Question 137
Let $a_{1}, \cdots, a_{n}$ ($n \in \mathbf{N}$) be positive real numbers such that $a_{1}+\cdots+a_{n}\lt 1$. Prove the inequality
$$
\frac{a_{1}+a_{2}+\cdots+a_{n}}{a_{1} a_{2} \cdots a_{n}} \cdot \frac{\left(1-a_{1}\right) \cdots\left(1-a_{n}\right)}{1-a_{1}-\cdots-a_{n}} \geq n^{n+1}
$$
$$
\frac{a_{1}+a_{2}+\cdots+a_{n}}{a_{1} a_{2} \cdots a_{n}} \cdot \frac{\left(1-a_{1}\right) \cdots\left(1-a_{n}\right)}{1-a_{1}-\cdots-a_{n}} \geq n^{n+1}
$$