IMO 1978 LL VIE50

Origin: VIE

Problem

A variable tetrahedron ABCD has the following properties: Its edge lengths can change as well as its vertices, but the opposite edges remain equal (BC = DA, CA = DB, AB = DC); and the vertices A, B, C lie respectively on three fixed spheres with the same center P and radii 3, 4, 12. What is the maximal length of PD?