BALKAN 2004

The sequence $\{a_n\}_{n\geq 0}$ of real numbers satisfies the relation: $$a_{m+n} + a_{m-n} - m + n -1 = \frac12 (a_{2m} + a_{2n})$$ for all non-negative integers $m$ and $n$ , $m \ge n$ . If $a_1 = 3$ find $a_{2004}$ .

Solve in prime numbers the equation $x^y - y^x = xy^2 - 19$ .

Let $O$ be an interior point of an acute triangle $ABC$ . The circles with centers the midpoints of its sides and passing through $O$ mutually intersect the second time at the points $K$ , $L$ and $M$ different from $O$ . Prove that $O$ is the incenter of the triangle $KLM$ if and only if $O$ is the circumcenter of the triangle $ABC$ .

The plane is partitioned into regions by a finite number of lines no three of which are concurrent. Two regions are called "neighbors" if the intersection of their boundaries is a segment, or half-line or a line (a point is not a segment). An integer is to be assigned to each region in such a way that:

i) the product of the integers assigned to any two neighbors is less than their sum; ii) for each of the given lines, and each of the half-planes determined by it, the sum of the integers, assigned to all of the regions lying on this half-plane equal to zero.

Prove that this is possible if and only if not all of the lines are parallel.