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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.
in Mathematics by Platinum (138,598 points) | 28 views

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