MathDB

1

edges with endpoint at v change color at most 2 times, in a planar graph

G

are colored either with blue or red. Prove that there is a vertex like

v

such that when we go around

v

through a complete cycle, edges with the endpoint at

v

change their color at most two times.
Clarifications for complete cycle: If all the edges with one endpoint at

v

are

(v,u_1),(v,u_2),\ldots,(v,u_k)

such that

u_1,u_2,\ldots,u_k

are clockwise with respect to

v

then in the sequence of

(v,u_1),(v,u_2),\ldots,(v,u_k),(v,u_1)

there are at most two

j

such that colours of

(v,u_j),(v,u_{j+1})

(

j \mod k

) differ.

3

1

tiling an infinite network with dominos

An infinite network is partitioned with dominos. Prove there exist three other tilings with dominos, have neither common domino with the existing tiling nor with each other.
Clarifications for network: It means an infinite board consisting of square cells.

2

1

Nice polynomial with gcd greater than 1

Let

n >1

be a natural number and

T_{n}(x)=x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + ... + a_1 x^1 + a_0

.\\ Assume that for each nonzero real number

t

we have

T_{n}(t+\frac {1}{t})=t^n+\frac {1}{t^n}

.\\ Prove that for each

0\le i \le n-1

gcd (a_i,n) >1

.
Proposed by Morteza Saghafian

4

1

prove lower bound for lcm

Let

a,b

be two relatively prime positive integers.Also let

m,n

be positive integers with

n> m

.\\ Prove that\\

lcm [am+b,a (m+1)+b,...,an+b]\ge (n+1)\cdot \binom {n}{m}

Proposed by Navid Safaei

1

Tangent geo with ratios

Let

ABC

be a triangle and

D

be the feet of

A

-altitude.\\

E,F

are defined on segments

AD,BC

,respectively such that

\frac {AE}{DE}=\frac{BF}{CF}

.\\ Assume that

G

lies on

AF

such that

BG\perp AF

.Prove that

EF

is tangent to the circumcircle of

CFG

.
Proposed by Mehdi Etesami Fard

6

1

MX passes through fixed point

Let

ABCD

be cyclic quadrilateral with circumcircle

\omega

and

M

be any point on

\omega

.\\ Let

E

and

F

be the intersection of

AB,CD

and

AD,BC

respectively.\\

ME

intersects lines

AD,BC

at

P,Q

and similarly

MF

intersects lines

AB,CD

at

R,S

.\\ Let the lines

PS

and

RQ

meet at

X

.\\ Prove that as

M

varies over

\omega

\\

MX

passes through fixed point.\\
Proposed by Mehdi Etesami Fard

2019 Iran RMM TST

Subcontests

edges with endpoint at v change color at most 2 times, in a planar graph

tiling an infinite network with dominos

Nice polynomial with gcd greater than 1

prove lower bound for lcm

Tangent geo with ratios

MX passes through fixed point