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]

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.

What is the smallest number of nodes that can be marked in a rectangular $n \times k$ grid so that each cell contains at least two marked nodes?

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?

Cells of a $2000\times2000$ board are colored according to the following rules: 1)At any moment a cell can be colored, if none of its neighbors are colored 2)At any moment a $1\times2$ rectangle can be colored, if exactly two of its neighbors are colored. 3)At any moment a $2\times2$ squared can be colored, if 8 of its neighbors are colored (Two cells are considered to be neighboring, if they share a common side). Can the entire $2000\times2000$ board be colored? [I]Proposed by K. Kohas[/i]

Let $n$ be a fixed positive integer and consider an $n\times n$ grid of real numbers. Determine the greatest possible number of cells $c$ in the grid such that the entry in $c$ is both strictly greater than the average of $c$'s column and strictly less than the average of $c$'s row. [i]Proposed by Holden Mui[/i]

Let $n \ge 2$ be a positive integer. An $n\times n$ grid of squares has been colored as a chessboard. Let a [i]move[/i] consist of picking a square from the board and then changing the colors to the opposite for all squares that lie in the same row as the chosen square, as well as for all squares that lie in the same column (the chosen square itself is also changed to the opposite color). Find all values of $n$ for which it is possible to make all squares of the grid be the same color in a finite sequence of moves.

Let $n\geq 2$ be a positive integer. We call a [i]vertex[/i] every point in the coordinate plane, whose $x$ and $y$ coordinates both are from the set $\{1,2,3,...,n\}$. We call a segment between two vertices an [i]edge[/i], if its length if $1$. We've colored some edges red, such that between any two vertices, there is a unique path of red edges (a path may contain each edge at most once). The red edge $f$ is [i]vital[/i] for an edge $e$, if the path of red edges connecting the two endpoints of $e$ contain $f$. Prove that there is a red edge, such that it is vital for at least $n$ edges.

In an $m \times n$ grid, each square is either filled or not filled. For each square, its [i]value[/i] is defined as $0$ if it is filled and is defined as the number of neighbouring filled cells if it is not filled. Here, two squares are neighbouring if they share a common vertex or side. Let $f(m,n)$ be the largest total value of squares in the grid. Determine the minimal real constant $C$ such that $$\frac{f(m,n)}{mn} \le C$$holds for any positive integers $m,n$ [i]CSJL[/i]

Find the smallest positive integer $n$ such that if $n$ squares of a $1000 \times 1000$ chessboard are colored, then there will exist three colored squares whose centers form a right triangle with sides parallel to the edges of the board.

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 $$A=B-C+m+n-1.$$

Each cell of an $m\times n$ board is filled with some nonnegative integer. Two numbers in the filling are said to be [i]adjacent[/i] if their cells share a common side. (Note that two numbers in cells that share only a corner are not adjacent). The filling is called a [i]garden[/i] if it satisfies the following two conditions: (i) The difference between any two adjacent numbers is either $0$ or $1$. (ii) If a number is less than or equal to all of its adjacent numbers, then it is equal to $0$. Determine the number of distinct gardens in terms of $m$ and $n$.

Given a 4×4 grid of points, the points at two opposite corners are denoted $A$ and $D$. We need to choose two other points $ B$ and $C$ such that the six pairwise distances of these four points are all distinct. (a) How many such quadruples of points are there? (b) How many such quadruples of points are non-congruent? (c) If each point is assigned a pair of coordinates $(x_i,y_i)$, prove that the sum of the expressions $|x_i-x_j |+|y_i-y_j|$ over all six pairs of points in a quadruple is constant.

Let $n$ be a positive integer. A [i]Nordic[/i] square is an $n \times n$ board containing all the integers from $1$ to $n^2$ so that each cell contains exactly one number. Two different cells are considered adjacent if they share a common side. Every cell that is adjacent only to cells containing larger numbers is called a [i]valley[/i]. An [i]uphill path[/i] is a sequence of one or more cells such that: (i) the first cell in the sequence is a valley, (ii) each subsequent cell in the sequence is adjacent to the previous cell, and (iii) the numbers written in the cells in the sequence are in increasing order. Find, as a function of $n$, the smallest possible total number of uphill paths in a Nordic square. Author: Nikola Petrović

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]

A rectangular grid whose side lengths are integers greater than $1$ is given. Smaller rectangles with area equal to an odd integer and length of each side equal to an integer greater than $1$ are cut out one by one. Finally one single unit is left. Find the least possible area of the initial grid before the cuttings. Ps. Collected [url=https://artofproblemsolving.com/community/c949611_2019_nigerian_senior_mo_round_3]here[/url]

For which positive integers $n$ can be covered a ladder like that of the figure (but with $n$ steps instead of $4$) with $n$ squares of integer sides, not necessarily the same size, without these squares overlapping and without standing out from the edge of the figure ?

