Problem

Source: Romanian IMO Team Selection Test TST 1996, problem 10

Tags: geometry, rectangle, combinatorics proposed, combinatorics



Let $ n $ and $ r $ be positive integers and $ A $ be a set of lattice points in the plane such that any open disc of radius $ r $ contains a point of $ A $. Show that for any coloring of the points of $ A $ in $ n $ colors there exists four points of the same color which are the vertices of a rectangle.