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Easy refinement of (a+b+c)(1/a+1/b+1/c)\ge 9

Source: Romanian NMO 2021 grade 8 P2

April 15, 2023
inequalities

Problem Statement

Prove that for all positive real numbers a,b,ca,b,c the following inequality holds: (a+b+c)(1a+1b+1c)≥2(a2+b2+c2)ab+bc+ca+7(a+b+c)\left(\frac1a+\frac1b+\frac1c\right)\ge\frac{2(a^2+b^2+c^2)}{ab+bc+ca}+7 and determine all cases of equality.
Lucian Petrescu
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