2005 Junior Balkan MO

Part of Junior Balkan MO

Subcontests

(4)

1

\overline{abc} = abc(a+b+c) in decimal base

Find all 3-digit positive integers

\overline{abc}

\overline{abc} = abc(a+b+c) ,

\overline{abc}

is the decimal representation of the number.

1

5 points in the plane so that among all there are 8 right-an

Prove that there exist (a) 5 points in the plane so that among all the triangles with vertices among these points there are 8 right-angled ones; (b) 64 points in the plane so that among all the triangles with vertices among these points there are at least 2005 right-angled ones.

1

tangent from $A$ to the circle meets the line $BC$ at point

ABC

be an acute-angled triangle inscribed in a circle

k

. It is given that the tangent from

A

to the circle meets the line

BC

P

M

be the midpoint of the line segment

AP

R

be the second intersection point of the circle

k

BM

PR

meets again the circle

k

S

R

. Prove that the lines

AP

CS

1

equation in two positive integers

Find all positive integers

x,y

satisfying the equation

9(x^2+y^2+1) + 2(3xy+2) = 2005 .