2017 JBMO Problems/Problem 2

Problem

Let $x,y,z$ be positive integers such that $x\neq y\neq z \neq x$ .Prove that $(x+y+z)(xy+yz+zx-2)\geq 9xyz.$ When does the equality hold?

Solution

Since the equation is symmetric and $x,y,z$ are distinct integers WLOG we can assume that $x\geq y+1\geq z+2$ . $\begin{align*} x+y+z\geq 3(z+1)\\ xy+yz+xz-2 = y(x+z)+xy-2 \geq (z+1)(2z+z)+z(z+2)-2 \\ xy+yz+xz-2 \geq 3z(z+2) \end{align*}$ Hence $(x+y+z)(xy+yz+xz-2)\geq 9(z)(z+1)(z+2)$