MathDB

1

3 coprime numbers satisfying gcd condition

Find all positive integers, such that there exist positive integers

a, b, c

, satisfying

\gcd(a, b, c)=1

and

n=\gcd(ab+c, ac-b)=a+b+c

.

N1

1

A number that is sum of two squares

For positive integers

a, b, c

(not necessarily distinct), suppose that

a+bc, b+ac, c+ab

are all perfect squares. Show that

a^2(b+c)+b^2(a+c)+c^2(a+b)+2abc

can be written as sum of two squares.

G6

1

Conditional geometry with a right angle

Let

ABC

be an acute triangle (

AB < BC < AC

) with circumcircle

\Gamma

. Assume there exists

X \in AC

satisfying

AB=BX

and

AX=BC

. Points

D, E \in \Gamma

are taken such that

\angle ADB<90^{\circ}

,

DA=DB

and

BC=CE

. Let

P

be the intersection point of

AE

with the tangent line to

\Gamma

at

B

, and let

Q

be the intersection point of

AB

with tangent line to

\Gamma

at

C

. Show that the projection of

D

onto

PQ

lies on the circumcircle of

\triangle PAB

.

G5

1

Another config geo with concurrent lines

Let

ABC

be a triangle with circumcenter

O

. Point

X

is the intersection of the parallel line from

O

to

AB

with the perpendicular line to

AC

from

C

. Let

Y

be the point where the external bisector of

\angle BXC

intersects with

AC

. Let

K

be the projection of

X

onto

BY

. Prove that the lines

AK, XO, BC

have a common point.

G4

1

Config geo with the Euler line

Let

O

and

H

be the circumcenter and orthocenter of a scalene triangle

ABC

, respectively. Let

D

be the intersection point of the lines

AH

and

BC

. Suppose the line

OH

meets the side

BC

at

X

. Let

P

and

Q

be the second intersection points of the circumcircles of

\triangle BDH

and

\triangle CDH

with the circumcircle of

\triangle ABC

, respectively. Show that the four points

P, D, Q

and

X

lie on a circle.

G2

1

Similar triangles and equal segments

Let

ABCD

be a cyclic quadrilateral with circumcenter

O

lying in the interior. Let

E

and

F

be the midpoints of the segments

BC

and

AD

, respectively. Let

X

be the point lying on the same side of the line

EF

as the vertex

C

such that

\triangle EXF

and

\triangle BOA

are similar. Prove that

XC = XD

.

G1

1

Circumscribed quadrilateral and incenters

Let

ABCD

be a circumscribed quadrilateral and let

X

be the intersection point of its diagonals

AC

and

BD

. Let

I_1, I_2, I_3, I_4

be the incenters of

\triangle DXC

,

\triangle BXC

,

\triangle AXB

, and

\triangle DXA

, respectively. The circumcircle of

\triangle CI_1I_2

intersects the sides

CB

and

CD

at points

P

and

Q

, respectively. The circumcircle of

\triangle AI_3I_4

intersects the sides

AB

and

AD

at points

M

and

N

, respectively. Prove that

AM+CQ=AN+CP

C4

1

Princesses and frogs

Once upon a time there are

n

pairs of princes and princesses who are in love with each other. One day a witch comes along and turns all the princes into frogs; the frogs can be distinguished by sight but the princesses cannot tell which frog corresponds to which prince. The witch tells the princesses that if any of them kisses the frog that corresponds to the prince very that she loves then that frog will immediately transform back into a prince. If each princess can stand kissing at most

k

frogs, what is the maximum number of princes they can be sure to save?
(The princesses may take turns kissing in any order, communicate with each other and vary their strategy for future kisses depending on information gained from past kisses.)

C3

1

Chordless graph with degrees at least 2

In a given community of people, each person has at least two friends within the community. Whenever some people from this community sit on a round table such that each adjacent pair of people are friends, it happens that no non-adjacent pair of people are friends. Prove that there exist two people in this community such that each has exactly two friends and they have at least one common friend.

C2

1

Swapping numbers that sum to a perfect square

For an integer

n>2

, the tuple

(1, 2, \ldots, n)

is written on a blackboard. On each turn, one can choose two numbers from the tuple such that their sum is a perfect square and swap them to obtain a new tuple. Find all integers

n > 2

for which all permutations of

\{1, 2,\ldots, n\}

can appear on the blackboard in this way.

C1

1

Classical game with stones

Joe and Penny play a game. Initially there are

5000

stones in a pile, and the two players remove stones from the pile by making a sequence of moves. On the

k

-th move, any number of stones between

1

and

k

inclusive may be removed. Joe makes the odd-numbered moves and Penny makes the even-numbered moves. The player who removes the very last stone is the winner. Who wins if both players play perfectly?

A6

1

Positive reals FE with x^2023

Find all functions

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

, such that

f(x^{2023}+f(x)f(y))=x^{2023}+yf(x)

for all

x, y>0

.

A5

1

Two polynomials with specific number of roots

Are there polynomials

P, Q

with real coefficients, such that

P(P(x))\cdot Q(Q(x))

has exactly

2023

distinct real roots and

P(Q(x)) \cdot Q(P(x))

has exactly

2024

distinct real roots?

A4

1

Infinitely many close pairs of members of a sequence

Prove that there exists a real

c<\frac{3}{4}

, such that for each sequence

x_1, x_2, \ldots

satisfying

0 \leq x_i \leq 1

for all

i

, there exist infinitely many

(m, n)

with

m>n

, such that

|x_m-x_n|\leq \frac{c} {m}.

A3

1

Reals to reals FE

Find all functions

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

, such that

f(xy+f(x^2))=xf(x+y)

for all reals

x, y

.

A2

1

Inequality about existence of permutation

Let

a, b, c, d

be non-negative reals such that

\frac{1}{a+3}+\frac{1}{b+3}+\frac{1}{c+3}+\frac{1}{d+3}=1

. Show that there exists a permutation

(x_1, x_2, x_3, x_4)

of

(a, b, c, d)

, such that

x_1x_2+x_2x_3+x_3x_4+x_4x_1 \geq 4.

2023 Balkan MO Shortlist

Subcontests

3 coprime numbers satisfying gcd condition

A number that is sum of two squares

Conditional geometry with a right angle

Another config geo with concurrent lines

Config geo with the Euler line

Similar triangles and equal segments

Circumscribed quadrilateral and incenters

Princesses and frogs

Chordless graph with degrees at least 2

Swapping numbers that sum to a perfect square

Classical game with stones

Positive reals FE with x^2023

Two polynomials with specific number of roots

Infinitely many close pairs of members of a sequence

Reals to reals FE

Inequality about existence of permutation