5. In a spring math camp, between 50 and 70 children arrived. On Pi Day (March 14), they decided to give each other squares if they were just acquaintances, and circles if they were friends. Andrey calculated that each boy received 3 circles and 8 squares, while each girl received 2 squares and 9 circles. And Katya noticed that the total number of circles and squares given was the same. How many children arrived at the camp?

(P. D. Mulyenko)
Answer: 60.

Solution. Let the number of boys be $m$, and the number of girls be $-d$. Then $3 m + 9 d = 8 m + 2 d$ (the number of circles equals the number of squares). Transforming, we get $5 m = 7 d$, which means the number of boys and girls are in the ratio $7: 5$. Therefore, the total number of children is divisible by 12. Between 50 and 70, only the number 60 fits.

Criteria. Incorrectly understood condition (for example, that an equal number of gifts were given to boys and girls) 0 points.

Correct answer without explanation -2 points. Correct answer with verification -3 points.