Here is an improvement of the bound in part b: b. Assuming that $n\ge2$, prove that there exist two distinct numbers $j,k$ in the set $\{1,2,\ldots,2n\}$ such that \[|a_j-a_k|<\frac{1}{\sqrt{n(n-1)}}.\] Prove that this is the best bound.

a) By AM-GM, we have $a_j + a_{n+j} \ge 2 \sqrt{a_j a_{n+j}} = 2$ for all $j \in [1,n]$. Adding everything up gives $a_1 + a_2 + \ldots + a_{2n} \ge 2n$. By Pigeonhole Principle (or something for reals), either $a_1 + a_2 + \ldots + a_n \ge n$ or $a_{n+1} + a_{n+2} + \ldots + a_{2n} \ge n$, and the conclusion follows.

What about part b?