"antipodal" points of a polyhaedron
Source: IMO Shortlist 2006, Combinatorics 7
June 28, 2007
geometry3D geometryEulerpolyhedronIMO Shortlist
Problem Statement
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 be the number of antipodal pairs of vertices, and let be the number of antipodal pairs of midpoint edges. Determine the difference in terms of the numbers of vertices, edges, and faces.Proposed by Kei Irei, Japan