IMO Shortlist 2006 problem C7

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Dodao/la: arhiva
2. travnja 2012.
Consider a convex polyhaedron 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 polyhaedron antipodal if there exist two parallel planes passing through these points and such that the polyhaedron 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 midpoint edges. Determine the difference A-B in terms of the numbers of vertices, edges, and faces.
Izvor: Međunarodna matematička olimpijada, shortlist 2006