Competition problem — Combinatorics

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Combinatorics Difficulty 5.7 AIME, harder Prove it China

$3n$ ( $n$ is a positive integer) girl students took part in a summer camp. There were three girl students to be on duty every day. When the summer camp ended, it was found that any two of the $3n$ girl students had just one time to be on duty on the same day.

(1) When $n = 3$ , is there any arrangement satisfying the requirement above? Prove your conclusion.

Solution

(1) When $n = 3$ , there is an arrangement satisfying the requirement. Now the concrete arrangement is given as follows (Denote the nine girl students by $1, 2, \ldots, 9$ ):

$(1, 2, 3), (1, 4, 5), (1, 6, 7), (1, 8, 9), \\ (2, 4, 6), (2, 7, 8), (2, 5, 9), (3, 4, 8), \\ (3, 5, 7), (3, 6, 9), (4, 7, 9), (5, 6, 8).$

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