In an exam 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 in this exam.
1. **Step 1: Prove that a student can solve at most 4 questions.**

   Suppose a student \( s \) solved 5 questions, denoted by \( q_1, q_2, q_3, q_4, q_5 \). According to the problem, each question is solved by exactly 4 students. Therefore, there are 3 other students who solve each of \( q_1, q_2, q_3, q_4, q_5 \) besides \( s \).

   Since \( s \) did not solve all the questions, there exists at least one question \( q_6 \) that \( s \) did not solve. Consider the pairs \( (q_6, q_i) \) for \( i = 1, 2, 3, 4, 5 \). Each pair must be solved by exactly one student. However, \( q_6 \) is solved by exactly 4 students, and there are 5 pairs involving \( q_6 \). This implies that at least one pair \( (q_6, q_i) \) is not solved by any student, which is a contradiction. Therefore, a student can solve at most 4 questions.

2. **Step 2: Prove that there are at most 13 questions.**

   Suppose there are \( n \) questions. Let \( q \) be one of these questions. It is solved by exactly 4 students, denoted by \( s_1, s_2, s_3, s_4 \). Note that \( q \) is the only common question solved by \( s_1, s_2, s_3, s_4 \), otherwise there would be a pair of questions solved by more than one student, which contradicts the problem's conditions.

   If \( p \) is a question other than \( q \), then \( p \) must be solved by at least one of \( s_1, s_2, s_3, s_4 \), otherwise no student would solve the pair of questions \( (q, p) \). Since each \( s_i \) can solve at most 4 questions, each \( s_i \) can solve at most 3 more questions other than \( q \).

   Therefore, the maximum number of questions is:
   \[
   1 + 3 \times 4 = 13
   \]

3. **Step 3: Verify the existence of a configuration with 13 questions.**

   The example for 13 questions can be found in the structure of a finite projective plane of order 3. In such a plane, there are 13 points (questions) and each point is incident with exactly 4 lines (students), and each pair of points is incident with exactly one line. This configuration satisfies all the given conditions.

The final answer is \(\boxed{13}\).