MathDB

3

Biggest worm on the table.

3

by

3

table, by a

k

-worm, we mean a path of different cells

(S_1,S_2,...,S_k)

such that each two consecutive cells have one side in common. The

k

-worm at each steep can go one cell forward and turn to the

(S,S_1,...,S_{k-1})

if

S

is an unfilled cell which is adjacent (has one side in common) with

S_1

. Find the maximum number of

k

such that there is a

k

-worm

(S_1,...,S_k)

such that after finitly many steps can be turned to

(S_k,...,S_1)

.

beautiful algebra problem

We call a polynomial

P(x)=a_dx^d+...+a_0

of degree

d

nice if

\frac{2021(|a_d|+...+|a_0|)}{2022}<max_{0 \le i \le d}|a_i|

Initially Shayan has a sequence of

d

distinct real numbers;

r_1,...,r_d \neq \pm 1

. At each step he choose a positive integer

N>1

and raises the

d

numbers he has to the exponent of

N

, then delete the previous

d

numbers and constructs a monic polynomial of degree

d

with these number as roots, then examine whether it is nice or not. Prove that after some steps, all the polynomials that shayan produces would be nice polynomials
Proposed by Navid Safaei

Hard number theory on cubes

Let

n

be an integer greater than

1

such that

n

could be represented as a sum of the cubes of two rational numbers, prove that

n

is also the sum of the cubes of two non-negative rational numbers.
Proposed by Navid Safaei

2

3

easy function on Tst

Let

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

satisfying

f(x)=f(x+2)+2f(x^2+2x)

. Prove that if for all

x>1400^{2021}

,

xf(x) \le 2021

, then

xf(x) \le 2021

for all

x \in \mathbb {R}^+

Proposed by Navid Safaei

GGG4 geometry in a Tst

Let

ABC

be a triangle with

AB \neq AC

and with incenter

I

. Let

M

be the midpoint of

BC

, and let

L

be the midpoint of the circular arc

BAC

. Lines through

M

parallel to

BI,CI

meet

AB,AC

at

E

and

F

, respectively, and meet

LB

and

LC

at

P

and

Q

, respectively. Show that

I

lies on the radical axis of the circumcircles of triangles

EMF

and

PMQ

.
Proposed by Andrew Wu

putting queen's on the chess board.

In a chess board we call a group of queens independant if no two are threatening each other. In an

n

by

n

grid, we put exaxctly one queen in each cell ofa greed. Let us denote by

M_n

the minimum number of independant groups that hteir union contains all the queens. Let

k

be a positive integer, prove that

M_{3k+1} \le 3k+2

Proposed by Alireza Haghi

1

3

line tangent and parallel at the same time.

Suppose that two circles

\alpha, \beta

with centers

P,Q

, respectively , intersect orthogonally at

A

,

B

. Let

CD

be a diameter of

\beta

that is exterior to

\alpha

. Let

E,F

be points on

\alpha

such that

CE,DF

are tangent to

\alpha

, with

C,E

on one side of

PQ

and

D,F

on the other side of

PQ

. Let

S

be the intersection of

CF,AQ

and

T

be the intersection of

DE,QB

. Prove that

ST

is parallel to

CD

and is tangent to

\alpha

an interesting $polyomina$

A polyomino is region with connected interior that is a union of a finite number of squares from a grid of unit squares. Do there exist a positive integer

n>4

and a polyomino

P

contained entirely within and

n

-by-

n

grid such that

P

contains exactly

3

unit squares in every row and every column of the grid?
Proposed by Nikolai Beluhov

Creating polynomial with property

Let

P(x)=x^{2016}+2x^{2015}+...+2017,Q(x)=1399x^{1398}+...+2x+1

. Prove that there are strictly increasing sequances

a_i,b_i, i=1,...

of positive integers such that

gcd(a_i,a_{i+1})=1

for each

i

. Moreover, for each even

i

,

P(b_i) \nmid a_i, Q(b_i) | a_i

and for each odd

i

,

P(b_i)|a_i,Q(b_i) \nmid a_i

Proposed by Shayan Talaei

2021 Iran RMM TST

Subcontests

Biggest worm on the table.

beautiful algebra problem

Hard number theory on cubes

easy function on Tst

GGG4 geometry in a Tst

putting queen's on the chess board.

line tangent and parallel at the same time.

an interesting $polyomina$

Creating polynomial with property