MathDB

1

Rubies, emeralds, and sapphires on an open necklace

An open necklace can contain rubies, emeralds, and sapphires. At every step we can perform any of the following operations:
[*]We can replace two consecutive rubies with an emerald and a sapphire, where the emerald is on the left of the sapphire.[/*] [*]We can replace three consecutive emeralds with a sapphire and a ruby, where the sapphire is on the left of the ruby. [/*] [*]If we find two consecutive sapphires then we can remove them.[/*] [*]If we find consecutively and in this order a ruby, an emerald, and a sapphire, then we can remove them.[/*]
Furthermore we can also reverse all of the above operations. For example by reversing 3. we can put two consecutive sapphires on any position we wish.
Initially the necklace has one sapphire (and no other precious stones). Decide, with proof, whether there is a finite sequence of steps such that at the end of this sequence the necklace contains one emerald (and no other precious stones).
Remark: A necklace is open if its precious stones are on a line from left to right. We are not allowed to move a precious stone from the rightmost position to the leftmost as we would be able to do if the necklace was closed.
Proposed by Demetres Christofides, Cyprus

N5

1

Divisibility holds for all naturals

Let

x,y

be positive integers. If for each positive integer

n

we have that

(ny)^2+1\mid x^{\varphi(n)}-1.

Prove that

x=1

.
(Silouanos Brazitikos, Greece)

N4

1

A polynomial with rational roots

Let

P(x)=a_d x^d+\dots+a_1 x+a_0

be a non-constant polynomial with non-negative integer coefficients having

d

rational roots.Prove that

\text{lcm} \left(P(m),P(m+1),\dots,P(n) \right)\geq m \dbinom{n}{m}

for all

n>m

(Navid Safaei, Iran)

N1

1

existence of a sequence

For positive integers

m

and

n

, let

d(m, n)

be the number of distinct primes that divide both

m

and

n

. For instance,

d(60, 126) = d(2^2 \cdot 3 \cdot 5, 2 \cdot 3^2 \cdot 7) = 2.

Does there exist a sequence

(a_n)

of positive integers such that: [*]

a_1 \geq 2018^{2018};

[*]

a_m \leq a_n

whenever

m \leq n

; [*]

d(m, n) = d(a_m, a_n)

for all positive integers

m\neq n

?
(Dominic Yeo, United Kingdom)

C1

1

Maximum of suprising games

Let

N

be an odd number,

N\geq 3

.

N

tennis players take part in a championship. Before starting the championship, a commission puts the players in a row depending on how good they think the players are. During the championship, every player plays with every other player exactly once, and each match has a winner. A match is called suprising if the winner was rated lower by the commission. At the end of the tournament, players are arranged in a line based on the number of victories they have achieved. In the event of a tie, the commission's initial order is used to decide which player will be higher.
It turns out that the final order is exactly the same as the commission's initial order. What is the maximal number of suprising matches that could have happened.

G5

1

prove that quadrilaterals are cyclic and their circumcenters coincide

Let

ABC

be an acute triangle with

AB<AC<BC

and let

D

be a point on it's extension of

BC

towards

C

. Circle

c_1

, with center

A

and radius

AD

, intersects lines

AC,AB

and

CB

at points

E,F

, and

G

respectively. Circumscribed circle

c_2

of triangle

AFG

intersects again lines

FE,BC,GE

and

DF

at points

J,H,H'

and

J'

respectively. Circumscribed circle

c_3

of triangle

ADE

intersects again lines

FE,BC,GE

and

DF

at points

I,K,K'

and

I'

respectively. Prove that the quadrilaterals

HIJK

and

H'I'J'K '

are cyclic and the centers of their circumscribed circles coincide.
by Evangelos Psychas, Greece

G3

1

max of \min (\frac{PA}{p-a},\frac{PB}{p-b},\frac{PC}{p-c} ).

