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Inequality in Crelle tetrahedron

Source: KoMaL A. 837

December 13, 2022
geometrylinear algebrageometric inequalityinequalities3D geometrykomaltetrahedron

Problem Statement

Let all the edges of tetrahedron A1A2A3A4A_1A_2A_3A_4 be tangent to sphere SS. Let ai\displaystyle a_i denote the length of the tangent from AiA_i to SS. Prove that (∑i=141ai) ⁣ ⁣2>2(∑i=141ai2).\bigg(\sum_{i=1}^4 \frac 1{a_i}\bigg)^{\!\!2}> 2\bigg(\sum_{i=1}^4 \frac1{a_i^2}\bigg).
Submitted by Viktor Vígh, Szeged
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