Olympiad problems

2019 Canada National Olympiad, 3

Tags: checkerboard , combinatorics , grid , tiling , bijection , induction

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.

2021 OMpD, 4

Tags: ratio , checkerboard , combinatorics

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 $\frac {B}{P} = \frac{2021}{4321}$.

1996 All-Russian Olympiad, 8

Tags: combinatorics , checkerboard

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]

2001 239 Open Mathematical Olympiad, 4

Tags: combinatorics , checkerboard

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.