2012 Iran MO (3rd Round)

Part of Iran MO (3rd Round)

Subcontests

(8)

1

Prime subsets of natural numbers

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

n

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

n

S

are coprime with

n

?
Proposed by Morteza Saghafian

1

Curves on the plane and bridges

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

1

Bounds for area of special convex polygons

a>0

exists such that for each natural number

n

, there exists a convex

n

P

in plane with lattice points as vertices such that the area of

P

an^3

.
b) Prove that there exists

b>0

such that for each natural number

n

n

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

n

P

in plane with lattice points as vertices, the area of

P

is not less than

cn^{2+\alpha}

.
Proposed by Mostafa Eynollahzade

4

7

7

7

7