Question 189
For a given positive integer $n \geq 2$, suppose positive integers $a_i$ ($i=1, 2, \dots, n$) satisfy $a_1 \lt a_2 \lt \dots \lt a_n$ and $\sum_{i=1}^{n} \frac{1}{a_i} \leq 1$.
Prove that, for any real number $x$, the following inequality holds,
$$
\left[ \sum_{i=1}^{n} \frac{1}{a_i^2 + x^2} \right]^2 \leq \frac{1}{2} \cdot \frac{1}{a_1 (a_1 - 1) + x^2}.
$$
Prove that, for any real number $x$, the following inequality holds,
$$
\left[ \sum_{i=1}^{n} \frac{1}{a_i^2 + x^2} \right]^2 \leq \frac{1}{2} \cdot \frac{1}{a_1 (a_1 - 1) + x^2}.
$$