This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 6

1983 Polish MO Finals, 3
Tags: chessboard , infinite chessboard , combinatorics
Consider the following one-player game on an infinite chessboard. If two horizontally or vertically adjacent squares are occupied by a pawn each, and a square on the same line that is adjacent to one of them is empty, then it is allowed to remove the two pawns and place a pawn on the third (empty) square. Prove that if in the initial position all the pawns were forming a rectangle with the number of squares divisible by $3$, then it is not possible to end the game with only one pawn left on the board.

1993 IMO Shortlist, 5
Tags: invariant , game , infinite chessboard , combinatorics
On an infinite chessboard, a solitaire game is played as follows: at the start, we have $n^2$ pieces occupying a square of side $n.$ The only allowed move is to jump over an occupied square to an unoccupied one, and the piece which has been jumped over is removed. For which $n$ can the game end with only one piece remaining on the board?

1993 IMO, 3
Tags: invariant , game , infinite chessboard , combinatorics
On an infinite chessboard, a solitaire game is played as follows: at the start, we have $n^2$ pieces occupying a square of side $n.$ The only allowed move is to jump over an occupied square to an unoccupied one, and the piece which has been jumped over is removed. For which $n$ can the game end with only one piece remaining on the board?

2019 Saint Petersburg Mathematical Olympiad, 6
Tags: chessboard , infinite chessboard , positive integer , combinatorial geometry , combinatorics
Is it possible to arrange everything in all cells of an infinite checkered plane all natural numbers (once) so that for each $n$ in each square $n \times n$ the sum of the numbers is a multiple of $n$?

1993 ITAMO, 3
Tags: sequence , infinite chessboard , combinatorics
Consider an infinite chessboard whose rows and columns are indexed by positive integers. At most one coin can be put on any cell of the chessboard. Let be given two arbitrary sequences ($a_n$) and ($b_n$) of positive integers ($n \in N$). Assuming that infinitely many coins are available, prove that they can be arranged on the chessboard so that there are $a_n$ coins in the $n$-th row and $b_n$ coins in the $n$-th column for all $n$.

2001 Tuymaada Olympiad, 2
Tags: divisible , infinite chessboard , sum , number theory
Is it possible to arrange integers in the cells of the infinite chechered sheet so that every integer appears at least in one cell, and the sum of any $10$ numbers in a row vertically or horizontal, would be divisible by $101$?