Problem

Source: Romania IMO TST 1992 p1

Tags: geometry, rectangle, combinatorics, collinear, combinatorial geometry



Let $S > 1$ be a real number. The Cartesian plane is partitioned into rectangles whose sides are parallel to the axes of the coordinate system. and whose vertices have integer coordinates. Prove that if the area of each triangle if at most $S$, then for any positive integer $k$ there exist $k$ vertices of these rectangles which lie on a line.