[Inequality problems. Case analysis]

A student passed 31 exams over 5 years of study. In each subsequent year, he passed more exams than in the previous year, and in the fifth year, he passed three times as many exams as in the first year. How many exams did he pass in the fourth year?

#
Try to determine the number of exams taken in the first (and, respectively, fifth) year.

## Solution

It is easy to determine that three exams were taken in the first year and, accordingly, nine in the fifth year. There are two possible distributions of the number of exams in the other years: $4+7+8$ and $5+6+8$. Thus, the student took 8 exams in the fourth year; in the third year, 6 or 7 exams.

## Answer

8 exams.