NO-ABEL 2010 Problem 3

$a)$ There are $25$ participants in a mathematics contest having four problems. Each problem is considered solved or not solved (that is, partial solutions are not possible). Show that either there are four contestants having solved the same problems (or not having solved any of them), or two contestants, one of which has solved exactly the problems that the other did not solve.

$b)$ There are $k$ sport clubs for the students of a secondary school. The school has $100$ students, and for any selection of three of them, there exists a club having at least one of them, but not all, as a member. What is the least possible value of $k$ ?