IMO 1970 LL ROM45

Origin: ROM

Problem

Let M be an interior point of tetrahedron V ABC. Denote by A1, B1, C1 the points of intersection of lines MA, MB, MC with the planes V BC, V CA, V AB, and by A2, B2, C2 the points of intersection of lines V A1, V B1, V C1 with the sides BC, CA, AB. (a) Prove that the volume of the tetrahedron V A2B2C2 does not exceed one-fourth of the volume of V ABC. (b) Calculate the volume of the tetrahedron V1A1B1C1 as a function of the volume of V ABC, where V1 is the point of intersection of the line V M with the plane ABC, and M is the barycenter of V ABC.