IR-MO 2012 Problem 6

a) Prove that $a>0$ exists such that for each natural number $n$ , there exists a convex $n$ -gon $P$ in plane with lattice points as vertices such that the area of $P$ is less than $an^3$ .b) Prove that there exists $b>0$ such that for each natural number $n$ and each $n$ -gon $P$ in plane with lattice points as vertices, the area of $P$ is not less than $bn^2$ .c) Prove that there exist $\alpha,c>0$ such that for each natural number $n$ and each $n$ -gon $P$ in plane with lattice points as vertices, the area of $P$ is not less than $cn^{2+\alpha}$ .

Proposed by Mostafa Eynollahzade