MathDB

Let

n \geq2

be an integer number and

D_n

the set of all the points

(x,y)

in the plane such that its coordinates are integer numbers with:

-n \le x \le n

and

-n \le y \le n

.
(a) There are three possible colors in which the points of

D_n

are painted with (each point has a unique color). Show that with any distribution of the colors, there are always two points of

D_n

with the same color such that the line that contains them does not go through any other point of

D_n

.
(b) Find a way to paint the points of

D_n

with 4 colors such that if a line contains exactly two points of

D_n

, then, this points have different colors.

3