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1/(x^2+y+z)+1/(y^2+z+x)+1/(z^2+x+y) <= 1 if x/yz+y/zx+z/xy<=1, for x,y,z>0

Source: Balkan BMO Shortlist 2016 A2

July 30, 2019
inequalitiesthree variable inequalityalgebra

Problem Statement

For all x,y,z>0x,y,z>0 satisfying xyz+yzx+zxy≤x+y+z\frac{x}{yz}+\frac{y}{zx}+\frac{z}{xy}\le x+y+z, prove that 1x2+y+z+1y2+z+x+1z2+x+y≤1\frac{1}{x^2+y+z}+\frac{1}{y^2+z+x}+\frac{1}{z^2+x+y} \le 1
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