IMO 2006 Shortlist C7
Consider a convex polyhedron without parallel edges and without an edge parallel to any face other than the two faces ad...
Category: Combinatorics
Problem
Consider a convex polyhedron without parallel edges and without an edge parallel to any face other than the two faces adjacent to it. Call a pair of points of the polyhedron antipodal if there exist two parallel planes passing through these points and such that the polyhedron is contained between these planes. Let A be the number of antipodal pairs of vertices, and let B be the number of antipodal pairs of midpoints of edges. Determine the difference A−B in terms of the numbers of vertices, edges and faces. (Japan)