In the coordinate plane a rectangle with vertices
is given where both
and
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
or
where
and
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.
![(0, 0),](/media/m/4/4/c/44cc792f617c5362cd1f502ffd01d2a5.png)
![(m, 0),](/media/m/a/0/f/a0f63f2bee739373326856710a02fbde.png)
![(0, n),](/media/m/a/b/7/ab7997a7a70584802662a11879c4431e.png)
![(m, n)](/media/m/4/f/4/4f46dcf2259fdaf3ff9891a4fa773ec3.png)
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
(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](/media/m/2/2/d/22d36aba893e3c79db1f7b241e3e3a11.png)
![y = k,](/media/m/3/f/1/3f1a2cad4d227d66f43dfccd3c1500ba.png)
![j](/media/m/7/9/e/79ebb10f98eb80d16b0c785d9d682a72.png)
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
(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.