AUSTRIAN-POLISH 2006 Problem 5

Prove that for all positive integers $n$ and all positive reals $a,b,c$ the following inequality holds: $$\frac{a^{n+1}}{a^{n}+a^{n-1}b+\ldots+b^{n}}+\frac{b^{n+1}}{b^{n}+b^{n-1}c+\ldots+c^{n}}+\frac{c^{n+1}}{c^{n}+c^{n-1}a+\ldots+a^{n}} \ge \frac{a+b+c}{n+1}$$