Problem

Source: Iranian National Math Olympiad (Final exam) 2006

Tags: geometry, 3D geometry, prism, sphere, tetrahedron, symmetry, perpendicular bisector



A regular polyhedron is a polyhedron that is convex and all of its faces are regular polygons. We call a regular polhedron a "Choombam" iff none of its faces are triangles. a) prove that each choombam can be inscribed in a sphere. b) Prove that faces of each choombam are polygons of at most 3 kinds. (i.e. there is a set $\{m,n,q\}$ that each face of a choombam is $n$-gon or $m$-gon or $q$-gon.) c) Prove that there is only one choombam that its faces are pentagon and hexagon. (Soccer ball) Invalid image file d) For $n>3$, a prism that its faces are 2 regular $n$-gons and $n$ squares, is a choombam. Prove that except these choombams there are finitely many choombams.