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Part of 1996 Japan MO Finals

Problems(1)

Every angle in triangle σ is greater than θ in partition

Source: Japanese MO Finals 1996

2/11/2011
A plane is partitioned into triangles. Let T0\mathcal{T}_0 denote the set of vertices of triangles in the partition. Let ABCABC be a triangle with A,B,C∈T0A,B,C\in\mathcal{T}_0 and θ\theta be its smallest angle. Assume that no point of T0\mathcal{T}_0 lies inside the circumcircle of △ABC\triangle ABC. Prove that there exists a triangle σ\sigma in the partition such that its intersection with △ABC\triangle ABC is nonempty and whose every angle is greater than θ\theta.
geometrycircumcirclecombinatorics unsolvedcombinatorics