Question 41
Given an integer $n \ge 2$. Suppose $A_1, A_2, \dots, A_n$ are $n$ nonempty finite sets satisfying: $|A_i \Delta A_j| = |i-j|$ for all $i, j \in \{1, 2, \dots, n\}$.
Find the minimum value of $|A_1| + |A_2| + \dots + |A_n|$. (Here $|X|$ denotes the number of elements of a finite set $X$ and $X \Delta Y = \{a \mid a \in X, a \notin Y\} \cup \{a \mid a \in Y, a \notin X\}$ for any sets $X$ and $Y$.)
Find the minimum value of $|A_1| + |A_2| + \dots + |A_n|$. (Here $|X|$ denotes the number of elements of a finite set $X$ and $X \Delta Y = \{a \mid a \in X, a \notin Y\} \cup \{a \mid a \in Y, a \notin X\}$ for any sets $X$ and $Y$.)