IMO Shortlist 1982 problem 18
Let be a point of three-dimensional space and let be mutually perpendicular straight lines passing through . Let denote the sphere with center and radius , and for every point of , let denote the sphere with center and radius . We denote by the intersection of with the straight lines , respectively, where we put if meets at two distinct points and otherwise (). What is the set of centers of gravity of the (possibly degenerate) triangles as runs through the points of ?