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n pairwise disjoint triangles

Source: Vietnam TST 1996 for the 37th IMO, problem 1

June 26, 2005
geometryrectanglecombinatorial geometryarea of a trianglecombinatorics unsolvedcombinatorics

Problem Statement

In the plane we are given 3⋅n3 \cdot n points (n>n>1) no three collinear, and the distance between any two of them is ≤1\leq 1. Prove that we can construct nn pairwise disjoint triangles such that: The vertex set of these triangles are exactly the given 3n points and the sum of the area of these triangles <1/2< 1/2.
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