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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 = 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.

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