1. A student passed 31 exams over 5 years of study. Each subsequent year, he passed more exams than 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?
# Answer: 8.

Solution: Let $a, b, c, d, e$ be the number of exams taken in each year of study. According to the problem, $a+b+c+d+e=31, a<b<c<d<e$.

Replace the numbers $b, c, d, e$ in the equation with definitely not larger values:

$a+(a+1)+(a+2)+(a+3)+3a \leq 31$. We get $7a \leq 25$. Replacing these numbers with definitely not smaller values, we get $a+(3a-3)+(3a-2)+(3a-1)+3a \geq 31$. We get $13a \geq 37$. Therefore, $a=3$. In this case, $e=9, b+c+d=19$.

If $d<8$, then this sum is no more than $5+6+7=18$. The numbers 5, 6, 8 fit. Evaluation criteria. The answer alone is not scored. An answer with an example of all five numbers - 2 points.