2020 Iranian Our MO

Part of Iran Contests

Subcontests

(7)

1

Points in frames

7501

96 \times 96

square is marked. We call the

4 \times 4

square without its central

2 \times 2

square a frame. Prove that there exist a frame with sides parallel to the

96 \times 96

square (not necessarily from the grid lines) that contains at least

10

marked points.
Proposed by Negar Babashah, Shima Amirbeygie Rated 5

1

FE with polynomial

Find all functions

f:\mathbb{R}^+ \to \mathbb{R}^+

P(x),Q(x),R(x)

with positive real coefficients such that

Q(-1)=-1

and for all positive reals

x,y

f(\frac{x}{y}+R(y))=\frac{f(x)}{Q(y)}+P(y).

Proposed by Alireza Danaie, Ali Mirazaie Anari Rated 2

1

On two AP

Concider two sequences

x_n=an+b

y_n=cn+d

a,b,c,d

are natural numbers and

gcd(a,b)=gcd(c,d)=1

, prove that there exist infinite

n

x_n

y_n

are both square-free.
Proposed by Siavash Rahimi Shateranloo, Matin Yadollahi Rated 3

1

Classical looking graph

In a school there are

n

k

student. We know that in this school every two students have attended exactly in one common class. Also due to smallness of school each class has less than

k

k-1

is not a perfect square, prove that there exist a student that has attended in at least

\sqrt k

classes.
Proposed by Mohammad Moshtaghi Far, Kian Shamsaie Rated 4

1

Classical looking NT

Does there exist a non-constant infinite sequence of distinct natural numbers such that for all large enough

k

we have that sum of any

k

-sized sub-sequence is square-free if and only if

k

is square-free.
Proposed by Alireza Haqi, Amir Rezaie Moghadam Rated 3

1

Constructing Geo

A

A

\omega_A

. Construct all possible triangles such that circumcenter of

\triangle ABC

coincide with centroid of the triangle formed by tangent points of

\omega_A

and triangle sides.
Proposed by Seyed Reza Hosseini Dolatabadi, Pooya Esmaeil Akhondy Rated 4

1

Tetromino

Find the maximum number of cells that can be coloured from a

4\times 3000

board such that no tetromino is formed.
Proposed by Arian Zamani, Matin Yousefi Rated 5