Problem
COM-B1-M12-P024 Set 6. Six People
#24
★★★★★ Level 5 of 5
Prove that among any \(6\) people, there are either \(3\) mutual acquaintances or \(3\) mutual strangers.
Choose one person and split the others into two groups.
Choose a person \(A\). Among the other \(5\), there are either \(3\) who know \(A\), or \(3\) who do not know \(A\). Suppose \(A\) knows \(B,C,D\). If among \(B,C,D\) there is an acquaintance pair, together with \(A\) we get three mutual acquaintances; if not, then \(B,C,D\) are mutual strangers. The case of three people not acquainted with \(A\) is analogous.
Final strong problem of the first level.