Simon plays a game on an $n\times n$ grid of cells. Initially, each cell is filled with an integer. Every minute, Simon picks a cell satisfying the following: [list] [*] The magnitude of the integer in the chosen cell is less than $n^{n^n}$ [*] The sum of all the integers in the neighboring cells (sharing one side with the chosen cell) is non-zero [/list] Simon then adds each integer in a neighboring cell to the chosen cell. Show that Simon will eventually not be able to make any valid moves.

Suppose $n$ is odd and each square of an $n \times n$ grid is arbitrarily filled with either by $1$ or by $-1$. Let $r_j$ and $c_k$ denote the product of all numbers in $j$-th row and $k$-th column respectively, $1 \le j, k \le n$. Prove that $$\sum_{j=1}^{n} r_j+ \sum_{k=1}^{n} c_k\ne 0$$

Joy has a square board of size $n \times n$. At every step, he colours a cell of the board. He cannot colour any cell more than once. He also counts points while colouring the cells. At first, he has $0$ points. Every step, after colouring a cell $c$, he takes the largest possible set $S$ that creates a "$+$" sign where all cells are coloured and $c$ lies in the centre. Then, he gets the size of set $S$ as points. After colouring the whole $n \times n$ board, what is the maximum possible amount of points he can get?

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]

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.

Let $a$ and $b$ be positive integers. The cells of an $(a+b+1)\times (a+b+1)$ grid are colored amber and bronze such that there are at least $a^2+ab-b$ amber cells and at least $b^2+ab-a$ bronze cells. Prove that it is possible to choose $a$ amber cells and $b$ bronze cells such that no two of the $a+b$ chosen cells lie in the same row or column.

Find all pairs of positive integers $(m, n)$, such that in a $m \times n$ table (with $m+1$ horizontal lines and $n+1$ vertical lines), a diagonal can be drawn in some unit squares (some unit squares may have no diagonals drawn, but two diagonals cannot be both drawn in a unit square), so that the obtained graph has an Eulerian cycle.

How many ways can $ 8$ mutually non-attacking rooks be placed on the $ 9\times9$ chessboard (shown here) so that all $ 8$ rooks are on squares of the same color? (Two rooks are said to be attacking each other if they are placed in the same row or column of the board.) [asy]unitsize(3mm); defaultpen(white); fill(scale(9)*unitsquare,black); fill(shift(1,0)*unitsquare); fill(shift(3,0)*unitsquare); fill(shift(5,0)*unitsquare); fill(shift(7,0)*unitsquare); fill(shift(0,1)*unitsquare); fill(shift(2,1)*unitsquare); fill(shift(4,1)*unitsquare); fill(shift(6,1)*unitsquare); fill(shift(8,1)*unitsquare); fill(shift(1,2)*unitsquare); fill(shift(3,2)*unitsquare); fill(shift(5,2)*unitsquare); fill(shift(7,2)*unitsquare); fill(shift(0,3)*unitsquare); fill(shift(2,3)*unitsquare); fill(shift(4,3)*unitsquare); fill(shift(6,3)*unitsquare); fill(shift(8,3)*unitsquare); fill(shift(1,4)*unitsquare); fill(shift(3,4)*unitsquare); fill(shift(5,4)*unitsquare); fill(shift(7,4)*unitsquare); fill(shift(0,5)*unitsquare); fill(shift(2,5)*unitsquare); fill(shift(4,5)*unitsquare); fill(shift(6,5)*unitsquare); fill(shift(8,5)*unitsquare); fill(shift(1,6)*unitsquare); fill(shift(3,6)*unitsquare); fill(shift(5,6)*unitsquare); fill(shift(7,6)*unitsquare); fill(shift(0,7)*unitsquare); fill(shift(2,7)*unitsquare); fill(shift(4,7)*unitsquare); fill(shift(6,7)*unitsquare); fill(shift(8,7)*unitsquare); fill(shift(1,8)*unitsquare); fill(shift(3,8)*unitsquare); fill(shift(5,8)*unitsquare); fill(shift(7,8)*unitsquare); draw(scale(9)*unitsquare,black);[/asy]