Problem

Source: 1999 Romania NMO X p2

Tags: geometry, 3D geometry, tetrahedron, geometric inequality, Geometric Inequalities



On the sides $(AB)$, $(BC)$, $(CD)$ and $(DA)$ of the regular tetrahedron $ABCD$, one considers the points $M$, $N$, $P$, $Q$, respectively Prove that $$MN \cdot NP \cdot PQ \cdot QM \ge AM \cdot BN \cdot CP \cdot DQ.$$