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.

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]

Consider a $N\times N$ square board where $N\geq 3$ is an odd integer. The caterpillar Carl sits at the center of the square; all other cells contain distinct positive integers. An integer $n$ weights $1\slash n$ kilograms. Carl wants to leave the board but can eat at most $2$ kilograms. Determine whether Carl can always find a way out when a) $N=2003.$ b) $N$ is an arbitrary odd integer.

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]

Ana draws a checkered board that has at least 20 rows and at least 24 columns. Then, Beto must completely cover that board, without holes or overlaps, using only pieces of the following two types: Each piece must cover exactly 4 or 3 squares of the board, as shown in the figure, without leaving the board. It is permitted to rotate the pieces and it is not necessary to use all types of pieces. Explain why, regardless of how many rows and how many columns Ana's board has, Beto can always complete his task.

Peter has a wooden square stamp divided into a grid. He coated some $102$ cells of this grid with black ink. After that, he pressed this stamp $100$ times on a list of paper so that each time just those $102$ cells left a black imprint on the paper. Is it possible that after his actions the imprint on the list is a square $101 \times 101$ such that all the cells except one corner cell are black? (Alexsandr Gribalko)

$40$ cells were marked on an infinite chessboard. Is it always possible to find a rectangle that contains $20$ marked cells? M. Evdokimov

Let $n\ge 3$ be a positive integer. Amy wrote all the integers from $1$ to $n^2$ on the $n\times n$ grid, so that each cell contains exactly one number. For $i=1,2,\cdots ,n^2-1$, the cell containing $i$ shares a common side with the cell containing $i+1$. Each turn, Bred can choose one cell, and check what number is written. Bred wants to know where $1$ is written by less than $3n$ turns. Determine whether $n$ such that Bred can always achieve his goal is infinite.

Let $n \geq 2$ be an integer. Lucia chooses $n$ real numbers $x_1,x_2,\ldots,x_n$ such that $\left| x_i-x_j \right|\geq 1$ for all $i\neq j$. Then, in each cell of an $n \times n$ grid, she writes one of these numbers, in such a way that no number is repeated in the same row or column. Finally, for each cell, she calculates the absolute value of the difference between the number in the cell and the number in the first cell of its same row. Determine the smallest value that the sum of the $n^2$ numbers that Lucia calculated can take.

Let $n$ be a positive integer. A grid of $n\times n$ has some black-colored cells. Drini can color a cell if at least three cells that share a side with it are also colored black. Drini discovers that by repeating this process, all the cells in the grid can be colored. Prove that if there are initially $k$ colored cells, then $$k\geq \frac{n^2+2n}{3}.$$

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.

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 $n$.

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

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]

Suppose a grid with $2m$ rows and $2n$ columns is given, where $m$ and $n$ are positive integers. You may place one pawn on any square of this grid, except the bottom left one or the top right one. After placing the pawn, a snail wants to undertake a journey on the grid. Starting from the bottom left square, it wants to visit every square exactly once, except the one with the pawn on it, which the snail wants to avoid. Moreover, it wants to finish in the top right square. It can only move horizontally or vertically on the grid. On which squares can you put the pawn for the snail to be able to finish its journey?

Alice and Bob play the following game on a $100\times 100$ grid, taking turns, with Alice starting first. Initially the grid is empty. At their turn, they choose an integer from $1$ to $100^2$ that is not written yet in any of the cells and choose an empty cell, and place it in the chosen cell. When there is no empty cell left, Alice computes the sum of the numbers in each row, and her score is the maximum of these $100$ numbers. Bob computes the sum of the numbers in each column, and his score is the maximum of these $100$ numbers. Alice wins if her score is greater than Bob's score, Bob wins if his score is greater than Alice's score, otherwise no one wins. Find if one of the players has a winning strategy, and if so which player has a winning strategy. [i]Théo Lenoir, France[/i]

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.

Is it possible to write positive integers from $1$ to $2025$ in the cells of a \( 45 \times 45 \) grid such that each number is used exactly once, and at the same time, each written number is either greater than all the numbers located in its side-adjacent cells or smaller than all the numbers located in its side-adjacent cells? [i]Proposed by Anton Trygub[/i]

In an $8 \times 8$ grid, $n$ disks, numbered $1$ to $n$ are stacked, with random order, in a pile in the bottom left comer. The disks can be moved one at a time to a neighbouring cell either to the right or top. The aim to move all the disks to the cell at the top right comer and stack them in the order $1,2,...,n$ from the bottom. Each cell, except the bottom left and top right cell, can have at most one disk at any given time. Find the largest value of $n$ so that the aim can be achieved.

A unit square is removed from the corner of an $n \times n$ grid, where $n \geq 2$. Prove that the remainder can be covered by copies of the figures of $3$ or $5$ unit squares depicted in the drawing below. [asy] import geometry; draw((-1.5,0)--(-3.5,0)--(-3.5,2)--(-2.5,2)--(-2.5,1)--(-1.5,1)--cycle); draw((-3.5,1)--(-2.5,1)--(-2.5,0)); draw((0.5,0)--(0.5,3)--(1.5,3)--(1.5,1)--(3.5,1)--(3.5,0)--cycle); draw((1.5,0)--(1.5,1)); draw((2.5,0)--(2.5,1)); draw((0.5,1)--(1.5,1)); draw((0.5,2)--(1.5,2)); [/asy] [b]Note:[/b] Every square must be covered once and figures must not go over the bounds of the grid.

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 $$\sum_{j=1}^{n} r_j+ \sum_{k=1}^{n} c_k\ne 0$$

There are $10000$ trees in a park, arranged in a square grid with $100$ rows and $100$ columns. Find the largest number of trees that can be cut down, so that sitting on any of the tree stumps one cannot see any other tree stump.

Let $n$ be a positive integer. In an $n\times n$ garden, a fountain is to be built with $1\times 1$ platforms covering the entire garden. Ana places all the platforms at a different height. Afterwards, Beto places water sources in some of the platforms. The water in each platform can flow to other platforms sharing a side only if they have a lower height. Beto wins if he fills all platforms with water. Find the least number of water sources that Beto needs to win no matter how Ana places the platforms.

Petya and Vasya are playing a game on an initially empty \(100 \times 100\) grid, taking turns. Petya goes first. On his turn, a player writes an uppercase Russian letter in an empty cell (each cell can contain only one letter). When all cells are filled, Petya is declared the winner if there are four consecutive cells horizontally spelling the word ``ПЕТЯ'' (PETYA) from left to right, or four consecutive cells vertically spelling ``ПЕТЯ'' from top to bottom. Can Petya guarantee a win regardless of Vasya's moves?

Bethan is playing a game on an $n\times n$ grid consisting of $n^2$ cells. A move consists of placing a counter in an unoccupied cell $C$ where the $2n-2$ other cells in the same row or column as $C$ contain an even number of counters. After making $M$ moves Bethan realises she cannot make any more moves. Determine the minimum value of $M$. [i]United Kingdom, Sam Bealing[/i]