Understanding a proof from the APMO 1998 on inequalities. I was having trouble with proving the following inequality.The question was from the book Secrets to Inequalities by Pham Kim Hung.
$\frac{x}{y} + \frac{y}{z} + \frac{z}{x} \geq \frac{x+y+z}{\sqrt[3]{xyz} }$
I know that the AM-GM inequality must be applied but I am having a hard time proving the the above inequality. In the book the author was able to show that by applying the trick shown below and then applying AM-GM we can prove the inequality holds. 
$3\left (\frac{x}{y}+\frac{y}{z}+\frac{z}{x}  \right ) = \left ( \frac{2x}{y} +\frac{y}{z}\right ) + \left ( \frac{2y}{z}+\frac{z}{x} \right ) + \left ( \frac{2z}{x}+\frac{x}{y} \right ) \geq \frac{3x}{\sqrt[3]{xyz}} + \frac{3y}{\sqrt[3]{xyz}} + \frac{3z}{\sqrt[3]{xyz}}$
The problem I had was after applying the AM-GM to the LHS I could not prove the RHS.

It is by AM-GM: $$\frac{x}{y}+\frac{x}{y}+\frac{y}{z}\geq 3\sqrt[3]{\frac{x^2y}{y^2z}}=3\sqrt[3]{\frac{x^3}{xyz}}$$
