RIOPLATENSE 1998 Problem 6

Let $k$ be a fixed positive integer. For each $n = 1, 2,...,$ we will call configuration of order $n$ any set of $kn$ points of the plane, which does not contain $3$ collinear, colored with $k$ given colors, so that there are $n$ points of each color. Determine all positive integers $n$ with the following property: in each configuration of order $n$ , it is possible to select three points of each color, such that the $k$ triangles with vertices of the same color that are determined are disjoint in pairs.