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Point M on surface of a regular tetrahedron

Source:

August 29, 2010
geometry3D geometrytetrahedronIMO Shortlist

Problem Statement

Prove that for every point MM on the surface of a regular tetrahedron there exists a point M′M' such that there are at least three different curves on the surface joining MM to M′M' with the smallest possible length among all curves on the surface joining MM to M′M'.
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