IMO Shortlist 2005 problem C6


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In a mathematical competition, in which 6 problems were posed to the participants, every two of these problems were solved by more than \frac 25 of the contestants. Moreover, no contestant solved all the 6 problems. Show that there are at least 2 contestants who solved exactly 5 problems each.

Radu Gologan and Dan Schwartz
Izvor: Međunarodna matematička olimpijada, shortlist 2005