x+y+z=3
Source: Iranian National Olympiad (3rd Round) 2008
August 30, 2008
inequalitiesinequalities proposedCauchy Inequality
Problem Statement
Let x,y,z\in\mathbb R^{\plus{}} and x\plus{}y\plus{}z\equal{}3. Prove that:
\frac{x^3}{y^3\plus{}8}\plus{}\frac{y^3}{z^3\plus{}8}\plus{}\frac{z^3}{x^3\plus{}8}\geq\frac19\plus{}\frac2{27}(xy\plus{}xz\plus{}yz)