Olympiad problems

2001 239 Open Mathematical Olympiad, 4

Integers are placed on every cell of an infinite checkerboard. For each cell if it contains integer $a$ then the sum of the numbers in the cell under it and the cell right to it is $2a+1$. Prove that in every infinite diagonal row of direction [i] top-right down-left [/i] all numbers are different.

checkerboard combinatorics

2021 OMpD, 4

Determine the smallest positive integer $n$ with the following property: on a board $n \times n$, whose squares are painted in checkerboard pattern (that is, for any two squares with a common edge, one of them is black and the other is white), it is possible to place the numbers $1,2,3 , ... , n^2$, a number in each square, so if $B$ is the sum of the numbers written in the white squares and $P$ is the sum of the numbers written in the black squares, then $\frac {B}{P} = \frac{2021}{4321}$.

checkerboard ratio combinatorics number theory

2019 Canada National Olympiad, 3

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.

checkerboard induction tiling combinatorics bijection grid casework mathematical induction parity pigeonhole principle recurrence relations stars and bars

1996 All-Russian Olympiad, 8

Can a $5\times 7$ checkerboard be covered by L's (figures formed from a $2\times2$ square by removing one of its four $1\times1$ corners), not crossing its borders, in several layers so that each square of the board is covered by the same number of L's? [i]M. Evdokimov[/i]

checkerboard combinatorics