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Inequality From JBMO

Source: JBMO 2016 Shortlist A5

June 25, 2017
inequalitiesalgebra

Problem Statement

Let x,y,zx,y,z be positive real numbers such that x+y+z=1x+1y+1z.x+y+z=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}. Prove that x+y+z≥xy+12+yz+12+zx+12 .x+y+z\geq \sqrt{\frac{xy+1}{2}}+\sqrt{\frac{yz+1}{2}}+\sqrt{\frac{zx+1}{2}} \ .
Proposed by Azerbaijan
[hide=Second Suggested Version]Let x,y,zx,y,z be positive real numbers such that x+y+z=1x+1y+1z.x+y+z=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}. Prove that x+y+z≥x2+12+y2+12+z2+12 .x+y+z\geq \sqrt{\frac{x^2+1}{2}}+\sqrt{\frac{y^2+1}{2}}+\sqrt{\frac{z^2+1}{2}} \ .
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