2012 IMO Shortlist A3

Let $a_{2}, \ldots, a_{n}$ be $n-1$ positive real numbers, where $n \geq 3$, such that $a_{2} a_{3} \cdots a_{n}=1$. Prove that $$\left(1+a_{2}\right)^{2}\left(1+a_{3}\right)^{3} \cdots\left(1+a_{n}\right)^{n}>n^{n} .$$

令 $a_{2}, \ldots, a_{n}$ 为 $n-1$ 正实数，其中 $n \geq 3$，使得 $a_{2} a_{3} \cdots a_{n}=1$。证明 $$\left(1+a_{2}\right)^{2}\left(1+a_{3}\right)^{3} \cdots\left(1+a_{n}\right)^{n}>n^{n} 。$$