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total number of languages used in KAUST is n

Source: 2015 Saudi Arabia IMO TST IV p2

July 24, 2020
combinatorics

Problem Statement

The total number of languages used in KAUST is nn. For each positive integer k≤nk \le n, let AkA_k be the set of all those people in KAUST who can speak at least kk languages; and let BkB_k be the set of all people PP in KAUST with the property that, for any kk pairwise different languages (used in KAUST), PP can speak at least one of these kk languages. Prove that (a) If 2k≥n+12k \ge n + 1 then Ak⊆BkA_k \subseteq B_k (b) If 2k≤n+12k \le n + 1 then Ak⊇Bk.A_k \supseteq B_k.
Nguyễn Duy Thái Sơn
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