## 273. Math Puzzler $2 / 88$

Of the participants from a school in the 1st stage of the Mathematics Olympiad, exactly $\frac{3}{40}$ were delegated to the 2nd stage. Of these students, exactly $\frac{2}{9}$ received prizes or certificates of recognition at the 2nd stage.

One student received a first prize in their grade, exactly one other student received a second prize in their grade; exactly two others received third prizes. In addition, exactly four other students were awarded certificates of recognition for particularly good solutions to a problem.

How many students from this school participated in the 1st stage of the Mathematics Olympiad?
8 students received prizes, which are $\frac{2}{9}$ of the students who reached the 2nd stage of the Math Olympiad. This leads to: $\frac{2}{9}: 8=\frac{1}{x} ; x=36$

36 students reached the 2nd stage, which are $\frac{3}{40}$ of all participants.

This leads to: $\frac{3}{40}: 36=\frac{1}{x} ; x=480$. 480 students participated in total.