KöMaL 2010 B4235

The sides of a convex quadrilateral $ABCD$ are divided into $n\ge 2$ equal parts. On the sides $AB$ , $BC$ , $CD$ , $DA$ , let $A_k$, $B_k$, $C_k$, $D_k$ denote the $k$ th points of division, counting from the vertices $A$ , $B$ , $C$ , $D$ , respectively. For which pairs $(n,k)$ is it true that the quadrilateral $ABCD$ is a parallelogram if and only if the quadrilateral $A_kB_kC_kD_k$ is also a parallelogram?