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Slovenia — National Math Olympiad

Combinatorics Difficulty 6.4 National Olympiad Prove it Slovenia

A teacher invited a group of children to sit down at a round table. There were three times as many boys as there were girls. The teacher walked around the table and observed the pairs of children sitting next to each other. She noticed that the number of same-sex pairs was twice the number of boy-girl pairs. At least how many children were sitting at the table?

Solution

Let $x$ be the number of girls and $y$ the number of boys. Then $y = 3x$ and there are $4x$ children altogether.

Let $a$ be the number of boy-girl pairs. Then the number of same-sex pairs is equal to $2a$ and there are $3a$ pairs altogether. The number of pairs equals the number of all children. Indeed, every child belongs to two pairs, so there are $2 \cdot 4x$ pairs, but every pair was counted twice, which gives us $\frac{2 \cdot 4x}{2} = 4x$ distinctive pairs. Hence, $3a = 4x$ . The smallest $x$ for which this equation can hold is

$\begin{array}{cccc} & & F & D \\ & & F & F \\ F & & & D \\ F & & & D \\ F & & & F \\ F & F & F & F \end{array}$

$x = 3$ . We conclude there was at least 12 children at the table. If they sat down as shown in the figure (D represents girls, F represents boys), then the number of same-sex pairs was twice the number of boy-girl pairs.

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