BALTIC-WAY 2007 Problem 17

Let $x,y,z$ be positive integers such that $\frac{x+1}{y}+\frac{y+1}{z}+\frac{z+1}{x}$ is an integer. Let $d$ be the greatest common divisor of $x,y$ and $z$ . Prove that $d\le \sqrt[3]{xy+yz+zx}$ .