MathDB

1

natural a is said to be contained in natural number b, (combo related)

a

is said to be contained in the natural number

b

if it is possible to obtain a by erasing some digits from

b

(in their decimal representations). For example,

123

is contained in

901523

, but not contained in

3412

. Does there exist an infinite set of natural numbers such that no number in the set is contained in any other number from the set?

9

1

5 elders around a large bonfire, hats rom 4 colours

Five elders are sitting around a large bonfire. They know that Oluf will put a hat of one of four colours (red, green, blue or yellow) on each elder’s head, and after a short time for silent reflection each elder will have to write down one of the four colours on a piece of paper. Each elder will only be able to see the colour of their two neighbours’ hats, not that of their own nor that of the remaining two elders’ hats, and they also cannot communicate after Oluf starts putting the hats on. Show that the elders can devise a strategy ahead of time so that at most two elders will end up writing down the colour of their own hat

8

1

n-3 internal diagonals in a non self-intersecting n-gon , cut into n-2 triangles

For a natural number

n \ge 3

, we draw

n - 3

internal diagonals in a non self-intersecting, but not necessarily convex, n-gon, cutting the

n

-gon into

n - 2

triangles. It is known that the value (in degrees) of any angle in any of these triangles is a natural number and no two of these angle values are equal. What is the largest possible value of

n

?

7

1

writer Arthur has n co-authors who write books with him, Fibonacci numbers

The writer Arthur has

n \ge1

co-authors who write books with him. Each book has a list of authors including Arthur himself. No two books have the same set of authors. At a party with all his co-author, each co-author writes on a note how many books they remember having written with Arthur. Inspecting the numbers on the notes, they discover that the numbers written down are the first

n

Fibonacci numbers (defined by

F_1 = F_2 = 1

and

F_{k+2}= F_{k+1} + F_k

). For which

n

is it possible that none of the co-authors had a lapse of memory?

6

1

badminton tournament for 40 players on 20 courts

Mattis is hosting a badminton tournament for

40

players on

20

courts numbered from

1

to

20

. The players are distributed with

2

players on each court. In each round a winner is determined on each court. Afterwards, the player who lost on court

1

, and the player who won on court

20

stay in place. For the remaining

38

players, the winner on court

i

moves to court

i + 1

and the loser moves to court

i - 1

. The tournament continues until every player has played every other player at least once. What is the minimal number of rounds the tournament can last?

15

1

XY is tangent to a fixed circle

Let

\Omega

be a circle, and

B, C

are two fixed points on

\Omega

. Given a third point

A

on

\Omega

, let

X

and

Y

denote the feet of the altitudes from

B

and

C

, respectively, in the triangle

ABC

. Prove that there exists a fixed circle

\Gamma

such that

XY

is tangent to

\Gamma

regardless of the choice of the point

A

.

14

1

Config geo with a weird circle

Let

\Gamma

denote the circumcircle and

O

the circumcentre of the acute-angled triangle

ABC

, and let

M

be the midpoint of the segment

BC

. Let

T

be the second intersection point of

\Gamma

and the line

AM

, and

D

the second intersection point of

\Gamma

and the altitude from

A

. Let further

X

be the intersection point of the lines

DT

and

BC

. Let

P

be the circumcentre of the triangle

XDM

. Prove that the circumcircle of the triangle

OPD

passes through the midpoint of

XD

.

13

1

Right angle at the midpoint of EF

Let

ABCD

be a cyclic quadrilateral with

AB < BC

and

AD < DC

. Let

E

and

F

be points on the sides

BC

and

CD

, respectively, such that

AB = BE

and

AD = DF

. Let further M denote the midpoint of the segment

EF

. Prove that

\angle BMD = 90^o

.

12

1

Two circles passing through a point on the altitude

An acute-angled triangle

ABC

has altitudes

AD, BE

and

CF

. Let

Q

be an interior point of the segment

AD

, and let the circumcircles of the triangles

QDF

and

QDE

meet the line

BC

again at points

X

and

Y

, respectively. Prove that

BX = CY

.

