2013-02-21, 08:15 PM
Assume that no problem was solved by more than two girls and more than two boys.
As given by the problem, for any one boy, there are 21 girl-boy pairings with him and another girl. Each pairing has a problem that is solved by both him and the girl.
If no problem was solved by more than two girls, then there must be at least 11 unique problems among these pairings. This means the boy has solved at least 11 problems which contradicts the fact that each contestant solved at most 6 problems. In fact, there must be a problem that is solved by at least four girls.
Same argument goes for the reversed gender.
Therefore there must be a problem that was solved by at least four girls and at least four boys.
As given by the problem, for any one boy, there are 21 girl-boy pairings with him and another girl. Each pairing has a problem that is solved by both him and the girl.
If no problem was solved by more than two girls, then there must be at least 11 unique problems among these pairings. This means the boy has solved at least 11 problems which contradicts the fact that each contestant solved at most 6 problems. In fact, there must be a problem that is solved by at least four girls.
Same argument goes for the reversed gender.
Therefore there must be a problem that was solved by at least four girls and at least four boys.