BALKAN-SL 2011 A3

Let $n$ be an integer number greater than $2$, let $x_{1},x_{2},\ldots ,x_{n}$ be $n$ positive real numbers such that $$\sum_{i=1}^{n}\frac{1}{x_{i}+1}=1$$

and let $k$ be a real number greater than $1$. Show that: $$\sum_{i=1}^{n}\frac{1}{x_{i}^{k}+1}\ge\frac{n}{(n-1)^{k}+1}$$

and determine the cases of equality.