$\underline{\text { Folklore }}$

Students from different cities came to the tournament. One of the organizers noticed that they could form 19 teams of 6 people each, and at the same time, less than a quarter of the teams would have a reserve player. Another suggested forming 22 teams of 5 or 6 people each, and then more than a third of the teams would consist of six players. How many students came to the tournament?
According to the first condition, the number of schoolchildren who arrived is no more than $19 \cdot 6+4=118$.

From the second condition, it follows that this number is no less than $22 \cdot 5+8=118$.

Thus, exactly 118 schoolchildren arrived at the tournament.

## Answer

118.

Author: Senderov V.A.

Natural numbers $a, b$, and $c$, where $c \geq 2$, are such that $1 / a + 1 / b = 1 / c$. Prove that at least one of the numbers $a+c, b+c$ is composite.

## Solution

It is sufficient to show that at least one of the two numbers $d_{a}=(a, c)$ and $d_{b}=(b, c)$ is greater than 1. Indeed, if, for example, $d_{a}>1$, then $a+c$ is divisible by $d_{a}$ and $a+c>d_{a}$, so $a+c$ is a composite number.

From the condition, it follows that $c(a+b)=a b$; hence, $a b$ is divisible by $c$. But then, if $d_{a}=d_{b}=1$, then $c=1$, which contradicts the condition.