4. There are 16 students taking an exam, and all questions are multiple-choice with 4 options each. After the exam, it is found that: any two students have at most one question with the same answer. How many questions are there at most? (33rd IMO


4. Let there be $n$ questions, and we prove that $n_{\text {max }}=5$. Clearly, $n_{\text {max }}>1$. When $n>1$, for a certain question $A$, if 5 people choose the same answer, then these 5 people's answers to any other question $B$ are all different. However, question $B$ only has 4 options, leading to a contradiction. Therefore, for any question, at most 4 people can choose the same option. On the other hand, for question $A$, the answers of 16 people are distributed among the 4 options, and each option is chosen by at most 4 people, so each option is chosen by exactly 4 people. Thus, for each person $x$, for the $i(i=1,2, \cdots, n)$-th question, there are exactly 3 people who choose the same answer, forming a group of 3 people $A_{i}$. Considering $A_{1}, A_{2}, \cdots, A_{n}$ as a whole, if two 3-person groups $A_{i} 、 A_{j}(i<j)$ intersect, let $y \in A_{i} \cap A_{j}$, then $x 、 y$ have the same answer for questions $i 、 j$, leading to a contradiction. Therefore, $A_{i}$ are pairwise disjoint. Thus, the number of people $S \geqslant 1+\left|A_{1}\right|+\left|A_{2}\right|+\cdots+\left|A_{n}\right|=3 n+1$. Therefore, $3 n+1 \leqslant 16$, $n \leqslant 5$. The table below shows that $n=5$ is possible.
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline \begin{tabular}{c} 
Student \\
Question Number
\end{tabular} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\
\hline 1 & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 2 & 3 & 3 & 3 & 3 & 4 & 4 & 4 & 4 \\
\hline 2 & 1 & 2 & 3 & 4 & 1 & 2 & 3 & 4 & 1 & 2 & 3 & 4 & 1 & 2 & 3 & 4 \\
\hline 3 & 1 & 2 & 3 & 4 & 4 & 3 & 2 & 1 & 3 & 4 & 1 & 2 & 2 & 1 & 4 & 3 \\
\hline 4 & 1 & 2 & 3 & 4 & 2 & 1 & 4 & 3 & 4 & 3 & 2 & 1 & 3 & 4 & 1 & 2 \\
\hline 5 & 1 & 2 & 3 & 4 & 3 & 4 & 1 & 2 & 2 & 1 & 4 & 3 & 4 & 3 & 2 & 1 \\
\hline
\end{tabular}