CMO 2014 Problem 1

Let $a_{1}, a_{2}, \ldots, a_{n}$ be positive real numbers whose product is 1 . Show that the sum $\frac{a_{1}}{1+a_{1}}+\frac{a_{2}}{\left(1+a_{1}\right)\left(1+a_{2}\right)}+\frac{a_{3}}{\left(1+a_{1}\right)\left(1+a_{2}\right)\left(1+a_{3}\right)}+\cdots+\frac{a_{n}}{\left(1+a_{1}\right)\left(1+a_{2}\right) \cdots\left(1+a_{n}\right)}$ is greater than or equal to $\frac{2^{n}-1}{2^{n}}$.