Difference between revisions of "1997 OIM Problems/Problem 3"
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Let <math>n \le 2</math> be an integer and <math>D_n</math> be the set of points <math>(x, y)</math> of the plane whose coordinates are integers with | Let <math>n \le 2</math> be an integer and <math>D_n</math> be the set of points <math>(x, y)</math> of the plane whose coordinates are integers with | ||
| − | <cmath>-n \le x \le n \text{ and } -n \le y \le n\text{.}</cmath> | + | <cmath>-n \le x \le n \text{, and } -n \le y \le n\text{.}</cmath> |
a. Three colors are available; each of the points of <math>D_n</math> is colored with one of them. Show that no matter how this coloring has been done, there are always two points of <math>D_n</math> of the same color such that the line containing them does not pass through any other point of <math>D_n</math>. | a. Three colors are available; each of the points of <math>D_n</math> is colored with one of them. Show that no matter how this coloring has been done, there are always two points of <math>D_n</math> of the same color such that the line containing them does not pass through any other point of <math>D_n</math>. | ||
Latest revision as of 14:26, 13 December 2023
Problem
Let 
be an integer and 
be the set of points 
of the plane whose coordinates are integers with
![\[-n \le x \le n \text{, and } -n \le y \le n\text{.}\]](http://latex.artofproblemsolving.com/e/7/9/e79b98a1ad1515f3f085c770b4a4e01cc643b1e9.png)
a. Three colors are available; each of the points of is colored with one of them. Show that no matter how this coloring has been done, there are always two points of of the same color such that the line containing them does not pass through any other point of .
b. Find a way to color the points of using 4 colors so that if a line contains exactly two points of , then those two points have different colors.
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
Solution
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