of an $n\times n$ table are painted in several colors so that there is no monochromatic $2\times2$ square. A sequence of different cells $a_1,a_2,\ldots,a_k$ is called a [i]colorful[/i] if any two consecutive cells are adjacent and are painted in different colors. What is the largest $k{}$ for which there is a colorful sequence of length $k{}$ regardless of the coloring of the cells of the table? [i]Proposed by N. Belukhov[/i]

A $1$ or $0$ is placed on each square of a $4 \times 4$ board. One is allowed to change each symbol in a row, or change each symbol in a column, or change each symbol in a diagonal (there are $14$ diagonals of lengths $1$ to $4$). For which arrangements can one make changes which end up with all $0$s?

Let there be a $n\times n$ board. Write down $0$ or $1$ in all $n^2$ squares. For $1 \le k \le n$, let $A_k$ be the product of all numbers in the $k$th row. How many ways are there to write down the numbers so that $A_1 + A_2 + ... + A_n$ is even?

Consider a checkered $3m\times 3m$ square, where $m$ is an integer greater than $1.$ A frog sits on the lower left corner cell $S$ and wants to get to the upper right corner cell $F.$ The frog can hop from any cell to either the next cell to the right or the next cell upwards. Some cells can be [i]sticky[/i], and the frog gets trapped once it hops on such a cell. A set $X$ of cells is called [i]blocking[/i] if the frog cannot reach $F$ from $S$ when all the cells of $X$ are sticky. A blocking set is [i] minimal[/i] if it does not contain a smaller blocking set.[list=a][*]Prove that there exists a minimal blocking set containing at least $3m^2-3m$ cells. [*]Prove that every minimal blocking set containing at most $3m^2$ cells.

Consider a game on an \( n \times n \) board, where each square starts with exactly one stone. A move consists of choosing $5$ consecutive squares in the same row or column of the board and toggling the state of each of those squares (removing the stone from squares with a stone and placing a stone in squares without a stone). For which positive integers \( n \geq 5 \) is it possible to end up with exactly one stone on the board after a finite number of moves?

Consider a checkered $3m\times 3m$ square, where $m$ is an integer greater than $1.$ A frog sits on the lower left corner cell $S$ and wants to get to the upper right corner cell $F.$ The frog can hop from any cell to either the next cell to the right or the next cell upwards. Some cells can be [i]sticky[/i], and the frog gets trapped once it hops on such a cell. A set $X$ of cells is called [i]blocking[/i] if the frog cannot reach $F$ from $S$ when all the cells of $X$ are sticky. A blocking set is [i] minimal[/i] if it does not contain a smaller blocking set.[list=a][*]Prove that there exists a minimal blocking set containing at least $3m^2-3m$ cells. [*]Prove that every minimal blocking set containing at most $3m^2$ cells.

The numbers $1$ to $1024$ are written one per square on a $32 \times 32$ board, so that the first row is $1, 2, ... , 32$, the second row is $33, 34, ... , 64$ and so on. Then the board is divided into four $16 \times 16$ boards and the position of these boards is moved round clockwise, so that $AB$ goes to $DA$ $DC \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \, CB$ then each of the $16 \times 16 $ boards is divided into four equal $8 \times 8$ parts and each of these is moved around in the same way (within the $ 16 \times 16$ board). Then each of the $8 \times 8$ boards is divided into four $4 \times 4$ parts and these are moved around, then each $4 \times 4$ board is divided into $2 \times 2$ parts which are moved around, and finally the squares of each $2 \times 2$ part are moved around. What numbers end up on the main diagonal (from the top left to bottom right)?

In a $44\times 44$ board, some of the cells are blue, and the rest are red. No blue cells borders another blue cell on the side. The red cells, on the other hand, form a connected component (one may get from any red cell to any other red cell only by traversing edge-adjacent red cells). Prove that less than one third of the cells on the board are blue. [i]Proposed by B. Frenkin[/i]

Arnaldo and Bernardo play a Super Naval Battle. Each has a board $n \times n$. Arnaldo puts boats on his board (at least one but not known how many). Each boat occupies the $n$ houses of a line or a column and the boats they can not overlap or have a common side. Bernardo marks $m$ houses (representing shots) on your board. After Bernardo marked the houses, Arnaldo says which of them correspond to positions occupied by ships. Bernardo wins, and then discovers the positions of all Arnaldo's boats. Determine the lowest value of $m$ for which Bernardo can guarantee his victory.

