2018 Balkan MO

Part of Balkan MO

Subcontests

(4)

1

Balkan Mathematical Olympiad 2018 p2

q

be a positive rational number. Two ants are initially at the same point

X

in the plane. In the

n

(n = 1,2,...)

each of them chooses whether to walk due north, east, south or west and then walks the distance of

q^n

metres. After a whole number of minutes, they are at the same point in the plane (not necessarily

X

), but have not taken exactly the same route within that time. Determine all possible values of

q

.
Proposed by Jeremy King, UK

1

Balkan Mathematical Olympiad 2018 p3

Alice and Bob play the following game: They start with non-empty piles of coins. Taking turns, with Alice playing first, each player choose a pile with an even number of coins and moves half of the coins of this pile to the other pile. The game ends if a player cannot move, in which case the other player wins.
Determine all pairs

(a,b)

of positive integers such that if initially the two piles have

a

b

coins respectively, then Bob has a winning strategy.
Proposed by Dimitris Christophides, Cyprus

1

Balkan Mathematical Olympiad 2018 P4

Find all primes

p

q

3p^{q-1}+1

11^p+17^p

Proposed by Stanislav Dimitrov,Bulgaria

1

Balkan Mathematical Olympiad 2018 P1

A quadrilateral

ABCD

is inscribed in a circle

k

AB

>

CD

AB

is not paralel to

CD

M

is the intersection of diagonals

AC

BD

, and the perpendicular from

M

AB

intersects the segment

AB

E

EM

bisects the angle

CED

AB

k

. Proposed by Emil Stoyanov,Bulgaria