The condition is essentially that there always exist students who can "collaborate" together to solve all problems. Since the condition is quite strong, we proceed by contradiction: assume none of the pairs of students can collaborate to solve all problems. This implies for each of the pairs, at most of the questions can be collaboratively solved.

Now we double count the number of such collaborations by questions. For question define as the number of participants that solved it, implying students didn't solve it. Thus, the number of pairs of students that can collaboratively solve the question is Thus
Since we bound the summation from below: But simplifies to which is false, hence contradiction!