5. The teacher invited a group of children to sit around a round table. There were three times as many boys as girls. She walked around the table and observed pairs of children sitting next to each other. She found that the number of pairs where both children were of the same gender was twice the number of pairs where the children were of different genders. What is the minimum number of children sitting around the table?

## 53rd Mathematical Competition for High School Students in Slovenia

Selection Competition, April 1, 2009

## Problems for 4th Year
III/5. 1. way. Let $x$ be the number of girls and $y$ be the number of boys. Then $y=3 x$ and therefore there are $4 x$ children in total.

Let $a$ be the number of pairs where the children are of different genders. Then the number of pairs where the children are of the same gender is $2 a$. Thus, there are $3 a$ pairs in total. The number of pairs is equal to the number of all children, since each child is in two pairs, which gives $2 \cdot 4 x$ pairs, but we have counted each pair twice, so the number of different pairs is $\frac{2 \cdot 4 x}{2}=4 x$. Therefore,

|  | B | G | B |  |
| :---: | :---: | :---: | :---: | :---: |
| B |  |  | G |  |
| B |  |  |  | G |
| B |  |  |  | B |
|  | $B$ |  | $B$ | $B$ |

$3 a=4 x$. The smallest number $x$ for which this equation can hold is $x=3$. Therefore, at least 12 children are seated around the table. If they sit as shown in the figure (with G representing girls and B representing boys), the number of pairs where the children are of the same gender is twice the number of pairs where the children are of different genders.
Observation that the total number of children is four times the number of girls (which is $4 x$) ..... 1 point
The number of all pairs is three times the number of pairs of children of different genders ..... 1 point
The number of all pairs is equal to the number of children ..... 1 point
Therefore, $3 a=4 x$ ..... 1 point
The smallest number $x$ is $x=3$ ..... 1 point
Writing an example with 12 children ..... 2 points
2. way. If only one girl is seated at the round table, then three boys are also seated.

$\mathrm{B}$ Then there are two pairs where the children are of different genders and two pairs where

$\mathrm{B}$

$\mathrm{G}$ the children are of the same gender. Since this does not satisfy the conditions of the problem, at least two girls must be seated at the table.

Let's count the number of pairs if two girls are seated at the table. If they sit together, there are only 2 pairs where a boy and a girl are together, and the remaining pairs are 6. Suppose the girls do not sit together. Consider the cases based on the smallest group of boys sitting together. Since there are 6 boys in total, the smallest group can have at most 3 boys. In all three cases, the number of pairs where the children are of the same gender is equal to the number of pairs where the children are of different genders.

None of the cases considered meet the conditions of the problem,

$$
\begin{aligned}
& \mathrm{B}_{\mathrm{G}}^{\mathrm{B}} \quad \mathrm{B}^{\mathrm{G}} \\
& \begin{array}{llll}
\mathrm{B} & \mathrm{G} & \mathrm{B} & \mathrm{G}
\end{array} \\
& \mathrm{B}_{\mathrm{B}} \mathrm{B} \quad \mathrm{B}_{\mathrm{B}} \mathrm{B} \\
& \mathrm{G}^{\mathrm{B}} \quad \mathrm{B} \quad \mathrm{B} \\
& \begin{array}{llll}
\mathrm{B} & \mathrm{G} & \mathrm{G} & \mathrm{G}
\end{array} \\
& B_{B} B_{B} \quad B_{B}
\end{aligned}
$$

Therefore, at least 3 girls are seated at the table. It is not difficult to find a suitable arrangement, for example, as shown in the last figure.

$$
\begin{array}{cccc} 
& \mathrm{B} & \mathrm{B} \\
\mathrm{B} & & & \mathrm{G} \\
\mathrm{B} & & & \mathrm{G} \\
& \mathrm{B} & \mathrm{B} & \mathrm{B}
\end{array}
$$

![](https://cdn.mathpix.com/cropped/2024_06_07_2dd3bccc19bc96d6d7dfg-16.jpg?height=54&width=1636&top_left_y=1024&top_left_x=216)

![](https://cdn.mathpix.com/cropped/2024_06_07_2dd3bccc19bc96d6d7dfg-16.jpg?height=57&width=1633&top_left_y=1071&top_left_x=220)

Examples where two girls are seated at the table and do not sit together............................. 2 points

Observation that at least 3 girls are seated at the table...................................................................................................

![](https://cdn.mathpix.com/cropped/2024_06_07_2dd3bccc19bc96d6d7dfg-16.jpg?height=51&width=1631&top_left_y=1225&top_left_x=218)