Use double counting.

My solution. Assume that in each problem, there are $a_i$ persons who answer choice $i$ and there are $k$ question in this exam Consider the number of pairs of students gave the same answer $S$ Since each problem has 5 choices. So the number $S$ is $\binom{a_1}{2}+\binom{a_2}{2}+...+\binom{a_5}{2}$ Use Cauchy-Schwartz and. $a_{1}+a_{2}+a_{3}+a_{4}+a_{5}=25$ Then we get $\binom{a_1}{2}+\binom{a_2}{2}+...+\binom{a_5}{2}\geq50$ Assume that there are $k$ problems in the contest Since every two students gave same answer for not more than 1 question. So $\binom{25}{2}\geq50k$ Thus $k\le6$ and this problem was solved.