RO-TST 1996 Problem 13

Let $x_1,x_2,\ldots,x_n$ be positive real numbers and $x_{n+1} = x_1 + x_2 + \cdots + x_n$ . Prove that $$\sum_{k=1}^n \sqrt { x_k (x_{n+1} - x_k)} \leq \sqrt { \sum_{k=1}^n x_{n+1}(x_{n+1}-x_k)}.$$ Mircea Becheanu