CGMO 2010 Problem 1

Let $n$ be an integer greater than two, and let $A_1,A_2, \cdots , A_{2n}$ be pairwise distinct subsets of $\{1, 2, ,n\}$ . Determine the maximum value of $$\sum_{i=1}^{2n} \dfrac{|A_i \cap A_{i+1}|}{|A_i| \cdot |A_{i+1}|}$$ Where $A_{2n+1}=A_1$ and $|X|$ denote the number of elements in $X.$