MathDB

1

Functional equation with polynomial from R^+ to R^+

\mathbb{R}^+ = (0, \infty)

be the set of all positive real numbers. Find all functions

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

and polynomials

P(x)

with non-negative real coefficients such that

P(0) = 0

which satisfy the equality

f(f(x) + P(y)) = f(x - y) + 2y

for all real numbers

x > y > 0

.
Proposed by Sardor Gafforov, Uzbekistan

3

1

number theory, divisibility, inequality

Let

a

and

b

be distinct positive integers such that

3^a + 2

is divisible by

3^b + 2

. Prove that

a > b^2

.
Proposed by Tynyshbek Anuarbekov, Kazakhstan

2

1

non-constant arithmetic progression, existence of a sequence

Let

n \ge k \ge 3

be integers. Show that for every integer sequence

1 \le a_1 < a_2 < . . . < a_k \le n

one can choose non-negative integers

b_1, b_2, . . . , b_k

, satisfying the following conditions:
[*]

0 \le b_i \le n

for each

1 \le i \le k

, [*] all the positive

b_i

are distinct, [*] the sums

a_i + b_i

,

1 \le i \le k

, form a permutation of the first

k

terms of a non-constant arithmetic progression.

1

Reflections of AB, AC with respect to BC and angle bisector of A

Let

ABC

be an acute-angled triangle with

AC > AB

and let

D

be the foot of the

A

-angle bisector on

BC

. The reflections of lines

AB

and

AC

in line

BC

meet

AC

and

AB

at points

E

and

F

respectively. A line through

D

meets

AC

and

AB

at

G

and

H

respectively such that

G

lies strictly between

A

and

C

while

H

lies strictly between

B

and

F

. Prove that the circumcircles of

\triangle EDG

and

\triangle FDH

are tangent to each other.

2024 Balkan MO

Subcontests

Functional equation with polynomial from R^+ to R^+

number theory, divisibility, inequality

non-constant arithmetic progression, existence of a sequence

Reflections of AB, AC with respect to BC and angle bisector of A