MathDB

1

German products

n

be a positive integer. A German set in an

n \times n

square grid is a set of

n

cells which contains exactly one cell in each row and column. Given a labelling of thecells with the integers from

1

to

n^2

using each integer exactly once, we say that an integer is a German product if it is the product of the labels of the cells in a German set.
(a) Let

n=8

. Determine whether there exists a labelling of an

8 \times 8

grid such that the following condition is fulfilled: The difference of any two German products is alwaysdivisible by

65

.
(b) Let

n=10

. Determine whether there exists a labelling of a

10 \times 10

grid such that the following condition is fulfilled: The difference of any two German products is always divisible by

101

.

19

1

Sum of digits NT part 2

Show that

S(2^{2^{2 \cdot 2023}})>2023

, where

S(m)

denotes the digit sum of

m

.

18

1

Combi NT

Let

p>7

be a prime and let

A

be subset of

\{0,1, \ldots, p-1\}

with size at least

\frac{p-1}{2}

. Show that for each integer

r

, there exist

a, b, c, d \in A

, not necessarily distinct, such that

ab-cd \equiv r \pmod p

.

17

1

Sum of digits NT part 1

Find all pairs of positive integers

(a, b)

, such that

S(a^{b+1})=a^b

, where

S(m)

denotes the digit sum of

m

.

16

1

Find two polynomials

Prove that there exist nonconstant polynomials

f, g

with integer coefficients, such that for infinitely many primes

p

,

p \nmid f(x)-g(y)

for any integers

x, y

.

15

1

Two circles and tangents

Let

\omega_1

and

\omega_2

be two circles with no common points, such that any of them is not inside the other one. Let

M, N

lie on

\omega_1, \omega_2

, such that the tangents at

M

to

\omega_1

and

N

to

\omega_2

meet at

P

, such that

PM=PN

. The circles

\omega_1

,

\omega_2

meet

MN

at

A, B

. The lines

PA, PB

meet

\omega_1, \omega_2

at

C, D

. Show that

\angle BCN=\angle ADM

.

14

1

Centroid geo

Let

ABC

be a triangle with centroid

G

. Let

D, E, F

be the circumcenters of triangles

BCG, CAG, ABG

. Let

X

be the intersection of the perpendiculars from

E

to

AB

and from

F

to

AC

. Prove that

DX

bisects

EF

.

13

1

Conditional geo with angle equality

Let

ABC

be an acute triangle with

AB<AC

and incenter

I

. Let

D

be the projection of

I

onto

BC

. Let

H

be the orthocenter of

ABC

and suppose that

\angle IDH=\angle CBA-\angle ACB

. Prove that

AH=2ID

.

12

1

Incenter of a weird triangle

Let

ABC

be an acute triangle with

AB>AC

. The internal angle bisector of

\angle BAC

meets

BC

at

D

. Let

O

be the circumcenter of

ABC

and let

AO

meet

BC

at

E

. Let

J

be the incenter of triangle

AED

. Show that if

\angle ADO=45^{\circ}

, then

OJ=JD

.

11

1

Config with reflection of the excenter

Let

ABC

be triangle with

A

-excenter

J

. The reflection of

J

in

BC

is

K

. The points

E

and

F

are on

BJ, CJ

such that

\angle EAB=\angle CAF=90^{\circ}

. Prove that

\angle FKE+\angle FJE=180^{\circ}

.

10

1

Colored points on a circle

On a circle,

n \geq 3

points are marked. Each marked point is coloured red, green or blue. In one step, one can erase two neighbouring marked points of different colours and mark a new point between the locations of the erased points with the third colour. In a final state, all marked points have the same colour which is called the colour of the final state. Find all

n

for which there exists an initial state of

n

marked points with one missing colour, from which one can reach a final state of any of the three colours by applying a suitable sequence of steps.

9

1

Partitioning a triangle

Determine if there exists a triangle that can be cut into

101

congruent triangles.

8

1

Catch the thief!

In the city of Flensburg there is a single, infinitely long, street with housesnumbered

2, 3, \ldots

. The police in Flensburg is trying to catch a thief who every night moves from the house where she is currently hiding to one of its neighbouring houses.
To taunt the local law enforcement the thief reveals every morning the highest prime divisor of the number of the house she has moved to.
Every Sunday afternoon the police searches a single house, and they catch the thief if they search the house she is currently occupying. Does the police have a strategy to catch the thief in finite time?

7

1

Path of a robot

A robot moves in the plane in a straight line, but every one meter it turns

90^{\circ}

to the right or to the left. At some point it reaches its starting point without having visited any other point more than once, and stops immediately. What are the possible path lengths of the robot?

6

1

Colored grid

Let

n

be a positive integer. Each cell of an

n \times n

table is coloured in one of

k

colours where every colour is used at least once. Two different colours

A

and

B

are said to touch each other, if there exists a cell coloured in

A

sharing a side with a cell coloured in

B

. The table is coloured in such a way that each colour touches at most

2

other colours. What is the maximal value of

k

in terms of

n

?

5

1

Two variable inequality

Find the smallest positive real

\alpha

, such that

\frac{x+y} {2}\geq \alpha\sqrt{xy}+(1 - \alpha)\sqrt{\frac{x^2+y^2}{2}}

for all positive reals

x, y

.

4

1

R to R FE

Find all functions

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

such that

f(f(x)+y)+xf(y)=f(xy+y)+f(x)

for reals

x, y

.

3

1

Flensburgian system

Denote a set of equations in the real numbers with variables

x_1, x_2, x_3 \in \mathbb{R}

Flensburgian if there exists an

i \in \{1, 2, 3\}

such that every solution of the set of equations where all the variables are pairwise different, satisfies

x_i>x_j

for all

j \neq i

.
Find all positive integers

n \geq 2

, such that the following set of two equations

a^n+b=a

and

c^{n+1}+b^2=ab

in three real variables

a,b,c

is Flensburgian.

2

1

Inequality with powers

Let

a_1, a_2, \ldots, a_{2023}

be positive reals such that

\sum_{i=1}^{2023}a_i^i=2023

. Show that

\sum_{i=1}^{2023}a_i^{2024-i}>1+\frac{1}{2023}.

1

Interesting sequence

Find all strictly increasing sequences of positive integers

a_1, a_2, \ldots

with

a_1=1

, satisfying

3(a_1+a_2+\ldots+a_n)=a_{n+1}+\ldots+a_{2n}

for all positive integers

n

.

2023 Baltic Way

Subcontests

German products

Sum of digits NT part 2

Combi NT

Sum of digits NT part 1

Find two polynomials

Two circles and tangents

Centroid geo

Conditional geo with angle equality

Incenter of a weird triangle

Config with reflection of the excenter

Colored points on a circle

Partitioning a triangle

Catch the thief!

Path of a robot

Colored grid

Two variable inequality

R to R FE

Flensburgian system

Inequality with powers

Interesting sequence