Geometry | Part IB, 2002

State Euler's formula for a graph G\mathcal{G} with FF faces, EE edges and VV vertices on the surface of a sphere.

Suppose that every face in G\mathcal{G} has at least three edges, and that at least three edges meet at every vertex of G\mathcal{G}. Let FnF_{n} be the number of faces of G\mathcal{G} that have exactly nn edges (n3)(n \geqslant 3), and let VmV_{m} be the number of vertices at which exactly mm edges meet (m3)(m \geqslant 3). By expressing 6FnnFn6 F-\sum_{n} n F_{n} in terms of the VjV_{j}, or otherwise, show that every convex polyhedron has at least four faces each of which is a triangle, a quadrilateral or a pentagon.

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