The Math Contest?

Question

Twenty-one girls and twenty-one boys entered a math contest.

Each contestand aolved at most six problems.
For each pair of a girl and a boy, there was at least one problem that was solved by both the giral and the boy.

Prove that there was a problem that was solved by at least three girls and at least three boys

Answer 1

Essentially, each boy must share a solved problem with each girl. For every boy, that's 20 solved problems shared... but he only solved 6. This means that each solved problem must average being shared with 3.3333 girls. Since the average is over 3, that means there must be AT LEAST one problem where the number of shares is >3 (in fact, there must be more, but we only need 3).

Answer 2

theres 6 problems and everyone solved alteast one of them. if you devide 21 by 6 you get a number that is more than 3 so there is no way that all of the problems were solved by less than 3 pairs. the fact that there are 21 boys and 21 girls reely doesnt mater in this problem. just look at each boy girl pair as 1 unit that answered 1 problem. then its pretty simple math from there. 21/3 is 3 and 3/7 which is more than 3 so since the average is more than 3 atleast one has to be over 3.