Question 130
Let $n$ be an integer greater than two, and let $A_1, A_2, \dots, A_{2n}$ be pairwise distinct nonempty subsets of $\{1, 2, \dots, n\}$.
Determine the maximum value of $\sum_{i=1}^{2n} \frac{|A_i \cap A_{i+1}|}{|A_i| \cdot |A_{i+1}|}$.
(Here, we set $A_{2n+1} = A_1$. For a set $X$, let $|X|$ denote the number of elements in $X$.)
Determine the maximum value of $\sum_{i=1}^{2n} \frac{|A_i \cap A_{i+1}|}{|A_i| \cdot |A_{i+1}|}$.
(Here, we set $A_{2n+1} = A_1$. For a set $X$, let $|X|$ denote the number of elements in $X$.)