IMO 1986 LL CHN12

Origin: CHN

Problem

Let O be an interior point of a tetrahedron A1A2A3A4. Let S1, S2, S3, S4 be spheres with centers A1, A2, A3, A4, respectively, and let U, V be spheres with centers at O. Suppose that for i, j = 1, 2, 3, 4, i ̸= j, the spheres Si and Sj are tangent to each other at a point Bij lying on AiAj. Suppose also that U is tangent to all edges AiAj and V is tangent to the spheres S1, S2, S3, S4. Prove that A1A2A3A4 is a regular tetrahedron.