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Part of 2017 Kazakhstan National Olympiad

Problems(1)

Inequality, Problem 2

Source: Kazakhstan National Olympiad 2017, March 14, P2 , matol.kz

3/15/2017
For positive reals x,y,z≥12x,y,z\ge \frac{1}{2} with x2+y2+z2=1x^2+y^2+z^2=1, prove this inequality holds
(1x+1y−1z)(1x−1y+1z)≥2(\frac{1}{x}+\frac{1}{y}-\frac{1}{z})(\frac{1}{x}-\frac{1}{y}+\frac{1}{z})\ge 2
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