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For an integer $n \geq 2$, let $D_n$ be the set of points $(x, y)$ of the plane with integer coordinates such that $-n \leq x, y \leq n$.

(a) Each of the points of $D_n$ is colored with one of three given colors. Prove that there always exist two points of $D_n$ of the same color such that the line passing through them contains no other point of $D_n$.

(b) Give an example of a coloring of points of $D_n$ with four colors in such a manner that if a line contains exactly two points of $D_n$, then these two points have different colors.
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