10*. A student passed 31 exams over 5 years of study. In each subsequent year, he passed more exams than in the previous year. In the fifth year, the number of exams was three times that of the first year. How many exams were there in the fourth year?

Could the student have had only one exam in the first year? Check and make a conclusion. Could he have started with two exams? Perform the check. What about three exams? Perform the calculations and make a conclusion. Could the student have had 4, 5, or more exams in the first year? Why?
10. Solutions. The student could not have even two exams in the first year, as in this case, the total number of exams over five years would be less than 31.4 exams, and he also could not have more than three exams initially. Indeed, in this case, he would have to have no less than $4+$ $+5+6+7+12=34$ exams, i.e., more than 31. The student could have only three exams in the first year and nine in the last. By checking, we find that in this case, he could have only eight exams in the fourth year. (VZMіSH)