1986 Balkan MO

Part of Balkan MO

Subcontests

(4)

1

classic sequence

a,b,c

be real numbers such that

ab\not= 0

c>0

(a_{n})_{n\geq 1}

be the sequence of real numbers defined by:

a_{1}=a, a_{2}=b

a_{n+1}=\frac{a_{n}^{2}+c}{a_{n-1}}

n\geq 2

. Show that all the terms of the sequence are integer numbers if and only if the numbers

a,b

\frac{a^{2}+b^{2}+c}{ab}

1

many points on a sphere :|

ABCD

be a tetrahedron and let

E,F,G,H,K,L

be points lying on the edges

AB,BC,CD

,DA,DB,DC

respectively, in such a way that

AE \cdot BE = BF \cdot CF = CG \cdot AG= DH \cdot AH=DK \cdot BK=DL \cdot CL.

Prove that the points

E,F,G,H,K,L

all lie on a sphere.

1

a line passing through the incenter

A line passing through the incenter

I

of the triangle

ABC

intersect its incircle at

D

E

and its circumcircle at

F

G

, in such a way that the point

D

I

F

DF \cdot EG \geq r^{2}

1

Beautiful geometry problem

ABC

P

a point such that the triangles

PAB, PBC, PCA

have the same area and the same perimeter. Prove that if: a)

P

is in the interior of the triangle

ABC

ABC

is equilateral. b)

P

is in the exterior of the triangle

ABC

ABC

is right angled triangle.