2007 IMO Shortlist C3

Find all positive integers $n$, for which the numbers in the set $S=\{1,2, \ldots, n\}$ can be colored red and blue, with the following condition being satisfied: the set $S \times S \times S$ contains exactly 2007 ordered triples $(x, y, z)$ such that (i) $x, y, z$ are of the same color and (ii) $x+y+z$ is divisible by $n$. (Netherlands) Answer. $n=69$ and $n=84$.

找到所有正整数 $n$，其中集合 $S=\{1,2, \ldots, n\}$ 中的数字可以被着色为红色和蓝色，且满足以下条件：集合 $S \times S \times S$ 恰好包含 2007 个有序三元组 $(x, y, z)$，使得 (i) $x, y, z$ 颜色相同，并且 (ii) $x+y+z$ 可以整除 $n$. (荷兰)回答。 $n=69$ and $n=84$.