IR-MO 2012

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

The city of Bridge Village has some highways. Highways are closed curves that have intersections with each other or themselves in $4$ -way crossroads. Mr.Bridge Lover, mayor of the city, wants to build a bridge on each crossroad in order to decrease the number of accidents. He wants to build the bridges in such a way that in each highway, cars pass above a bridge and under a bridge alternately. By knowing the number of highways determine that this action is possible or not.

Proposed by Erfan Salavati

a) Does there exist an infinite subset $S$ of the natural numbers, such that $S\neq \mathbb{N}$ , and such that for each natural number $n\not \in S$ , exactly $n$ members of $S$ are coprime with $n$ ?b) Does there exist an infinite subset $S$ of the natural numbers, such that for each natural number $n\in S$ , exactly $n$ members of $S$ are coprime with $n$ ?

Proposed by Morteza Saghafian