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cyc sum (x^2+y^2+z)/(x^2+2) \geq3 [Macedonia JBMO TST 2017, P3]

Source: Macedonia JBMO TST 2017, Problem 3

June 26, 2018
inequalities

Problem Statement

Let x,y,zx,y,z be positive reals such that xyz=1xyz=1. Show that x2+y2+zx2+2+y2+z2+xy2+2+z2+x2+yz2+2≥3.\frac{x^2+y^2+z}{x^2+2} + \frac{y^2+z^2+x}{y^2+2} + \frac{z^2+x^2+y}{z^2+2} \geq 3. When does equality happen?
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