MathDB

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} \equal{} \frac{1}{N} \cdot \left(n \cdot \overrightarrow{AB} \plus{} m \cdot \overrightarrow{AC} \right) with

m, n

integers,

0 \leq n \leq N,

0 \leq m \leq N

and n \plus{} 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.

Problem Statement