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Triangle subdivided into triangles

Source: ILL 1979 - Problem 62.

June 5, 2011
combinatorics unsolvedcombinatorics

Problem Statement

TT is a given triangle with vertices P1,P2,P3P_1,P_2,P_3. Consider an arbitrary subdivision of TT into finitely many subtriangles such that no vertex of a subtriangle lies strictly between two vertices of another subtriangle. To each vertex VV of the subtriangles there is assigned a number n(V)n(V) according to the following rules: (i)(\text{i}) If VV = PiP_i, then n(V)=in(V) = i. (ii)(\text{ii}) If VV lies on the side PiPjP_i P_j of TT, then n(V)=in(V) = i or jj. (iii)(\text{iii}) If VV lies inside the triangle TT, then n(V)n(V) is any of the numbers 1,2,31,2,3. Prove that there exists at least one subtriangle whose vertices are numbered 1,2,31, 2, 3.
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