2019 Balkan MO

Part of Balkan MO

Subcontests

(4)

1

winning strategy for Balkans (BMO 2019 p4)

A grid consists of all points of the form

(m, n)

m

n

are integers with

|m|\le 2019,|n| \le 2019

|m| +|n| < 4038

. We call the points

(m,n)

of the grid with either

|m| = 2019

|n| = 2019

the boundary points. The four lines

x = \pm 2019

y= \pm 2019

are called boundary lines. Two points in the grid are called neighbours if the distance between them is equal to

1

. Anna and Bob play a game on this grid. Anna starts with a token at the point

(0,0)

. They take turns, with Bob playing first. 1) On each of his turns. Bob deletes at most two boundary points on each boundary line. 2) On each of her turns. Anna makes exactly three steps , where a step consists of moving her token from its current point to any neighbouring point, which has not been deleted. As soon as Anna places her token on some boundary point which has not been deleted, the game is over and Anna wins. Does Anna have a winning strategy?
Proposed by Demetres Christofides, Cyprus

1

Concurrence, Isogonality

ABC

be an acute scalene triangle. Let

X

Y

be two distinct interior points of the segment

BC

\angle{CAX} = \angle{YAB}

. Suppose that:

1)

K

S

are the feet of the perpendiculars from from

B

AX

AY

2)

T

L

are the feet of the perpendiculars from

C

AX

AY

respectively. Prove that

KL

ST

intersect on the line

BC

1

Function with primes

\mathbb{P}

be the set of all prime numbers. Find all functions

f:\mathbb{P}\rightarrow\mathbb{P}

f(p)^{f(q)}+q^p=f(q)^{f(p)}+p^q

p,q\in\mathbb{P}

.
Proposed by Dorlir Ahmeti, Albania

1

Inequality with condition a+b+c = ab+bc+ca (and special equality case)

a,b,c

be real numbers such that

0 \leq a \leq b \leq c

a+b+c=ab+bc+ca >0.

\sqrt{bc}(a+1) \geq 2

and determine the equality cases.
(Edit: Proposed by sir Leonard Giugiuc, Romania)