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sum \sqrt{z + xy} >=\sqrt{xyz}+\sqrt{x }+\sqrt{y} +\sqrt{z} if 1/x+1/y+1/z=1

Source: 2019 Grand Duchy of Lithuania, Mathematical Contest p1 (Baltic Way TST)

October 3, 2020

algebrainequalities

Problem Statement

Let

x, y, z

be positive numbers such that

\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 1

. Prove that

\sqrt{x + yz} +\sqrt{y + zx} +\sqrt{z + xy} \ge\sqrt{xyz}+\sqrt{x }+\sqrt{y} +\sqrt{z}