Let $n \ge 2$ be an integer. In each cell of a $4n \times 4n$ table we write the sum of the cell row index and the cell column index. Initially, no cell is colored. A move consists of choosing two cells which are not colored and coloring one of them in red and one of them in blue. Show that, however Alex perfors $n^2$ moves, Jane can afterwards perform a number of moves (eventually none) after which the sum of the numbers written in the red cells is the same as the sum of the numbers written in the blue ones.

Each of the ${4n^2}$ unit squares of a ${2n \times 2n}$ board ${(n \ge 1) }$ has been colored blue or red. A set of four different unit squares of the board is called [i]pretty [/i]if these squares can be labeled ${A,B,C,D}$ in such a way that ${A}$ and ${B}$ lie in the same row, ${C}$ and ${D}$ lie in the same row, ${A}$ and ${C}$ lie in the same column, ${B}$ and ${D}$ lie in the same column, ${A}$ and ${D}$ are blue, and ${B}$ and ${C}$ are red. Determine the largest possible number of different [i]pretty [/i]sets on such a board. (Poland)

A lame rook lies on a $9\times 9$ chessboard. It can move one cell horizontally or vertically. The rook made $n{}$ moves, visited each cell at most once, and did not make two moves consecutively in the same direction. What is the largest possible value of $n{}$? [i]From the folklore[/i]

Alice and Bob play a game together as a team on a $100 \times 100$ board with all unit squares initially white. Alice sets up the game by coloring exactly $k$ of the unit squares red at the beginning. After that, a legal move for Bob is to choose a row or column with at least $10$ red squares and color all of the remaining squares in it red. What is the smallest $k$ such that Alice can set up a game in such a way that Bob can color the entire board red after finitely many moves? Proposed by [i]Nikola Velov, Macedonia[/i]

Maria has a board of size $n \times n$, initially with all the houses painted white. Maria decides to paint black some houses on the board, forming a mosaic, as shown in the figure below, as follows: she paints black all the houses from the edge of the board, and then leaves white the houses that have not yet been painted. Then she paints the houses on the edge of the next remaining board again black, and so on. a) Determine a value of $n$ so that the number of black houses equals $200$. b) Determine the smallest value of $n$ so that the number of black houses is greater than $2012$.

The numbers from $1$ to $100$ are placed without repetition in each cell of a \(10 \times 10\) board. An increasing path of length \(k\) on this board is a sequence of cells \(c_1, c_2, \ldots, c_k\) such that, for each \(i = 2, 3, \ldots, k\), the following properties are satisfied: • The cells \(c_i\) and \(c_{i-1}\) share a side or a vertex; • The number in \(c_i\) is greater than the number in \(c_{i-1}\). What is the largest positive integer \(k\) for which we can always find an increasing path of length \(k\), regardless of how the numbers from 1 to 100 are arranged on the board?

Every square of $1000 \times 1000$ board is colored black or white. It is known that exists one square $10 \times 10$ such that all squares inside it are black and one square $10 \times 10$ such that all squares inside are white. For every square $K$ $10 \times 10$ we define its power $m(K)$ as an absolute value of difference between number of white and black squares $1 \times 1$ in square $K$. Let $T$ be a square $10 \times 10$ which has minimum power among all squares $10 \times 10$ in this board. Determine maximal possible value of $m(T)$

What is the largest number of cells that can be marked on a $100 \times 100$ board in such a way that a chess king from any cell attacks no more than two marked ones? (The cell on which a king stands is also considered to be attacked by this king.)

On a $100 \times 100$ chessboard, the plan is to place several $1 \times 3$ boards and $3 \times 1$ board, so that [list] [*] Each tile of the initial chessboard is covered by at most one small board. [*] The boards cover the entire chessboard tile, except for one tile. [*] The sides of the board are placed parallel to the chessboard. [/list] Suppose that to carry out the instructions above, it takes $H$ number of $1 \times 3$ boards and $V$ number of $3 \times 1$ boards. Determine all possible pairs of $(H,V)$. [i]Proposed by Muhammad Afifurrahman, Indonesia[/i]

Consider a checkered $3m\times 3m$ square, where $m$ is an integer greater than $1.$ A frog sits on the lower left corner cell $S$ and wants to get to the upper right corner cell $F.$ The frog can hop from any cell to either the next cell to the right or the next cell upwards. Some cells can be [i]sticky[/i], and the frog gets trapped once it hops on such a cell. A set $X$ of cells is called [i]blocking[/i] if the frog cannot reach $F$ from $S$ when all the cells of $X$ are sticky. A blocking set is [i] minimal[/i] if it does not contain a smaller blocking set.[list=a][*]Prove that there exists a minimal blocking set containing at least $3m^2-3m$ cells. [*]Prove that every minimal blocking set containing at most $3m^2$ cells.