6

Part of 1985 Polish MO Finals

Problems(1)

then the polyhedron cannot have an inscribed sphere

Source: 1985 Polish MO Finals p6

There is a convex polyhedron with

k

faces. Show that if more than

k/2

of the faces are such that no two have a common edge, then the polyhedron cannot have an inscribed sphere.

3D geometrygeometrypolyhedronsphereinscribed