2023 Balkan MO

Part of Balkan MO

Subcontests

(4)

1

Sum of some numbers of a set is equal to the sum of some of the rests

Find the greatest integer

k\leq 2023

for which the following holds: whenever Alice colours exactly

k

numbers of the set

\{1,2,\dots, 2023\}

in red, Bob can colour some of the remaining uncoloured numbers in blue, such that the sum of the red numbers is the same as the sum of the blue numbers.
Romania

1

Beautiful number theory problem

For each positive integer

n

\omega(n)

the number of distinct prime divisors of

n

\omega(1)=0

\omega(12)=2

). Find all polynomials

P(x)

with integer coefficients, such that whenever

n

is a positive integer satisfying

\omega(n)>2023^{2023}

P(n)

is also a positive integer with

\omega(n)\ge\omega(P(n)).

Greece (Minos Margaritis - Iasonas Prodromidis)

1

Geometry of BMO 2023

ABC

, the incircle touches sides

BC,CA,AB

D,E,F

respectively. Assume there exists a point

X

EF

\angle{XBC} = \angle{XCB} = 45^{\circ}.

M

be the midpoint of the arc

BC

on the circumcircle of

ABC

A

. Prove that the line

MD

E

F

.
United Kingdom

1

Functional xf(x+f(y))=(y-x)f(f(x)) for all reals x,y

Find all functions

f\colon \mathbb{R} \rightarrow \mathbb{R}

such that for all

x,y \in \mathbb{R}

xf(x+f(y))=(y-x)f(f(x)).

Proposed by Nikola Velov, Macedonia