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n persons, each of them speaking at most 3 languages

Source: 2011 Romania JBMO TST 1.5

June 1, 2020
combinatorics

Problem Statement

Consider nn persons, each of them speaking at most 33 languages. From any 33 persons there are at least two which speak a common language. i) For n≤8n \le 8, exhibit an example in which no language is spoken by more than two persons. ii) For n≥9n \ge 9, prove that there exists a language which is spoken by at least three persons
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