IMO 1989 LL FRA22

Origin: FRA

Problem

Let ABC be an equilateral triangle with side length equal to a natural number N. Consider the set S of all points M inside the triangle ABC such that −−\to AM = N (n−−\to AB + m−\to AC), where m, n are integers and 0 \leqm, n, m+n \leqN. Every point of S is colored in one of the three colors blue, white, red such that no point on AB is colored blue, no point on AC is colored white, and no point on BC is colored red. Prove that there exists an equilateral triangle with vertices in S and side length 1 whose three vertices are colored blue, white, and red.