Show that there exists a set S of 15 distinct circles on the surface of a sphere, all having the same radius and such that 5 touch exactly 5 others, 5 touch exactly 4 others, and 5 touch exactly 3 others.[General Problem: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=384764] geometry3D geometryspheregeometry unsolved