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Prove this trigonometric inequality

Source: 2019 Jozsef Wildt International Math Competition

May 19, 2020

inequalitiestrigonometry

Problem Statement

Let

x

y

z > 0

such that

x^2 + y^2 + z^2 = 3

. Then

x^3\tan^{-1}\frac{1}{x}+y^3\tan^{-1}\frac{1}{y}+z^3\tan^{-1}\frac{1}{z}<\frac{\pi \sqrt{3}}{2}