VN TST 1996 Question 1

In a plane, $3n$ points ($n>1$) are given, no three collinear, and every pair has distance at most $1$. Prove that one can construct $n$ pairwise disjoint triangles such that (1) every given point is a vertex of exactly one triangle and (2) their total area is less than $\tfrac12$. Two triangles are disjoint if they have no common point in their interiors or on their boundaries.

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