GULF 2013 Problem 1

Let $a_1,a_2,\ldots,a_{2n}$ be positive real numbers such that $a_ja_{n+j}=1$ for the values $j=1,2,\ldots,n$ .

[list] a. Prove that either the average of the numbers $a_1,a_2,\ldots,a_n$ is at least 1 or the average of the numbers $a_{n+1},a_{n+2},\ldots,a_{2n}$ is at least 1.

b. Assuming that $n\ge2$ , prove that there exist two distinct numbers $j,k$ in the set $\{1,2,\ldots,2n\}$ such that $$|a_j-a_k|<\frac{1}{n-1}.$$ [/list]