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An inequality with square roots and minima

Source: European Mathematical Cup 2020, Problem J4

December 22, 2020
inequalitiesemc

Problem Statement

Let a,b,ca,b,c be positive real numbers such that ab+bc+ac=a+b+cab+bc+ac = a+b+c. Prove the following inequality: a+bc+b+ca+c+ab≤2⋅min⁡{ab+bc+ca, ba+cb+ac}.\sqrt{a+\frac{b}{c}} + \sqrt{b+\frac{c}{a}} + \sqrt{c+\frac{a}{b}} \leq \sqrt{2} \cdot \min \left\{ \frac{a}{b}+\frac{b}{c}+\frac{c}{a},\ \frac{b}{a}+\frac{c}{b}+\frac{a}{c} \right\}. \\ \\ Proposed by Dorlir Ahmeti.
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