Inside the convex polyhedron, the point P and several lines ℓ1,ℓ2,...,ℓn passing through P and not lying in the same plane. To each face of the polyhedron we associate one of the lines l1,l2,...,ln that forms the largest angle with the plane of this face (if there are there are several direct ones, we will choose any of them). Prove that there is a face that intersects with its corresponding line.
Problem
Source: 2012 Oral Moscow Geometry Olympiad grades 10-11 p4
Tags: geometry, 3D geometry, polyhedron