ILL 1990 Problem 47

In the coordinate plane a rectangle with vertices $(0, 0),$ $(m, 0),$ $(0, n),$ $(m, n)$ is given where both $m$ and $n$ are odd integers. The rectangle is partitioned into triangles in such a way that

(i) each triangle in the partition has at least one side (to be called a "good" side) that lies on a line of the form $x = j$ or $y = k,$ where $j$ and $k$ are integers, and the altitude on this side has length 1;

(ii) each "bad" side (i.e., a side of any triangle in the partition that is not a "good" one) is a common side of two triangles in the partition.

Prove that there exist at least two triangles in the partition each of which has two good sides.