Let

P

be an interior point of triangle

ABC

. Let

a,b,c

be the sidelengths of triangle

ABC

and let

p

be it's semiperimeter. Find the maximum possible value of

\min\left(\frac{PA}{p-a},\frac{PB}{p-b},\frac{PC}{p-c}\right)

taking into consideration all possible choices of triangle

ABC

and of point

P

.
by Elton Bojaxhiu, Albania

G2

1

perpendicularity wanted, intersections of tangents with perp. bisectors related

Let

ABC

be a triangle inscribed in circle

\Gamma

with center

O

. Let

H

be the orthocenter of triangle

ABC

and let

K

be the midpoint of

OH

. Tangent of

\Gamma

at

B

intersects the perpendicular bisector of

AC

at

L

. Tangent of

\Gamma

at

C

intersects the perpendicular bisector of

AB

at

M

. Prove that

AK

and

LM

are perpendicular.
by Michael Sarantis, Greece

G1

1

equal segments wanted, 2 circumcircles and 2 excirles given

Let

ABC

be an acute triangle and let

M

be the midpoint of side

BC

. Let

D,E

be the excircles of triangles

AMB,AMC

respectively, towards

M

. Circumcirscribed circle of triangle

ABD

intersects line

BC

at points

B

and

F

. Circumcirscribed circles of triangle

ACE

intersects line

BC

at points

C

and

G

. Prove that

BF=CG

.
by Petru Braica, Romania

G6

1

Isoceles triangle, cyclic quadrilateral

In a triangle

ABC

with

AB=AC

,

\omega

is the circumcircle and

O

its center. Let

D

be a point on the extension of

BA

beyond

A

. The circumcircle

\omega_{1}

of triangle

OAD

intersects the line

AC

and the circle

\omega

again at points

E

and

G

, respectively. Point

H

is such that

DAEH

is a parallelogram. Line

EH

meets circle

\omega_{1}

again at point

J

. The line through

G

perpendicular to

GB

meets

\omega_{1}

again at point

N

and the line through

G

perpendicular to

GJ

meets

\omega

again at point

L

. Prove that the points

L, N, H, G

lie on a circle.

A5

1

prove that f is a second degree polynomial.

Let

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

be a concave function and

g: \mathbb {R} \to \mathbb {R}

be a continuous function . If

f (x + y) + f (x-y) -2f (x) = g (x) y^2

for all

x, y \in \mathbb {R},

prove that

f

is a second degree polynomial.

A6

1

A inequality

Let

x_1, x_2, \cdots, x_n

be positive real numbers . Prove that:

\sum_ {i = 1}^n x_i ^2\geq \frac {1} {n + 1} \left (\sum_ {i = 1}^n x_i \right)^2+\frac{12(\sum_ {i = 1}^n i x_i)^2}{n (n + 1) (n + 2) (3n + 1)}.

A4

1

Simple inequality

Let

a, b, c

be positive real numbers such that

abc = 1.

Prove that:

2 (a^ 2 + b^ 2 + c^ 2) \left (\frac 1 {a^ 2} + \frac 1{b^ 2}+ \frac 1{c^2}\right)\geq 3(a+ b + c + ab + bc + ca).

N2

1

Function with factorial

Find all functions

f:\mathbb{N}\rightarrow\mathbb{N}

such that

n!+f(m)!|f(n)!+f(m!)

for all

m,n\in\mathbb{N}

Proposed by Valmir Krasniqi and Dorlir Ahmeti, Albania

A3

1

Inequality with positive integer

Show that for every positive integer

n

we have:

\sum_{k=0}^{n}\left(\frac{2n+1-k}{k+1}\right)^k=\left(\frac{2n+1}{1}\right)^0+\left(\frac{2n}{2}\right)^1+...+\left(\frac{n+1}{n+1}\right)^n\leq 2^n

Proposed by Dorlir Ahmeti, Albania

2018 Balkan MO Shortlist

Subcontests

Rubies, emeralds, and sapphires on an open necklace

Divisibility holds for all naturals

A polynomial with rational roots

existence of a sequence

Maximum of suprising games

prove that quadrilaterals are cyclic and their circumcenters coincide

max of \min (\frac{PA}{p-a},\frac{PB}{p-b},\frac{PC}{p-c} ).

perpendicularity wanted, intersections of tangents with perp. bisectors related

equal segments wanted, 2 circumcircles and 2 excirles given

Isoceles triangle, cyclic quadrilateral

prove that f is a second degree polynomial.

A inequality

Simple inequality

Function with factorial

Inequality with positive integer