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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 \equal{} j or y \equal{} 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.

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