BALKAN 2003

Can one find 4004 positive integers such that the sum of any 2003 of them is not divisible by 2003?

Let $ABC$ be a triangle, and let the tangent to the circumcircle of the triangle $ABC$ at $A$ meet the line $BC$ at $D$ . The perpendicular to $BC$ at $B$ meets the perpendicular bisector of $AB$ at $E$ . The perpendicular to $BC$ at $C$ meets the perpendicular bisector of $AC$ at $F$ . Prove that the points $D$ , $E$ and $F$ are collinear.

Valentin Vornicu

Find all functions $f: \mathbb{Q}\to\mathbb{R}$ which fulfill the following conditions:

a) $f(1)+1>0$ ;

b) $f(x+y) -xf(y) -yf(x) = f(x)f(y) -x-y +xy$ , for all $x,y\in\mathbb{Q}$ ;

c) $f(x) = 2f(x+1) +x+2$ , for every $x\in\mathbb{Q}$ .

A rectangle $ABCD$ has side lengths $AB = m$ , $AD = n$ , with $m$ and $n$ relatively prime and both odd. It is divided into unit squares and the diagonal AC intersects the sides of the unit squares at the points $A_1 = A, A_2, A_3, \ldots , A_k = C$ . Show that $$A_1A_2 - A_2A_3 + A_3A_4 - \cdots + A_{k-1}A_k = {\sqrt{m^2+n^2}\over mn}.$$