The following question has been posed in Yahoo! Answers and although I've proved it for a regular tetrahedron, the general case is much more difficult. Any suggestions?
Here's the question:
Let O be any point within a tetrahedron ABCD. Lines joining the vertices A, B, C, D to O when extended meet the opposite faces in P, Q, R and S respectively. Prove that OP/AP + OQ/BQ + OR/CR + OS/DS = 1.