Question 80
Let $a_1, a_2, \dots, a_n$ be positive real numbers with $a_1 + a_2 + \dots + a_n = 1$ ($n \ge 2$). Prove that
$$
\sum_{k=2}^{n} \frac{a_k}{1 - a_k} (a_1 + a_2 + \dots + a_{k-1})^2 \lt \frac{1}{3}.
$$
$$
\sum_{k=2}^{n} \frac{a_k}{1 - a_k} (a_1 + a_2 + \dots + a_{k-1})^2 \lt \frac{1}{3}.
$$