MathDB

In the Cartesian coordinate plane define the strips S_n \equal{} \{(x,y)|n\le x < n \plus{} 1\},

n\in\mathbb{Z}

and color each strip black or white. Prove that any rectangle which is not a square can be placed in the plane so that its vertices have the same color.
IMO Shortlist 2007 Problem C5 as it appears in the official booklet: In the Cartesian coordinate plane define the strips S_n \equal{} \{(x,y)|n\le x < n \plus{} 1\} for every integer

n.

Assume each strip

S_n

is colored either red or blue, and let

a

and

b

be two distinct positive integers. Prove that there exists a rectangle with side length

a

and

b

such that its vertices have the same color. (Edited by Orlando Döhring)
Author: Radu Gologan and Dan Schwarz, Romania

Problem Statement