Problem 3
Twenty-one girls and twenty-one boys took part in a mathematical contest. Each contestant solved at most six problems. For each girl and each boy, at least one problem was solved by both of them. Prove that there was a problem solved by at least three girls and at least three boys.
Step 5 of 5: Contradict the count
In plain words
Every cell belongs to one of the two classes.
Detailed analysis
Every labelled cell has a problem with at most two girls or at most two boys, so , a contradiction. Thus some problem has at least three girls and at least three boys.