RO-TST 1999 Problem 4

Show that for all positive real numbers $x_1,x_2,\ldots,x_n$ with product 1, the following inequality holds $$\frac 1{n-1+x_1 } +\frac 1{n-1+x_2} + \cdots + \frac 1{n-1+x_n} \leq 1.$$