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Inequality where a^x=bc,b^y=ca, c^z=ab

Source: 2017 Azerbaijan Senior National Olympiad

April 29, 2022

Inequalityalgebrainequalities

Problem Statement

a,b,c \in (0,1)

and

x,y,z \in ( 0, \infty)

reals satisfies the condition

a^x=bc,b^y=ca,c^z=ab

. Prove that

\dfrac{1}{2+x}+\dfrac{1}{2+y}+\dfrac{1}{2+z} \leq \dfrac{3}{4}