RO-NMO 2003 Problem 92

An integer $n$ , $n\ge2$ is called friendly if there exists a family $A_1,A_2,\ldots,A_n$ of subsets of the set $\{1,2,\ldots,n\}$ such that: (1) $i\not\in A_i$ for every $i=\overline{1,n}$ ; (2) $i\in A_j$ if and only if $j\not\in A_i$ , for every distinct $i,j\in\{1,2,\ldots,n\}$ ; (3) $A_i\cap A_j$ is non-empty, for every $i,j\in\{1,2,\ldots,n\}$ . Prove that: (a) 7 is a friendly number; (b) $n$ is friendly if and only if $n\ge7$ .

Valentin Vornicu