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Volumes in Tetrahedrons

Source: IMO LongList 1970 - P45

May 22, 2011
geometry3D geometrytetrahedronfunctiongeometry unsolved

Problem Statement

Let MM be an interior point of tetrahedron VABCV ABC. Denote by A1,B1,C1A_1,B_1, C_1 the points of intersection of lines MA,MB,MCMA,MB,MC with the planes VBC,VCA,VABVBC,V CA,V AB, and by A2,B2,C2A_2,B_2, C_2 the points of intersection of lines VA1,VB1,VC1V A_1, VB_1, V C_1 with the sides BC,CA,ABBC,CA,AB.
(a) Prove that the volume of the tetrahedron VA2B2C2V A_2B_2C_2 does not exceed one-fourth of the volume of VABCV ABC.
(b) Calculate the volume of the tetrahedron V1A1B1C1V_1A_1B_1C_1 as a function of the volume of VABCV ABC, where V1V_1 is the point of intersection of the line VMVM with the plane ABCABC, and MM is the barycenter of VABCV ABC.
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