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China Mathematical Olympiad 1987 problem5

Source: China Mathematical Olympiad 1987 problem5

January 9, 2014
geometry3D geometryspheretetrahedrongeometry unsolved

Problem Statement

Let A1A2A3A4A_1A_2A_3A_4 be a tetrahedron. We construct four mutually tangent spheres S1,S2,S3,S4S_1,S_2,S_3,S_4 with centers A1,A2,A3,A4A_1,A_2,A_3,A_4 respectively. Suppose that there exists a point QQ such that we can construct two spheres centered at QQ satisfying the following conditions: i) One sphere with radius rr is tangent to S1,S2,S3,S4S_1,S_2,S_3,S_4; ii) One sphere with radius RR is tangent to every edges of tetrahedron A1A2A3A4A_1A_2A_3A_4. Prove that A1A2A3A4A_1A_2A_3A_4 is a regular tetrahedron.
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