11

1

Two circles through A and O

Let

ABC

be a triangle with circumcircle

\Gamma

and circumcentre

O

. The circle with centre on the line

AB

and passing through the points

A

and

O

intersects

\Gamma

again in

D

AC

and passing through the points

A

and

O

intersects

\Gamma

again in

E

. Prove that

BD

is parallel with

CE

.

1

Another FE

Let

\mathbb{R^+}

denote the set of positive real numbers. Assume that

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

is a function satisfying the equations: f(x^3)=f(x)^3 \text{and} f(2x)=f(x) for all

x \in \mathbb{R^+}

. Find all possible values of

f(\sqrt[2022]{2})

.

2

1

Sequence with floor

We define a sequence of natural numbers by the initial values

a_0 = a_1 = a_2 = 1

and the recursion

a_n = \bigg \lfloor \frac{n}{a_{n-1}a_{n-2}a_{n-3}} \bigg \rfloor

for all

n \ge 3

. Find the value of

a_{2022}

.

3

1

Secret one variable polynomials

We call a two-variable polynomial

P(x, y)

secretly one-variable, if there exist polynomials

Q(x)

and

R(x, y)

such that

\deg(Q) \ge 2

and

P(x, y) = Q(R(x, y))

(e.g.

x^2 + 1

and

x^2y^2 +1

are secretly one-variable, but

xy + 1

is not).
Prove or disprove the following statement: If

P(x, y)

is a polynomial such that both

P(x, y)

and

P(x, y) + 1

can be written as the product of two non-constant polynomials, then

P

is secretly one-variable.
Note: All polynomials are assumed to have real coefficients.

4

1

Inequality with xy+yz+zx=1

The positive real numbers

x,y,z

satisfy

xy+yz+zx=1

. Prove that:

2(x^2+y^2+z^2)+\frac{4}{3}\bigg (\frac{1}{x^2+1}+\frac{1}{y^2+1}+\frac{1}{z^2+1}\bigg) \ge 5

5

1

Standart looking FE

Let

\mathbb{R}

be the set of real numbers. Determine all functions

f: \mathbb{R} \rightarrow \mathbb{R}

such that

f(0)+1=f(1)

and for any real numbers

x

and

y

,

f(xy-x)+f(x+f(y))=yf(x)+3

16

1

Classical NT FE

Let

\mathbb{Z^+}

denote the set of positive integers. Find all functions

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

satisfying the condition

f(a) + f(b) \mid (a + b)^2

for all

a,b \in \mathbb{Z^+}

17

1

Sum of divisors

Let

n

be a positive integer such that the sum of its positive divisors is at least

2022n

. Prove that

n

has at least

2022

distinct prime factors.

18

1

GCD relation

Find all pairs

(a, b)

of positive integers such that

a \le b

and

\gcd(x, a) \gcd(x, b) = \gcd(x, 20) \gcd(x, 22)

holds for every positive integer

x

.

19

1

Classical equation with exponents

Find all triples

(x, y, z)

of nonnegative integers such that

x^5+x^4+1=3^y7^z

20

1

NT game with products

Ingrid and Erik are playing a game. For a given odd prime

p

, the numbers

1, 2, 3, ..., p-1

are written on a blackboard. The players take turns making moves with Ingrid starting. A move consists of one of the players crossing out a number on the board that has not yet been crossed out. If the product of all currently crossed out numbers is

1 \pmod p

after the move, the player whose move it was receives one point, otherwise, zero points are awarded. The game ends after all numbers have been crossed out.
The player who has received the most points by the end of the game wins. If both players have the same score, the game ends in a draw. For each

p

, determine which player (if any) has a winning strategy

2022 Baltic Way

Subcontests

natural a is said to be contained in natural number b, (combo related)

5 elders around a large bonfire, hats rom 4 colours

n-3 internal diagonals in a non self-intersecting n-gon , cut into n-2 triangles

writer Arthur has n co-authors who write books with him, Fibonacci numbers

badminton tournament for 40 players on 20 courts

XY is tangent to a fixed circle

Config geo with a weird circle

Right angle at the midpoint of EF

Two circles passing through a point on the altitude

Two circles through A and O

Another FE

Sequence with floor

Secret one variable polynomials

Inequality with xy+yz+zx=1

Standart looking FE

Classical NT FE

Sum of divisors

GCD relation

Classical equation with exponents

NT game with products