MathDB

a) Prove that for any natural number

n\ge 1,

there is a set

\mathcal{M}

n

points from the Cartesian plane such that the barycenter of every subset of

\mathcal{M}

has integral coordinates (both coordinates are integer numbers).
b) Show that if a set

\mathcal{N}

formed by an infinite number of points from the Cartesian plane is given such that no three of them are collinear, then there exists a finite subset of

\mathcal{N} ,

the barycenter of which has non-integral coordinates.

Problem Statement