2009 Serbia National Math Olympiad

Part of Serbia National Math Olympiad

Subcontests

(5)

1

Serbian MO 2009

x

y

z

be arbitrary positive numbers such that

xy+yz+zx=x+y+z

\frac{1}{x^2+y+1} + \frac{1}{y^2+z+1} + \frac{1}{z^2+x+1} \leq 1

. When does equality occur? Proposed by Marko Radovanovic

1

Serbian MO 2009

Determine the largest positive integer

n

for which there exist pairwise different sets

\mathbb{S}_1 , ..., \mathbb{S}_n

with the following properties:

1

|\mathbb{S}_i \cup \mathbb{S}_j | \leq 2004

for any two indices

1 \leq i, j\leq n

2

\mathbb{S}_i \cup \mathbb{S}_j \cup \mathbb{S}_k = \{ 1,2,...,2008 \}

1 \leq i < j < k \leq n

Proposed by Ivan Matic

1

Serbian MO 2009

Find the smallest natural number which is a multiple of

2009

and whose sum of (decimal) digits equals

2009

Proposed by Milos Milosavljevic

1

Serbian MO 2009

In a scalene triangle

ABC

\alpha

\beta

respectively denote the interior angles at vertixes

A

B

. The bisectors of these two angles meet the opposite sides of the triangle at points

D

E

, respectively. Prove that the acute angle between the lines

DE

AB

does not exceed

\frac{ | \alpha - \beta |}{3}

. Proposed by Dusan Djukic

1

Kazakhstan olympiad 2009

Triangle ABC has incircle w centered as S that touches the sides BC,CA and AB at P,Q and R respectively. AB isn't equal AC, the lines QR and BC intersects at point M, the circle that passes through points B and C touches the circle w at point N, circumcircle of triangle MNP intersects with line AP at L (L isn't equal to P). Then prove that S,L and M lie on the same line