Problem

Source: 2013 China Mathematical Olympaid P4

Tags: floor function, induction, inequalities, combinatorics proposed, combinatorics



Let $n \geqslant 2$ 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|} $.