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3d geo helps provin 6-var ineq xyz+uv(1- x)+(1-y)(1-v)t +(1- z)(1- w)(1- t)<1

Source: 1996 Romania NMO VIII p4

August 13, 2024
inequalitiesalgebrageometric inequalityGeometric Inequalities3D geometrygeometry

Problem Statement

a) Let ABCDAB CD be a regular tetrahedron. On the sides ABAB, ACAC and ADAD, the points MM, NN and PP, are considered. Determine the volume of the tetrahedron AMNPAMNP in terms of x,y,zx, y, z, where x=AMx=AM, y=ANy=AN, z=APz=AP.
b) Show that for any real numbers x,y,z,t,u,v∈(0,1)x, y, z, t, u, v \in (0, 1) : xyz+uv(1−x)+(1−y)(1−v)t+(1−z)(1−w)(1−t)<1.xyz + uv(1- x) + (1- y)(1- v)t + (1- z)(1- w)(1- t) < 1.
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