Problem
COM-B1-M04-P015 Shared Memberships
#15
★★★☆☆ Level 3 of 5
A school has \(40\) students and \(5\) clubs. Each student attends exactly \(2\) clubs. Prove that some club has at least \(16\) students.
Count total memberships.
Total memberships: \(40\cdot2=80\). If every club had at most \(15\) students, the total would be at most \(5\cdot15=75\), contradiction. Hence some club has at least \(16\) students.
Average plus integrality.