Question 166
Let $a_1, a_2, \ldots, a_n$ be $n \ge 2$ real numbers such that $0 \le a_i \le \frac{\pi}{2}$. Prove that
$$
\left( \frac{1}{n} \sum_{i=1}^{n} \frac{1}{1 + \sin a_i} \right) \left( 1 + \prod_{i=1}^{n} (\sin a_i)^{\frac{1}{n}} \right) \le 1.
$$
$$
\left( \frac{1}{n} \sum_{i=1}^{n} \frac{1}{1 + \sin a_i} \right) \left( 1 + \prod_{i=1}^{n} (\sin a_i)^{\frac{1}{n}} \right) \le 1.
$$