Let $n\ge 1$ be a positive integer. A square of side length $n$ is divided by lines parallel to each side into $n^2$ squares of side length $1$. Find the number of parallelograms which have vertices among the vertices of the $n^2$ squares of side length $1$, with both sides smaller or equal to $2$, and which have tha area equal to $2$. (Greece)

Let $p$ be a prime number. Determine the largest possible $n$ such that the following holds: it is possible to fill an $n\times n$ table with integers $a_{ik}$ in the $i$th row and $k$th column, for $1\le i,k\le n$, such that for any quadruple $i,j,k,l$ with $1\le i<j\le n$ and $1\le k<l\le n$, the number $a_{ik}a_{jl}-a_{il}a_{jk}$ is not divisible by $p$. [i]Proposed by oneplusone[/i]

A rook stands in one cell of an infinite square grid. A different cell was colored blue and mines were placed in $n$ additional cells: the rook cannot stand on or pass through them. It is known that the rook can reach the blue cell in finitely many moves. Can it do so (for every $n$ and such a choice of mines, starting point, and blue cell) in at most [b](a)[/b] $1.99n+100$ moves? [b](b)[/b] $2n-2\sqrt{3n}+100$ moves? [b]Remark.[/b] In each move, the rook goes in a vertical or horizontal line.

Petya and Vasya live in neighboring houses (see the plan in the figure). Vasya lives in the fourth entrance. It is known that Petya runs to Vasya by the shortest route (it is not necessary walking along the sides of the cells) and it does not matter from which side he runs around his house. Determine in which entrance he lives Petya . [img]https://cdn.artofproblemsolving.com/attachments/b/1/741120341a54527b179e95680aaf1c4b98ff84.png[/img]

In an urban area whose street plan is a grid, a person started walking from an intersection and turned right or left at every intersection he reached until he ended up in the same initial intersection. [b]a)[/b] Show that the number of intersections (not necessarily distinct) in which he were is equivalent to $ 1 $ modulo $ 4. $ [b]b)[/b] Enunciate and prove a reciprocal statement. [i]Marius Beceanu[/i]

A "Hishgad" lottery ticket contains the numbers $1$ to $mn$, arranged in some order in a table with $n$ rows and $m$ columns. It is known that the numbers in each row increase from left to right and the numbers in each column increase from top to bottom. An example for $n=3$ and $m=4$: [asy] size(3cm); Label[][] numbers = {{"$1$", "$2$", "$3$", "$9$"}, {"$4$", "$6$", "$7$", "$10$"}, {"$5$", "$8$", "$11$", "$12$"}}; for (int i=0; i<5;++i) { draw((i,0)--(i,3)); } for (int i=0; i<4;++i) { draw((0,i)--(4,i)); } for (int i=0; i<4;++i){ for (int j=0; j<3;++j){ label(numbers[2-j][i], (i+0.5, j+0.5)); }} [/asy] When the ticket is bought the numbers are hidden, and one must "scratch" the ticket to reveal them. How many cells does it always suffice to reveal in order to determine the whole table with certainty?

The teacher reported to Peter an odd integer $m \le 2013$ and gave the guy a homework. Petrick should star the cells in the $2013 \times 2013$ table so to make the condition true: if there is an asterisk in some cell in the table, then or in row or column containing this cell should be no more than $m$ stars (including this one). Thus in each cell of the table the guy can put at most one star. The teacher promised Peter that his assessment would be just the number of stars that the guy will be able to place. What is the greatest number will the stars be able to place in the table Petrick?

For some positive integer $n$, consider the board $n\times n$. On this board you can put any rectangles with sides along the sides of the grid. What is the smallest number of such rectangles that must be placed so that all the cells of the board are covered by distinct numbers of rectangles (possibly $0$)? The rectangles are allowed to have the same sizes. [i]Proposed by Anton Trygub[/i]

Let $p$ be a prime number. Determine the largest possible $n$ such that the following holds: it is possible to fill an $n\times n$ table with integers $a_{ik}$ in the $i$th row and $k$th column, for $1\le i,k\le n$, such that for any quadruple $i,j,k,l$ with $1\le i<j\le n$ and $1\le k<l\le n$, the number $a_{ik}a_{jl}-a_{il}a_{jk}$ is not divisible by $p$. [i]Proposed by oneplusone[/i]

Points $A,B,C,D$ have been marked on checkered paper (see fig.). Find the tangent of the angle $ABD$. [img]https://cdn.artofproblemsolving.com/attachments/6/1/eeb98ccdee801361f9f66b8f6b2da4714e659f.png[/img]

Let $n{}$ be a natural number. The playing field for a "Master Sudoku" is composed of the $n(n+1)/2$ cells located on or below the main diagonal of an $n\times n$ square. A teacher secretly selects $n{}$ cells of the playing field and tells his student [list] [*]the number of selected cells on each row, and [*]that there is one selected cell on each column. [/list]The teacher's selected cells form a Master Sudoku if his student can determine them with the given information. How many Master Sudokus are there? [i]Proposed by T. Amdeberkhan, M. Ruby and F. Petrov[/i]

You have a $2m$ by $2n$ grid of squares coloured in the same way as a standard checkerboard. Find the total number of ways to place $mn$ counters on white squares so that each square contains at most one counter and no two counters are in diagonally adjacent white squares.

