1. In a class, there are boys and girls, such that the number of boys divides the number of girls. Florin distributes 1000 candies equally among the students in the class. Knowing that there are at least 10 boys in the class, and if 10 more students were to join, Florin could, again, distribute the 1000 candies equally among the students, find out how many candies each student in the class receives.

Romeo Ilie
## Solution.

Let $b$ be the number of boys, $f$ be the number of girls, and $n=b+f$ be the number of students in the class. According to the problem, $n \mid 1000$. (1p)

From $b \mid f$ and $b \geq 10$, it follows that $n=b+f \geq 2b \geq 20$. (1p)

Therefore, $n \in \{20, 25, 40, 50, 100, 125, 200, 250, 500, 1000\}$. (1p)

From $(n+10) \mid 1000$, it follows that $n=40$. (2p)

Then each student receives $1000 \div 40 = 25$ candies. (1p)

It remains to show that the situation indicated by the problem is possible. Thus, from $b \mid f$ and $b+f=n=40$, we deduce $b \mid 40$. Since $b \geq 10$, we obtain two possible variants: a) $b=10$ and $f=30$; b) $b=20$ and $f=20$. (1p)