n-variable inequality

jhsinutopia

$a,b,c\in\mathbb{R}^+,n\in\mathbb{N}$
Prove that
$\sum_{cyc}\frac{a}{b+c}\le \left(\sum_{cyc}\frac{a^n}{b^n+b^{n-1}c}\right) \left( \sum_{cyc}\frac{a}{c+a}\right)^{n-1}$