RO-NMO 2005 Problem 94

Let $x_1,x_2,\ldots,x_n$ be positive reals. Prove that $$\frac 1{1+x_1} + \frac 1{1+x_1+x_2} + \cdots + \frac 1{1+x_1+\cdots + x_n} < \sqrt { \frac 1{x_1} + \frac 1{x_2} + \cdots + \frac 1{x_n}} .$$

Bogdan Enescu