MathDB

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.

Problem Statement