Task 2 - 190822 In a mathematics AG, a member of the patron brigade presented the participants with the following problem:

"Our brigade has more than 20 but fewer than 35 members. Of them, three times as many took their annual leave in July last year as in May, and twice as many in February as in May. In January, three fewer people took leave than in July, and in August, one more person than in May. In the months not mentioned this year, no member of our brigade took leave. Among the mentioned leave-takers, each member of our brigade is represented exactly once.

Determine whether you can determine the number of our brigade members from these details alone!"
The number of members of this brigade who took vacation in May is $x$. Then, in July $3x$, in February $2x$, in January $3x-3$, and in August $x+1$ brigade members took vacation. Together, these are $(10x-2)$ people.

Now, $20<10x-2<35$, so $22<10x<37$.

Since $x$ is a natural number, it follows that $x=3$. Therefore, the brigade had 28 members.