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in a school n students take part in m clubs

Source: 2016 Saudi Arabia GMO TST level 4+, II p3

August 1, 2020
combinatorics

Problem Statement

In a school, there are totally nn students, with n≥2n \ge 2. The students take part in mm clubs and in each club, there are at least 22 members (a student may take part in more than 11 club). Eventually, the Principal notices that: If 22 clubs share at least 22 common members then they have different numbers of members. Prove that m≤(n−1)2m \le (n - 1)^2
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