IMO Shortlist 1962 problem 7


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2. travnja 2012.
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The tetrahedron SABC has the following property: there exist five spheres, each tangent to the edges SA, SB, SC, BC, CA, AB, or to their extensions.

a) Prove that the tetrahedron SABC is regular.

b) Prove conversely that for every regular tetrahedron five such spheres exist.
Izvor: Međunarodna matematička olimpijada, shortlist 1962