Algebraic inequality involving n positive numbers with a unity product.

Question

I came across this inequality on another website but there was no solution, so I am posting it here. Let $x_1,x_2,\dots,x_n$ (with $n\ge 3$) be positive numbers such that $\prod_i x_i = 1$. Prove that $$\left(1 + \sum x_i\right)\sum \frac{1}{1+nx_i}\ge\sum x_i.$$ I tried playing around with the individual terms on the l.h.s. to no avail, e.g. the inequality is equivalent to $\sum\frac{1+nx_i}{1+nx_j}\ge n\sum x_i$ but I don't see a continuation from here. Note that the original inequality is stronger than $\sum\frac{1}{1+nx_i}\ge \frac{n}{1+n}$, which in itself is not obvious. Any ideas?