1999 Junior Balkan MO

Part of Junior Balkan MO

Subcontests

(4)

1

Find the greatest comon divisor of the numbers A_1,..,A_1999

For each nonnegative integer

n

A_n = 2^{3n}+3^{6n+2}+5^{6n+2}

. Find the greatest common divisor of the numbers

A_0,A_1,\ldots, A_{1999}

1

1999 points in the interior of a square S

S

be a square with the side length 20 and let

M

be the set of points formed with the vertices of

S

and another 1999 points lying inside

S

. Prove that there exists a triangle with vertices in

M

and with area at most equal with

\frac 1{10}

1

Prove that a + b + c = 0

a,b,c,x,y

be five real numbers such that a^3 \plus{} ax \plus{} y \equal{} 0, b^3 \plus{} bx \plus{} y \equal{} 0 and c^3 \plus{} cx \plus{} y \equal{} 0. If

a,b,c

are all distinct numbers prove that their sum is zero. Ciprus

1

independent of the position of D

ABC

be a triangle with

AB=AC

D\in[BC]

be a point such that

BC>BD>DC>0

\mathcal{C}_1,\mathcal{C}_2

be the circumcircles of the triangles

ABD

ADC

respectively. Let

BB'

CC'

be diameters in the two circles, and let

M

be the midpoint of

B'C'

. Prove that the area of the triangle

MBC

is constant (i.e. it does not depend on the choice of the point

D