SILK-ROAD 2004

Find all $f: \mathbb{R} \to \mathbb{R}$ , such that $(x+y)(f(x)-f(y))=(x-y)f(x+y)$ for all real $x,y$ .

find all primes $p$ , for which exist natural numbers, such that $p=m^2+n^2$ and $p|(m^3+n^3-4)$ .

In-circle of $ABC$ with center $I$ touch $AB$ and $AC$ at $P$ and $Q$ respectively. $BI$ and $CI$ intersect $PQ$ at $K$ and $L$ respectively. Prove, that circumcircle of $ILK$ touch incircle of $ABC$ iff $|AB|+|AC|=3|BC|$ .

Natural $n \geq 2$ is given. Group of people calls $n-compact$ , if for any men from group, we can found $n$ people (without he), each two of there are familiar. Find maximum $N$ such that for any $n-compact$ group, consisting $N$ people contains subgroup from $n+1$ people, each of two of there are familiar.