ÖMO 2011 Problem 5

Let $k$ and $n$ be positive integers. Show that if $x_j$ ( $1\leqslant j\leqslant n$ ) are real numbers with $\sum_{j=1}^n\frac{1}{x_j^{2^k}+k}=\frac{1}{k}$ , then $$\sum_{j=1}^n\frac{1}{x_j^{2^{k+1}}+k+2}\leqslant\frac{1}{k+1}\text{.}$$