MathNet 2012 Question 6

Let $n\ge3$ be an integer and let $a_2,a_3,\ldots,a_n$ be positive real numbers with $a_2a_3\cdots a_n=1$. Prove that
$$(1+a_2)^2(1+a_3)^3\cdots(1+a_n)^n\gt n^n.$$