International Mathematical Olympiad 2012 — Official contest Question 2

Let $a_2$, $a_3$, …, $a_n$ be positive reals with product $1$, where $n \ge 3$. Show that $$
(1+a_2)^2 (1+a_3)^3 \dots (1+a_n)^n \gt n^n.
$$