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Tetrahedron and centroid inequality

Source: IMO Shortlist 1995, G6

August 10, 2008
geometry3D geometrytetrahedrongeometric inequalitysphereIMO Shortlist

Problem Statement

Let A1A2A3A4 A_1A_2A_3A_4 be a tetrahedron, G G its centroid, and A1′,A2′,A3′, A'_1, A'_2, A'_3, and A4′ A'_4 the points where the circumsphere of A1A2A3A4 A_1A_2A_3A_4 intersects GA1,GA2,GA3, GA_1,GA_2,GA_3, and GA4, GA_4, respectively. Prove that GA1⋅GA2⋅GA3⋅GA⋅4≤GA1′⋅GA2′⋅GA3′⋅GA4′ GA_1 \cdot GA_2 \cdot GA_3 \cdot GA_ \cdot4 \leq GA'_1 \cdot GA'_2 \cdot GA'_3 \cdot GA'_4 and \frac{1}{GA'_1} \plus{} \frac{1}{GA'_2} \plus{} \frac{1}{GA'_3} \plus{} \frac{1}{GA'_4} \leq \frac{1}{GA_1} \plus{} \frac{1}{GA_2} \plus{} \frac{1}{GA_3} \plus{} \frac{1}{GA_4}.
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