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3-variable inequality with min(ab,bc,ca)>=1

Source: 2016 IMO Shortlist A1

July 19, 2017
IMO Shortlistinequalitiesthree variable inequalityHi

Problem Statement

Let aa, bb, cc be positive real numbers such that min⁡(ab,bc,ca)≥1\min(ab,bc,ca) \ge 1. Prove that (a2+1)(b2+1)(c2+1)3≤(a+b+c3)2+1.\sqrt[3]{(a^2+1)(b^2+1)(c^2+1)} \le \left(\frac{a+b+c}{3}\right)^2 + 1.
Proposed by Tigran Margaryan, Armenia
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