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lines forming largeast angles with polyhedron faces and concurrent lines

Source: 2012 Oral Moscow Geometry Olympiad grades 10-11 p4

September 25, 2019
geometry3D geometrypolyhedron

Problem Statement

Inside the convex polyhedron, the point PP and several lines 1,2,...,n\ell_1,\ell_2, ..., \ell_n passing through PP and not lying in the same plane. To each face of the polyhedron we associate one of the lines l1,l2,...,lnl_1, l_2, ..., l_n 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.