BY-MO 2001 Problem 4

The problem committee of a mathematical olympiad prepares some variants of the contest. Each variant contains $4$ problems, chosen from a shortlist of $n$ problems, and any two variants have at most one problem in common. (a) If $n = 14$ , determine the largest possible number of variants the problem committee can prepare. (b) Find the smallest value of n such that it is possible to prepare ten variants of the contest.