A man disposes of sufficiently many metal bars of length $2$ and wants to construct a grill of the shape of an $n \times n$ unit net. He is allowed to fold up two bars at an endpoint or to cut a bar into two equal pieces, but two bars may not overlap or intersect. What is the minimum number of pieces he must use?

A [i]stick[/i] is defined as a $1 \times k$ or $k\times 1$ rectangle for any integer $k \ge 1$. We wish to partition the cells of a $2022 \times 2022$ chessboard into $m$ non-overlapping sticks, such that any two of these $m$ sticks share at most one unit of perimeter. Determine the smallest $m$ for which this is possible. [i]Holden Mui[/i]

Two player are playing in an $100 \times 100$ grid. Initially the whole board is black. On $A$'s move, he selects $4 \times 4$ subgrid and color it white. On $B$'s move, he selects a $3 \times 3$ subgrid and colors it black. $A$ wants to make the whole board white. Can he do it? [i]Proposed by S M A Nahian[/i]

On grid paper, mark three nodes so that in the triangle they formed, the sum of the two smallest medians equals to half-perimeter.

Each cell of a $3 $ × $3$ grid is labeled with a digit in the set {$1, 2, 3, 4, 5$} Then, the maximum entry in each row and each column is recorded. Compute the number of labelings for which every digit from $1$ to $5$ is recorded at least once.

Petya and Vasya live in neighboring houses (see the plan in the figure). Vasya lives in the fourth entrance. It is known that Petya runs to Vasya by the shortest route (it is not necessary walking along the sides of the cells) and it does not matter from which side he runs around his house. Determine in which entrance he lives Petya . [img]https://cdn.artofproblemsolving.com/attachments/b/1/741120341a54527b179e95680aaf1c4b98ff84.png[/img]

Consider a $10 \times 10$ grid. On every move, we colour $4$ unit squares that lie in the intersection of some two rows and two columns. A move is allowed if at least one of the $4$ squares is previously uncoloured. What is the largest possible number of moves that can be taken to colour the whole grid?

The cells of a $100 \times 100$ table are colored white. In one move, it is allowed to select some $99$ cells from the same row or column and recolor each of them with the opposite color. What is the smallest number of moves needed to get a table with a chessboard coloring? [i]S. Berlov[/i]

Consider a table with $n$ rows and $2n$ columns. we put some blocks in some of the cells. After putting blocks in the table we put a robot on a cell and it starts moving in one of the directions right, left, down or up. It can change the direction only when it reaches a block or border. Find the smallest number $m$ such that we can put $m$ blocks on the table and choose a starting point for the robot so it can visit all of the unblocked cells. (the robot can't enter the blocked cells.) Proposed by Seyed Mohammad Seyedjavadi and Alireza Tavakoli

Let $n \geq 2$ be an integer. Switzerland and Liechtenstein are performing their annual festive show. There is a field divided into $n \times n$ squares, in which the bottom-left square contains a red house with $k$ Swiss gymnasts, and the top-right square contains a blue house with $k$ Liechtensteiner gymnasts. Every other square only has enough space for a single gymnast at a time. Each second either a Swiss gymnast or a Liechtensteiner gymnast moves. The Swiss gymnasts move to either the square immediately above or to the right and the Liechtensteiner gymnasts move either to the square immediately below or to the left. The goal is to move all the Swiss gymnasts to the blue house and all the Liechtensteiner gymnasts to the red house, with the caveat that a gymnast cannot enter a house until all the gymnasts of the other nationality have left. Determine the largest $k$ in terms of $n$ for which this is possible.

Let $n \geq 2$ be an integer. A lead soldier is moving across the unit squares of a $n \times n$ grid, starting from the corner square. Before each move to the neighboring square, the lead soldier can (but doesn't need to) turn left or right. Determine the smallest number of turns, which it needs to do, to visit every square of the grid at least once. At the beginning the soldier's back is faced at the edge of the grid.

Given a $m\times n$ grid rectangle with $m,n \ge 4$ and a closed path $P$ that is not self intersecting from inner points of the grid, let $A$ be the number of points on $P$ such that $P$ does not turn in them and let $B$ be the number of squares that $P$ goes through two non-adjacent sides of them furthermore let $C$ be the number of squares with no side in $P$. Prove that $$A=B-C+m+n-1.$$

Let $n$ be an odd positive integer and consider an $n \times n$ chessboard of $n^2$ unit squares. In some of the cells of the chessboard, we place a knight. A knight in a cell $c$ is said to [i]attack [/i] a cell $c'$ if the distance between the centers of $c$ and $c'$ is exactly $\sqrt{5}$ (in particular, a knight does not attack the cell which it occupies). Suppose each cell of the board is attacked by an even number of knights (possibly zero). Show that the configuration of knights is symmetric with respect to all four axes of symmetry of the board (i.e. the configuration of knights is both horizontally and vertically symmetric, and also unchanged by reflection along either diagonal of the chessboard). [i]NIkolai Beluhov[/i]