1997 OIM Problems/Problem 3

Problem

Let $n \le 2$ be an integer and $D_n$ be the set of points $(x, y)$ of the plane whose coordinates are integers with

$-n \le x \le n \text{, and } -n \le y \le n\text{.}$

a. Three colors are available; each of the points of $D_n$ is colored with one of them. Show that no matter how this coloring has been done, there are always two points of $D_n$ of the same color such that the line containing them does not pass through any other point of $D_n$ .

b. Find a way to color the points of $D_n$ using 4 colors so that if a line contains exactly two points of $D_n$ , then those two points have different colors.

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

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1997 OIM Problems/Problem 3

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