1988 Balkan MO

Part of Balkan MO

Subcontests

(4)

1

interesting sequence

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

be a sequence defined by

a_{n}=2^{n}+49

. Find all values of

n

a_{n}=pg, a_{n+1}=rs

p,q,r,s

are prime numbers with

p<q, r<s

q-p=s-r

1

tetrahedron included in a region

ABCD

be a tetrahedron and let

d

be the sum of squares of its edges' lengths. Prove that the tetrahedron can be included in a region bounded by two parallel planes, the distances between the planes being at most

\frac{\sqrt{d}}{2\sqrt{3}}

1

find all polynomials

Find all polynomials of two variables

P(x,y)

P(a,b) P(c,d) = P (ac+bd, ad+bc), \forall a,b,c,d \in \mathbb{R}.

1

another "strange" geometry problem of Bulgaria, but nice

ABC

be a triangle and let

M,N,P

be points on the line

BC

AM,AN,AP

are the altitude, the angle bisector and the median of the triangle, respectively. It is known that

\frac{[AMP]}{[ABC]}=\frac{1}{4}

\frac{[ANP]}{[ABC]}=1-\frac{\sqrt{3}}{2}

. Find the angles of triangle

ABC