IMO 1979 LL SWE62

Origin: SWE

Problem

T is a given triangle with vertices P1, P2, P3. Consider an arbi- trary subdivision of T into finitely many subtriangles such that no vertex of a subtriangle lies strictly between two vertices of another subtriangle. To each vertex V of the subtriangles there is assigned a number n(V ) according to the following rules: (i) If V = Pi, then n(V ) = i. (ii) If V lies on the side PiPj of T , then n(V ) = i or j. (iii) If V lies inside the triangle T , then n(V ) is any of the numbers 1,2,3. Prove that there exists at least one subtriangle whose vertices are num- bered 1, 2, and 3.