2002 Balkan MO

Part of Balkan MO

Subcontests

(4)

1

Integer functions fulfilling a double inequality

Determine all functions

f: \mathbb N\to \mathbb N

such that for every positive integer

n

2n+2001\leq f(f(n))+f(n)\leq 2n+2002.

1

4 orthocenters form a rectangle

Two circles with different radii intersect in two points

A

B

. Let the common tangents of the two circles be

MN

ST

M,S

lie on the first circle, and

N,T

on the second. Prove that the orthocenters of the triangles

AMN

AST

BMN

BST

are the four vertices of a rectangle.

1

Graph with even cycles

n

A_1,A_2,A_3,\ldots, A_n

n\geq 4

) in the plane, such that any three are not collinear. Some pairs of distinct points among

A_1,A_2,A_3,\ldots, A_n

are connected by segments, such that every point is connected with at least three different points. Prove that there exists

k>1

and the distinct points

X_1,X_2,\ldots, X_{2k}

\{A_1,A_2,A_3,\ldots, A_n\}

, such that for every

i\in \overline{1,2k-1}

X_i

is connected with

X_{i+1}

X_{2k}

is connected with

X_1

1

Let sequence

Let the sequence

\{a_n\}_{n\geq 1}

be defined by a_1 \equal{} 20, a_2 \equal{} 30 and a_{n \plus{} 2} \equal{} 3a_{n \plus{} 1} \minus{} a_n for all

n\geq 1

. Find all positive integers

n

such that 1 \plus{} 5a_n a_{n \plus{} 1} is a perfect square.