MathDB

1

Integers with neighbouring digits differing by one (BxMO 2023, Problem 4)

n

is friendly if the difference of each pair of neighbouring digits of

n

, written in base

10

, is exactly

1

. For example, 6787 is friendly, but 211 and 901 are not.
Find all odd natural numbers

m

for which there exists a friendly integer divisible by

64m

.

3

1

Lines intersecting on the circumcircle (BxMO 2023, Problem 3)

Let

ABC

be a triangle with incentre

I

and circumcircle

\omega

. Let

N

denote the second point of intersection of line

AI

and

\omega

. The line through

I

perpendicular to

AI

intersects line

BC

, segment

[AB]

, and segment

[AC]

at the points

D

,

E

, and

F

, respectively. The circumcircle of triangle

AEF

meets

\omega

again at

P

, and lines

PN

and

BC

intersect at

Q

. Prove that lines

IQ

and

DN

intersect on

\omega

.

2

1

Colourings of integers and power-of-two differences (BxMO 2023, Problem 2)

Determine all integers

k\geqslant 1

with the following property: given

k

different colours, if each integer is coloured in one of these

k

colours, then there must exist integers

a_1<a_2<\cdots<a_{2023}

of the same colour such that the differences

a_2-a_1,a_3-a_2,\dots,a_{2023}-a_{2022}

are all powers of

2

.

1

Functional inequality (BxMO 2023, Problem 1)

Find all functions

f\colon\mathbb{R}\to\mathbb{R}

such that

(x-y)\bigl(f(x)+f(y)\bigr)\leqslant f\bigl(x^2-y^2\bigr)

for all

x,y\in\mathbb{R}

.

2023 Benelux

Subcontests

Integers with neighbouring digits differing by one (BxMO 2023, Problem 4)

Lines intersecting on the circumcircle (BxMO 2023, Problem 3)

Colourings of integers and power-of-two differences (BxMO 2023, Problem 2)

Functional inequality (BxMO 2023, Problem 1)