ISL 1970 Problem 45

Let $M$ be an interior point of tetrahedron $V ABC$ . Denote by $A_1,B_1, C_1$ the points of intersection of lines $MA,MB,MC$ with the planes $VBC,V CA,V AB$ , and by $A_2,B_2, C_2$ the points of intersection of lines $V A_1, VB_1, V C_1$ with the sides $BC,CA,AB$ .(a) Prove that the volume of the tetrahedron $V A_2B_2C_2$ does not exceed one-fourth of the volume of $V ABC$ .(b) Calculate the volume of the tetrahedron $V_1A_1B_1C_1$ as a function of the volume of $V ABC$ , where $V_1$ is the point of intersection of the line $VM$ with the plane $ABC$ , and $M$ is the barycenter of $V ABC$ .