Problem
COM-B2-M01-P012 Many Solved Problems
#12
★★★☆☆ Level 3 of 5
At an olympiad, there are \(30\) participants and \(6\) problems. Each participant solved at least \(3\) problems. Prove that some problem was solved by at least \(15\) participants.
Count pairs \((participant, problem)\).
The total number of solved-problem incidences is at least \(30\cdot3=90\). If each problem were solved by at most \(14\) participants, the total would be at most \(6\cdot14=84\), a contradiction. Hence some problem was solved by at least \(15\) participants.
A simple but useful transition to olympiad-style settings.