AoPS - Problem

Source: Bundeswettbewerb Mathematik 2021, Round 2 - Problem 4

Tags: analytic geometry, geometry, rectangle

Tintarn

22.09.2021 14:34

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.

Call a triangle with one tame side good and bad if it has two. If we add the number of wild sides for each triangle, the result is even, since every wild side is counted twice by (3). Thus, the number of bad triangles is always even, so it suffices to show there exists one bad triangle. Assume $M$ contains only good triangles. Colour a triangle grey if its tame side is parallel to the $x$-axis and white if it's parallel to the $y$-axis. This gives a colouring of the rectangle. Call the sides which border both white and grey triangles special. It's obvious that special sides have to be wild and condition (2) implies that they are $(1,1)$ or $(1,-1)$ (as vectors). Now consider the grey region which borders the edge from $(0,0)$ to $(m,0)$. This region is bounded by a path of special sides. However, since $m$ is odd, there can be no such path.