MathDB
Friendly numbers

Source: RMO 2003, Grade 9, Problem 2

October 23, 2008
modular arithmetic

Problem Statement

An integer n n, n≥2 n\ge2 is called friendly if there exists a family A1,A2,…,An A_1,A_2,\ldots,A_n of subsets of the set {1,2,…,n} \{1,2,\ldots,n\} such that: (1) i∉Ai i\not\in A_i for every i\equal{}\overline{1,n}; (2) i∈Aj i\in A_j if and only if j∉Ai j\not\in A_i, for every distinct i,j∈{1,2,…,n} i,j\in\{1,2,\ldots,n\}; (3) Ai∩Aj A_i\cap A_j is non-empty, for every i,j∈{1,2,…,n} i,j\in\{1,2,\ldots,n\}. Prove that: (a) 7 is a friendly number; (b) n n is friendly if and only if n≥7 n\ge7. Valentin Vornicu
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