IMO 2012 Problem 2

Let $n\ge 3$ be an integer, and let $a_2,a_3,\ldots ,a_n$ be positive real numbers such that $a_{2}a_{3}\cdots a_{n}=1$ . Prove that $$(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.$$

Proposed by Angelo Di Pasquale, Australia