2007 Balkan MO

Part of Balkan MO

Subcontests

(4)

1

Convex polygons

For a given positive integer

n >2

C_{1},C_{2},C_{3}

be the boundaries of three convex

n-

gons in the plane , such that

C_{1}\cap C_{2}, C_{2}\cap C_{3},C_{1}\cap C_{3}

are finite. Find the maximum number of points of the sets

C_{1}\cap C_{2}\cap C_{3}

1

Quadrilateral and sum of angles

ABCD

a convex quadrilateral with

AB=BC=CD

AC

BD

E

be the intersection point of it's diagonals. Prove that

AE=DE

\angle BAD+\angle ADC = 120

1

Many square roots, but rational !

Find all positive integers

n

such that there exist a permutation

\sigma

\{1,2,3, \ldots, n\}

\sqrt{\sigma(1)+\sqrt{\sigma(2)+\sqrt{\ldots+\sqrt{\sigma(n-1)+\sqrt{\sigma(n)}}}}}

is a rational number.

1

Find function f

Find all real functions

f

\mathbb R

f(f(x)+y) = f(f(x)-y)+4f(x)y ,

for all real numbers

x,y