Question 97
Suppose $a_1, \dots, a_n \gt 0$, where $n \gt 1$ and $\sum_{i=1}^n a_i = 1$. For $i = 1, 2, \dots, n$, let $b_i = a_i^2 / \sum_{j=1}^n a_j^2$. Prove that
$$
\sum_{i=1}^{n} \frac{a_i}{1 - a_i} \le \sum_{i=1}^{n} \frac{b_i}{1 - b_i}.
$$
When does equality occur?
$$
\sum_{i=1}^{n} \frac{a_i}{1 - a_i} \le \sum_{i=1}^{n} \frac{b_i}{1 - b_i}.
$$
When does equality occur?