A convex polygon having precisely 2N (N ≥2) sides has each of its vertices at lattice points.
Can the area of the polygon be < N^{3}/100?
If so, give an example.
If not, prove it.
Before even looking for a solution it stands to reason that since the limit grows as a cubic, all we need to do is find a way of increasing the area as a lower degree function of N and a solution will occur.
This is easy to do with a zigzag.
(0,0) (1,0)
(1,1) (2,1)
(0,2) (1,2)
(1,3) (2,3)
etc..
When N=10, the area of this 20gon is 9 < 10^3/100 = 10

Posted by Jer
on 20160901 13:25:49 