2009 JBMO TST - Macedonia

Part of JBMO TST - Macedonia

Subcontests

(4)

1

Macedonian JBMO TST 2009 - Problem 4

1\times1

cell of a rectangle board a natural number is written. In one step it is allowed the numbers written in every cell of arbitrary chosen row, to be doubled, or the numbers written in the cells of the arbitrary chosen column to be decreased by 1. Will after final number of steps all the numbers on the board be

0

1

Macedonian JBMO TST 2009 - Problem 3

\triangle ABC

be equilateral. On the side

AB

C_{1}

C_{2}

AC

B_{1}

B_{2}

are chosen, and on the side

BC

A_{1}

A_{2}

are chosen. The following condition is given :

A_{1}A_{2}

B_{1}B_{2}

C_{1}C_{2}

. Let the intersection lines

A_{2}B_{1}

B_{2}C_{1}

B_{2}C_{1}

C_{2}A_{1}

C_{2}A_{1}

A_{2}B_{1}

E

F

G

respectively. Show that the triangle formed by

B_{1}A_{2}

A_{1}C_{2}

C_{1}B_{2}

\triangle EFG

1

Macedonian JBMO TST 2009 - Problem 2

a

b

be integer numbers. Let

a = a^{2}+b^{2}-8b-2ab+16

a

is a perfect square.

1

Macedonian JBMO TST 2009 - Problem 1

On a board, the numbers from 1 to 2009 are written. A couple of them are erased and instead of them, on the board is written the remainder of the sum of the erased numbers divided by 13. After a couple of repetition of this erasing, only 3 numbers are left, of which two are 9 and 999. Find the third number.