ILL 1992 Problem 20

Let $X$ and $Y$ be two sets of points in the plane and $M$ be a set of segments connecting points from $X$ and $Y$ . Let $k$ be a natural number. Prove that the segments from $M$ can be painted using $k$ colors in such a way that for any point $x \in X \cup Y$ and two colors $\alpha$ and $\beta$ $(\alpha \neq \beta)$ , the difference between the number of $\alpha$ -colored segments and the number of $\beta$ -colored segments originating in $X$ is less than or equal to $1$ .