2013 CMO 第 4 题

Let $n \geq 2$ be an integer. There are $n$ finite sets ${A_1},{A_2},\ldots,{A_n}$ which satisfy the condition

$$\left| {{A_i}\Delta {A_j}} \right| = \left| {i - j} \right| \quad \forall i,j \in \left\{ {1,2,...,n} \right\}.$$

Find the minimum of $\sum\limits_{i = 1}^n {\left| {{A_i}} \right|}$ .

设 $n \geq 2$ 为整数。有$n$个有限集${A_1},{A_2},\ldots,{A_n}$满足条件

\左| {{A_i}\Delta {A_j}}\right| = \左| {i - j} \右| \quad \forall i,j \in \left\{ {1,2,...,n} \right\}.

求 $\sum\limits_{i = 1}^n {\left| 的最小值{{A_i}} \right|}$ .