ISL 1991 Problem 7

$ABCD$ is a terahedron: $AD+BD=AC+BC,$ $BD+CD=BA+CA,$ $CD+AD=CB+AB,$ $M,N,P$ are the mid points of $BC,CA,AB.$ $OA=OB=OC=OD.$ Prove that $\angle MOP = \angle NOP =\angle NOM.$