Two types of tiles, depicted on the figure below, are given. [img]https://wiki-images.artofproblemsolving.com//2/23/Izrezak.PNG[/img] Find all positive integers $n$ such that an $n\times n$ board consisting of $n^2$ unit squares can be covered without gaps with these two types of tiles (rotations and reflections are allowed) so that no two tiles overlap and no part of any tile covers an area outside the $n\times n$ board. \\ [i]Proposed by Art Waeterschoot[/i]

Let $n \ge 3$ be an integer, and consider a $n \times n$-board, divided into $n^2$ unit squares. For all $m \ge 1$, arbitrarily many $1\times m$-rectangles (type I) and arbitrarily many $m\times 1$-rectangles (type II) are available. We cover the board with $N$ such rectangles, without overlaps, and such that every rectangle lies entirely inside the board. We require that the number of type I rectangles used is equal to the number of type II rectangles used.(Note that a $1 \times 1$-rectangle has both types.) What is the minimal value of $N$ for which this is possible?

Consider an integer \(n \ge 2\) and write the numbers \(1, 2, \ldots, n\) down on a board. A move consists in erasing any two numbers \(a\) and \(b\), then writing down the numbers \(a+b\) and \(\vert a-b \vert\) on the board, and then removing repetitions (e.g., if the board contained the numbers \(2, 5, 7, 8\), then one could choose the numbers \(a = 5\) and \(b = 7\), obtaining the board with numbers \(2, 8, 12\)). For all integers \(n \ge 2\), determine whether it is possible to be left with exactly two numbers on the board after a finite number of moves. [i]Proposed by China[/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.

A $8 \times 8$ board is given, with sides directed north-south and east-west. It is divided into $1 \times 1$ cells in the usual manner. In each cell, there is most one [i]house[/i]. A house occupies only one cell. A house is [i] in the shade[/i] if there is a house in each of the cells in the south, east and west sides of its cell. In particular, no house placed on the south, east or west side of the board is in the shade. Find the maximal number of houses that can be placed on the board such that no house is in the shade.

Let $n$ be a positive with $n\geq 3$. Consider a board of $n \times n$ boxes. In each step taken the colors of the $5$ boxes that make up the figure bellow change color (black boxes change to white and white boxes change to black) The figure can be rotated $90°, 180°$ or $270°$. Firstly, all the boxes are white.Determine for what values of $n$ it can be achieved, through a series of steps, that all the squares on the board are black.

An equilateral triangle of side $n$ has been divided into little equilateral triangles of side $1$ in the usual way. We draw a path over the segments of this triangulation, in such a way that it visits exactly once each one of the $\frac{(n+1)(n+2)}{2}$ vertices. What is the minimum number of times the path can change its direction? The figure below shows a valid path on a triangular board of side $4$, with exactly $9$ changes of direction. [asy] unitsize(30); pair h = (1, 0); pair v = dir(60); pair d = dir(120); for(int i = 0; i < 4; ++i) { draw(i*v -- i*v + (4 - i)*h); draw(i*h -- i*h + (4 - i)*v); draw((i + 1)*h -- (i + 1)*h + (i + 1)*d); } draw(h + v -- v -- (0, 0) -- 2*h -- 2*h + v -- h + 2*v -- 2*v -- 4*v -- 3*h + v -- 3*h -- 4*h, linewidth(2)); draw(3*h -- 4*h, EndArrow); fill(circle(h + v, 0.1)); [/asy] [i]Proposed by Oriol Solé[/i]

The entries on an $ n$ × $ n$ board are colored black and white like it is usually done in a chessboard, and the upper left hand corner is black. We color the entries on the chess board black according to the following rule: In each step we choose an arbitrary $ 2$×$ 3$ or $ 3$× $ 2$ rectangle that still contains $ 3$ white entries, and we color these three entries black. For which values of $ n$ can the whole board be colored black in a finite number of steps

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.

