IMO Shortlist 1999 problem G3

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Dodao/la: arhiva
2. travnja 2012.
A set S of points from the space will be called completely symmetric if it has at least three elements and fulfills the condition that for every two distinct points A and B from S, the perpendicular bisector plane of the segment AB is a plane of symmetry for S. Prove that if a completely symmetric set is finite, then it consists of the vertices of either a regular polygon, or a regular tetrahedron or a regular octahedron.
Izvor: Međunarodna matematička olimpijada, shortlist 1999