ZA-SAMO 2012

Sixty points, of which thirty are coloured red, twenty are coloured blue and ten are coloured green, are marked on a circle. These points divide the circle into sixty arcs. Each of these arcs is assigned a number according to the colours of its endpoints: an arc between a red and a green point is assigned a number $1$ , an arc between a red and a blue point is assigned a number $2$ , and an arc between a blue and a green point is assigned a number $3$ . The arcs between two points of the same colour are assigned a number $0$ . What is the greatest possible sum of all the numbers assigned to the arcs?

Let $p$ and $k$ be positive integers such that $p$ is prime and $k>1$ . Prove that there is at most one pair $(x,y)$ of positive integers such that $x^k+px=y^k$ .

Let $ABC$ be a triangle such that $AB\neq AC$ . We denote its orthocentre by $H$ , its circumcentre by $O$ and the midpoint of $BC$ by $D$ . The extensions of $HD$ and $AO$ meet in $P$ . Prove that triangles $AHP$ and $ABC$ have the same centroid.

Find all functions $f:\mathbb{N}\to\mathbb{R}$ such that $f(km)+f(kn)-f(k)f(mn)\ge 1$ for all $k,m,n\in\mathbb{N}$ .