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 3 of 5: Bound labels with few girls
In plain words
Each boy solves at most six problems.
Detailed analysis
Fix a boy. At least one of his at most six problems is solved by at least three girls, since . Thus at most five of his problems have at most two girl solvers, contributing at most cells. Hence .