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Non cyclic inequality

Source: Turkey National Mathematical Olympiad 2020 P3

March 8, 2021

inequalitiesinequalities proposedTurkey

Problem Statement

x, y, z

are positive real numbers find the minimum value of

2\sqrt{(x+y+z) \left( \frac{1}{x}+ \frac{1}{y} + \frac{1}{z} \right)} - \sqrt{ \left( 1+ \frac{x}{y} \right) \left( 1+ \frac{y}{z} \right)}