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Find the minimum

Source: 2013 China Mathematical Olympaid P4

January 13, 2013
floor functioninductioninequalitiescombinatorics proposedcombinatorics

Problem Statement

Let n⩾2n \geqslant 2 be an integer. There are nn finite sets A1,A2,…,An{A_1},{A_2},\ldots,{A_n} which satisfy the condition \left| {{A_i}\Delta {A_j}} \right| = \left| {i - j} \right|   \forall i,j \in \left\{ {1,2,...,n} \right\}. Find the minimum of ∑i=1n∣Ai∣\sum\limits_{i = 1}^n {\left| {{A_i}} \right|} .
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