2013 Balkan MO

Part of Balkan MO

Subcontests

(4)

1

Graph Theory problem

In a mathematical competition, some competitors are friends; friendship is mutual, that is, when

A

B

B

is also a friend of

A

n \geq 3

different competitors

A_1, A_2, \ldots, A_n

form a weakly-friendly cycle if

A_i

is not a friend of

A_{i+1}

1 \leq i \leq n

A_{n+1} = A_1

), and there are no other pairs of non-friends among the components of the cycle.
The following property is satisfied:
"for every competitor

C

and every weakly-friendly cycle

\mathcal{S}

of competitors not including

C

, the set of competitors

D

\mathcal{S}

which are not friends of

C

has at most one element"
Prove that all competitors of this mathematical competition can be arranged into three rooms, such that every two competitors in the same room are friends.
(Serbia)

1

Functional equation in three variables

S

be the set of positive real numbers. Find all functions

f\colon S^3 \to S

such that, for all positive real numbers

x

y

z

k

, the following three conditions are satisfied:
(a)

xf(x,y,z) = zf(z,y,x)

f(x, ky, k^2z) = kf(x,y,z)

f(1, k, k+1) = k+1

.
(United Kingdom)

1

An exponential Diophantine

Determine all positive integers

x

y

z

x^5 + 4^y = 2013^z

1

Excircles

ABC

\omega_a

A

AB

P

AC

Q

, while the excircle

\omega_b

B

BA

M

BC

N

K

be the projection of

C

MN

L

be the projection of

C

PQ

. Show that the quadrilateral

MKLP

is cyclic.
(Bulgaria)