Question 107
Suppose $n \ge 2$, and $a_1, a_2, \dots, a_n$ are positive numbers that sum to $1$. Prove that
$$
\frac{n}{n-1} \le \sum_{i=1}^{n} \frac{a_i}{1-a_i},
$$
with equality iff $a_i = 1/n$, $i = 1, 2, \dots, n$.
$$
\frac{n}{n-1} \le \sum_{i=1}^{n} \frac{a_i}{1-a_i},
$$
with equality iff $a_i = 1/n$, $i = 1, 2, \dots, n$.