We received the following solution fram Juan Mir Pieras (Spain).
Given n > 4, any convex n-sided polygon with lattice points as vertices has at least one lattice point either inside or on the boundary.
Demo:
There are four different kinds of lattice points:
(even, even)
(even, odd)
(odd, even)
(odd, odd)
There are only four kinds of lattice points. So, in any n-sided polygon P, being n > 4, at least two of those vertices belong to the same kind. Let's call them A(a1,a2) and B(b1,b2) and have a look at the point M( (a1+b1)/2 , (a2+b2)/2 ):
M is a lattice point, because a1+b1 and a2+b2 are even numbers.
M is the center of the segment AB, and AB is either a side of the polygon P (in that case, M would be on the boundary of P), or a diagonal of P (in that case, and if P is convex, M would be inside P).
So, given n > 4, any convex n-sided polygon with lattice points as vertices has at least one lattice point either inside or on the boundary. (QED)