Let ABC be an equilateral triangle with side length equal to N∈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≤n≤N, 0≤m≤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∩[AB] is coloured blue
(ii) no point of S∩[AC] is coloured white
(iii) no point of S∩[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. geometry3D geometrytetrahedroninductioncombinatorics unsolvedcombinatorics