## Task 1 - 090911

At the award ceremony of a district olympiad, the following was announced:

Exactly one ninth of all participants in this district olympiad won a prize. Exactly one tenth of all participants in the district olympiad are members of the district club Young Mathematicians. Of the prize winners, exactly 75 percent are from the district club. Exactly 6 of the students who participated in the district olympiad and are members of the district club did not receive a prize.

Determine the total number of participants in this district olympiad!
Let $T$ be the number of all participants in the district olympiad. According to the problem, $\frac{1}{9}$ of the participants are prize winners, and $75\%$, or $\frac{3}{4}$, of the prize winners are club members.

Thus, $\frac{1}{9} \cdot \frac{3}{4} \cdot T = \frac{1}{12} \cdot T$ of the club members receive a prize, meaning that one twelfth of the participants are both prize winners and club members.

The number of club members who receive a prize $\left(\frac{1}{12} \cdot T\right)$ and those who do not receive a prize (6) is equal to $\frac{1}{10}$ of the participants. Therefore,

$$
\frac{1}{12} \cdot T + 6 = \frac{1}{10} \cdot T \quad \text{and thus} \quad T = 360
$$

The number of participants in the district olympiad was 360.