On a quiz, every question is solved by exactly four students, every pair of questions is solved by exactly one student, and none of the students solved all of the questions. Find the maximum possible number of questions on the quiz.
Number the students and the questions so that student $S_{1}$ solved questions $Q_{1}, \ldots, Q_{k}$ but not $Q_{k+1}, \ldots, Q_{n}$. Assume first that $k>4$. Since $Q_{k+1}$ is solved by exactly 4 students, while $Q_{i}$ and $Q_{k+1}$ are solved by exactly one student for each $1 \leq i \leq k$, there must be another student who solved two of the questions $Q_{1}, \ldots, Q_{k}$ besides $S_{1}$, a contradiction. Therefore each student solved at most four questions. We conclude that there are at most 13 questions- $Q_{1}$ plus at most 3 other questions solved by each of the 4 students who solved $Q_{1}$.

The following example shows that a quiz with 13 questions is possible.

![](https://cdn.mathpix.com/cropped/2024_04_17_55326006cf43f36c336eg-3.jpg?height=715&width=724&top_left_y=1716&top_left_x=690)