Problem 9.2. In four classes of a school, there are more than 70 children, all of whom came to the grade meeting (no other children were present at the meeting).

Each girl who came was asked: "How many people from your class, including you, came to the meeting?"

Each boy who came was asked: "How many boys from your class, including you, came to the meeting?"

Among the answers, the numbers $7,9,10,12,15,16,19$ and 21 were encountered (all children answered correctly).

(a) (1 point) How many children are there in the largest class of the grade?

(b) (3 points) How many girls came to the grade meeting?
# Answer:

(a) (1 point) 21 students.

(b) (3 points) 33 girls.

Solution. (a) Since all 8 numbers listed in the condition are distinct, exactly 4 of them represent the number of children in the classes, and the other 4 represent the number of boys in the classes. The number 21, being the largest of the listed numbers, cannot be the number of boys in any class (otherwise, there would be no number corresponding to the total number of students in that class). Therefore, the largest class has 21 students.
(b) It follows that the number of students in the second largest class does not exceed 19, in the third largest does not exceed 16, and in the fourth largest does not exceed 15. Then the total number of students does not exceed $15+16+19+21=71$, and according to the problem, more than 70 students attended the grade meeting. Therefore, the class sizes are exactly $15, 16, 19, 21$, and the total number of students is 71.

Then the total number of boys in the classes is $7+9+10+12=38$, and the total number of girls is $71-38=33$.