The cells of an $n \times n$ board are labelled with the numbers $1$ through $n^2$ in the usual way. Let $n$ of these cells be selected, no two of which are in the same row or column. Find all possible values of the sum of their labels.

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]

Board has written numbers: $5$, $7$ and $9$. In every step we do the following: for every pair $(a,b)$, $a>b$ numbers from the board, we also write the number $5a-4b$. Is it possible that after some iterations, $2003$ occurs at the board ?

There are several dominoes on a board such that each domino occupies two adjacent cells and none of the dominoes are adjacent by side or vertex. The bottom left and top right cells of the board are free. A token starts at the bottom left cell and can move to a cell adjacent by side: one step to the right or upwards at each turn. Is it always possible to move from the bottom left to the top right cell without passing through dominoes if the size of the board is a) $100 \times 101$ cells and b) $100 \times 100$ cells? [i]Nikolay Chernyatiev[/i]

The numbers $1$ through $16$ are to be written in the cells of a $4\times 4$ board. (a) Prove that this can be done in such a way that any two numbers in cells that share a side differ by at most $4$. (b) Prove that this cannot be done in such a way that any two numbers in cells that share a side differ by at most $3$.

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 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.

There was a rook at some square of a $10 \times 10{}$ chessboard. At each turn it moved to a square adjacent by side. It visited each square exactly once. Prove that for each main diagonal (the diagonal between the corners of the board) the following statement is true: in the rook’s path there were two consecutive steps at which the rook first stepped away from the diagonal and then returned back to the diagonal. [i]Alexandr Gribalko[/i]

We have a 7x7 board. We want to color some 1x1 squares such that any 3x3 sub-board have more painted 1x1 than no painted 1x1. What is the smallest number of 1x1 that we need to color?

Board with dimesions $2018 \times 2018$ is divided in unit cells $1 \times 1$. In some cells of board are placed black chips and in some white chips (in every cell maximum is one chip). Firstly we remove all black chips from columns which contain white chips, and then we remove all white chips from rows which contain black chips. If $W$ is number of remaining white chips, and $B$ number of remaining black chips on board and $A=min\{W,B\}$, determine maximum of $A$

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?

Esmeralda chooses two distinct positive integers \(a\) and \(b\), with \(b > a\), and writes the equation \[ x^2 - ax + b = 0 \] on the board. If the equation has distinct positive integer roots \(c\) and \(d\), with \(d > c\), she writes the equation \[ x^2 - cx + d = 0 \] on the board. She repeats the procedure as long as she obtains distinct positive integer roots. If she writes an equation for which this does not occur, she stops. a) Show that Esmeralda can choose \(a\) and \(b\) such that she will write exactly 2024 equations on the board. b) What is the maximum number of equations she can write knowing that one of the initially chosen numbers is 2024?

Let $n$ be a positive integer. Ana and Beto play a game on a $2 \times n$ board (with 2 rows and $n$ columns). First, Ana writes a digit from 1 to 9 in each cell of the board such that in each column the two written digits are different. Then, Beto erases a digit from each column. Reading from left to right, a number with $n$ digits is formed. Beto wins if this number is a multiple of $n$; otherwise, Ana wins. Determine which of the two players has a winning strategy in the following cases: $\bullet$ (a) $n = 1001$. $\bullet$ (b) $n = 1003$.

$\bullet$ Determine a natural $n$ such that the constant sum $S$ of a magic square of $ n \times n$ (that is, the sum of its elements in any column, or the diagonal) differs as little as possible from $1992$. $\bullet$ Construct or describe the construction of this magic square.

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.

There are $n$ pieces on the squares of a $5 \times 9$ board, at most one on each square at any time during the game. A move in the game consists of simultaneously moving each piece to a neighboring square by side, under the restriction that a piece having been moved horizontally in the previous move must be moved vertically and vice versa. Find the greatest value of $n$ for which there exists an initial position starting at which the game can be continued until the end of the world.