CMO 1987 Problem 5

Let $A_1A_2A_3A_4$ be a tetrahedron. We construct four mutually tangent spheres $S_1,S_2,S_3,S_4$ with centers $A_1,A_2,A_3,A_4$ respectively. Suppose that there exists a point $Q$ such that we can construct two spheres centered at $Q$ satisfying the following conditions: i) One sphere with radius $r$ is tangent to $S_1,S_2,S_3,S_4$ ; ii) One sphere with radius $R$ is tangent to every edges of tetrahedron $A_1A_2A_3A_4$ . Prove that $A_1A_2A_3A_4$ is a regular tetrahedron.