DE-BUNDESWETTBEWERB 2021 Problem 8

In the Cartesian plane, a line segment is called tame if it lies parallel to one of the coordinate axes and its distance to this axis is an integer. Otherwise it is called wild.

Let $m$ and $n$ be odd positive integers. The rectangle with vertices $(0,0),(m,0),(m,n)$ and $(0,n)$ is partitioned into finitely many triangles. Let $M$ be the set of these triangles. Assume that

(1) Each triangle from $M$ has at least one tame side. (2) For each tame side of a triangle from $M$ , the corresponding altitude has length $1$ . (3) Each wild side of a triangle from $M$ is a common side of exactly two triangles from $M$ .

Show that at least two triangles from $M$ have two tame sides each.