Školjka

%V0 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 $\left(x, y, z\right)$ such that: (i) the numbers $x$, $y$, $z$ are of the same color, and (ii) the number $x + y + z$ is divisible by $n$. Author: Gerhard Wöginger, Netherlands