Question 32
Let $a_{1}, a_{2}, \ldots, a_{2 n}$ be positive real numbers such that $a_{i}+a_{n+i}=1$, for all $i=1, \cdots, n$. Prove that there exist two different integers $1 \leq j, k \leq 2 n$ for which
$$
\sqrt{a_{j}^{2}-a_{k}^{2}}\lt \frac{1}{\sqrt{n}+\sqrt{n-1}}
$$
$$
\sqrt{a_{j}^{2}-a_{k}^{2}}\lt \frac{1}{\sqrt{n}+\sqrt{n-1}}
$$