Question 62
Let $a_1$, $a_2$, $\dots$, $a_n$ be positive real numbers whose product is $1$. Show that the sum
$$
\frac{a_1}{1+a_1} + \frac{a_2}{(1+a_1)(1+a_2)} + \frac{a_3}{(1+a_1)(1+a_2)(1+a_3)} + \dots + \frac{a_n}{(1+a_1)(1+a_2)\dots(1+a_n)}
$$
is greater than or equal to $\frac{2^n - 1}{2^n}$.
$$
\frac{a_1}{1+a_1} + \frac{a_2}{(1+a_1)(1+a_2)} + \frac{a_3}{(1+a_1)(1+a_2)(1+a_3)} + \dots + \frac{a_n}{(1+a_1)(1+a_2)\dots(1+a_n)}
$$
is greater than or equal to $\frac{2^n - 1}{2^n}$.