2018 Serbia JBMO TST

Part of Serbia JBMO TST

Subcontests

(4)

1

Game on the board

Two players are playing the following game. They are alternatively putting blue and red coins on the board

2018

2018

. If first player creates

n

blue coins in a row or column, he wins. Second player wins if he can prevent it. Who will win if:

a)n=4

b)n=5

? Note: first player puts only blue coins, and second only red.

1

Exponential diophantine equation

Solve the equation in positive integers:

2^x-3^y 5^z=1009

1

Inequality

a,b,c > 0

the following inequality holds:

\frac{\sqrt{ab}}{a+b+2c}+\frac{\sqrt{bc}}{b+c+2a}+\frac{\sqrt{ca}}{c+a+2b} \le \frac {3}{4}

1

Three collinear points

AD

be an internal angle bisector in triangle

\Delta ABC

. An arbitrary point

M

is chosen on the closed segment

AD

. A parallel to

BC

M

AB

N

AD, CM

cut circumcircle of

\Delta ABC

K, L

, respectively. Prove that

K,N,L