NMC 1999 Problem 4

Let $a_{1}, a_{2}, \ldots, a_{n}$ be positive real numbers and $n \geq 1$. Show that

$$\begin{aligned} & n\left(\frac{1}{a_{1}}+\cdots+\frac{1}{a_{n}}\right) \\ & \geq\left(\frac{1}{1+a_{1}}+\cdots+\frac{1}{1+a_{n}}\right)\left(n+\frac{1}{a_{1}}+\cdots+\frac{1}{a_{n}}\right) \end{aligned}$$

When does equality hold?