## Task 4 - 210824

Regarding the membership status of a company sports club (BSG), which consists of exactly five sections, the following statements have been made:

- Exactly 22 members of the BSG belong to the Chess section.
- Exactly one third of all BSG members belong to the Football section.
- Exactly one fifth of all BSG members belong to the Athletics section.
- Exactly three sevenths of all BSG members belong to the Table Tennis section.
- Exactly two ninths of all BSG members belong to the Gymnastics section.
- Exactly 8 members of the BSG belong to exactly three different sections.
- Exactly 72 members of the BSG belong to at least two different sections.
- No member of the BSG belongs to more than three sections, but each member belongs to at least one section.

Investigate whether there is a composition of membership numbers both for the entire BSG and for the five individual sections such that all these statements are true! Investigate whether these membership numbers are uniquely determined by the statements! If this is the case, provide the membership numbers!
I. If for a compilation of membership numbers all the statements mentioned apply, then:

If $x$ is the membership number of the BSG, then the membership numbers of the mentioned sections are

$$
22, \quad \frac{x}{3}, \quad \frac{x}{5}, \quad \frac{3}{7} x, \quad \frac{2}{9} x
$$

If these numbers are added, then each member of the BSG is counted as many times as the number of sections to which the respective member belongs. The same type of recording can be achieved as follows:

Record each of the $x$ members once, then the 72 members mentioned in the problem text a second time, and finally the previously mentioned 8 members a third time. Therefore, we have

$$
22+\frac{x}{3}+\frac{x}{5}+\frac{3}{7} x+\frac{2}{9} x=x+80
$$

By subtracting $22+x$ and multiplying by 315, we get

$$
105 x+63 x+135 x+70 x-315 x=18270 \quad \Rightarrow \quad x=315
$$

Therefore, according to (1), the mentioned statements can only be true if the membership number of the BSG is 315 and 22, 105, 63, 135, 70 are the membership numbers of the sections.

II. Conversely, if these membership numbers for the sections are achieved and it is additionally achieved that exactly 72 members belong to at least two sections and of these exactly 8 belong to exactly three sections, then the membership number of the BSG is $22+105+63+135+70-72-8=315$.

Thus, the statements that the membership numbers 105, 63, 135, 70 are the same as $\frac{1}{3}, \frac{1}{5}, \frac{3}{7}$, and $\frac{2}{9}$ of the membership number of the BSG also hold true.

From I. and II. it follows: There is a compilation of membership numbers of the BSG and the sections such that all the mentioned statements are true. It is uniquely determined by the statements and reads:

BSG: 315; Sections: 22, 105, 63, 135, 70.

Solutions of the 2nd Round 1981 adopted from [5]

### 5.23.3 3rd Round 1981, Class 8