Question 14
(JAP 2) 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.
(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.