2020 Iran Team Selection Test

Part of Iran Team Selection Test

Subcontests

(6)

2

Diameter length in triangulation

n

positive numbers are given. Is it always possible to find a convex polygon with

n+3

edges and a triangulation of it so that the length of the diameters used in the triangulation are the given

n

numbers?
Proposed by Morteza Saghafian

Number Theory

p

is an odd prime number. Find all

\frac{p-1}2

\left(x_1,x_2,\dots,x_{\frac{p-1}2}\right)\in \mathbb{Z}_p^{\frac{p-1}2}

\sum_{i = 1}^{\frac{p-1}{2}} x_{i} \equiv \sum_{i = 1}^{\frac{p-1}{2}} x_{i}^{2} \equiv \cdots \equiv \sum_{i = 1}^{\frac{p-1}{2}} x_{i}^{\frac{p - 1}{2}} \pmod p.

Proposed by Ali Partofard

2

s(2n) and s(n^2)

k \in \mathbb{Z}

prove that there exist infinite pairs of distinct natural numbers such that \begin{align*} n+s(2n)=m+s(2m) \\ kn+s(n^2)=km+s(m^2). \end{align*} (

s(n)

denotes the sum of digits of

n

.)
Proposed by Mohammadamin Sharifi

Fractional part

For every positive integer

k>1

prove that there exist a real number

x

so that for every positive integer

n<1398

\left\{x^n\right\}<\left\{x^{n-1}\right\} \Longleftrightarrow k\mid n.

Proposed by Mohammad Amin Sharifi

2

Wired-looking FE

Given a function

g:[0,1] \to \mathbb{R}

satisfying the property that for every non empty dissection of the trivial

[0,1]

A,B

\exists x \in A; g(x) \in B

\exists x \in B; g(x) \in A

and we have furthermore

g(x)>x

x \in [0,1]

. Prove that there exist infinite

x \in [0,1]

g(x)=1

.
Proposed by Ali Zamani

Concurrency

ABC

be an isosceles triangle (

AB=AC

) with incenter

I

\omega

C

I

and is tangent to

AI

\omega

AC

and circumcircle of

ABC

Q

D

, respectively. Let

M

be the midpoint of

AB

N

be the midpoint of

CQ

AD

MN

BC

are concurrent.
Proposed by Alireza Dadgarnia

2

Polynomial permutation

We call a number

n

interesting if for each permutation

\sigma

1,2,\ldots,n

there exist polynomials

P_1,P_2,\ldots ,P_n

\epsilon > 0

i)

P_1(0)=P_2(0)=\ldots =P_n(0)

ii)

P_1(x)>P_2(x)>\ldots >P_n(x)

-\epsilon<x<0

iii)

P_{\sigma (1)} (x)>P_{\sigma (2)}(x)> \ldots >P_{\sigma (n)} (x)

0<x<\epsilon

Find all interesting

n

.
Proposed by Mojtaba Zare Bidaki

Iran geometry

Given a triangle

ABC

with circumcircle

\Gamma

E

F

are the foot of angle bisectors of

B

C

I

is incenter and

K

is the intersection of

AI

EF

T

be the midpoint of arc

BAC

\Gamma

A

-median and circumcircle of

AEF

for the second time at

X

S

S'

be the reflection of

S

AI

J

be the second intersection of circumcircle of

AS'K

AX

. Prove that quadrilateral

TJIX

is cyclic.
Proposed by Alireza Dadgarnia and Amir Parsa Hosseini

2

A crowded geometry

O

be the circumcenter of the triangle

ABC

D,E

AC,AB

P,Q,R,S

are given in plane such that

P,C

R,C

are on different sides of

AB

Q,B

S,B

are on different sides of

AC

R,S

lie on circumcircle of

DAP,EAQ

\triangle BCE \sim \triangle ADQ , \triangle CBD \sim \triangle AEP

(In that order),

\angle ARE=\angle ASD=\angle BAC

RS\| PQ

RE ,DS

are concurrent on

AO

.
Proposed by Alireza Dadgarnia

A game on a board

Alice and Bob take turns alternatively on a

2020\times2020

board with Alice starting the game. In every move each person colours a cell that have not been coloured yet and will be rewarded with as many points as the coloured cells in the same row and column. When the table is coloured completely, the points determine the winner. Who has the wining strategy and what is the maximum difference he/she can grantees?
Proposed by Seyed Reza Hosseini

2

Weighted complete graph

A weighted complete graph with distinct positive wights is given such that in every triangle is degenerate that is wight of an edge is equal to sum of two other. Prove that one can assign values to the vertexes of this graph such that the wight of each edge is the difference between two assigned values of the endpoints.
Proposed by Morteza Saghafian

square free mod n

We call a monic polynomial

P(x) \in \mathbb{Z}[x]

square-free mod n if there dose not exist polynomials

Q(x),R(x) \in \mathbb{Z}[x]

Q

being non-constant and

P(x) \equiv Q(x)^2 R(x) \mod n

. Given a prime

p

m \geq 2

. Find the number of monic square-free mod p

P(x)

m

and coeeficients in

\{0,1,2,3,...,p-1\}

.
Proposed by Masud Shafaie