ILL 1989 Problem 21

Let $ABC$ be an equilateral triangle with side length equal to $N \in \mathbb{N}.$ Consider the set $S$ of all points $M$ inside the triangle $ABC$ satisfying

$$\overrightarrow{AM} = \frac{1}{N} \cdot \left(n \cdot \overrightarrow{AB} + m \cdot \overrightarrow{AC} \right)$$

with $m, n$ integers, $0 \leq n \leq N,$ $0 \leq m \leq N$ and $n + m \leq N.$

Every point of S is colored in one of the three colors blue, white, red such that

**(i)**no point of $S \cap [AB]$ is coloured blue (ii) no point of $S \cap [AC]$ is coloured white (iii) no point of $S \cap [BC]$ is coloured red

Prove that there exists an equilateral triangle the following properties:

(1) the three vertices of the triangle are points of $S$ and coloured blue, white and red, respectively. (2) the length of the sides of the triangle is equal to 1.

Variant: Same problem but with a regular tetrahedron and four different colors used.