KöMaL 2011 A548

Prove that $\prod_{i=1}^n \left(1+\frac1{x_1+\ldots+x_i}\right) + \prod_{i=1}^n \left(1-\frac1{x_i+\ldots+x_n}\right) \le n+1$ holds for arbitrary real numbers $x_1,\ldots,x_n\ge 1$.