, 2024

A non self-intersecting polygon is given in a Cartesian coordinate system such that its perimeter contains no lattice points, and its vertices have no integer coordinates. A point is called semi-integer if exactly one of its coordinates is an integer. Let $P_1$, $P_2$, $\ldots$, $P_k$ denote the semi-integer points on the perimeter of the polygon. Let $n_i$ denote the floor of the non-integer coordinate of $P_i$. Prove that integers $n_1, n_2, \ldots , n_k$ can be divided into two groups with the same sum.