There are a total of $ nk$ student-question pairs, and call such a pair "good" if the student solves the question. If every question is easy, then (counting over questions) more than $ \frac{n}{2} \cdot k$ student-question pairs are good, while if every student fails then (counting over students) fewer than $ n \cdot \frac{k}{2}$ student-question pairs are good. So this situation is impossible. When we replace strict inequalities with weak inequalities in part (b), a similar analysis shows that we must have that every question is solved by exactly half the students and every student solves exactly half the questions. This is possible whenever $ k, n$ are both even: e.g., have the first half of the students solve the first half of the questions and the second half of the students solve the second half of the questions. Question related to part (b): how many (labeled, or up to isomorphism) bipartite graphs are there with partition sizes $ k, n$ even integers such that every vertex on each side is connected to exactly half the vertices on the other?