ZA-SAMO 2011

An airline company is planning to introduce a network of connections between the ten different airports of Sawubonia. The airports are ranked by priority from first to last (with no ties). We call such a network feasible if it satisfies the following conditions:

• All connections operate in both directions
• If there is a direct connection between two airports A and B, and C has higher priority than B, then there must also be a direct connection between A and C.

Some of the airports may not be served, and even the empty network (no connections at all) is allowed. How many feasible networks are there?

Let $\mathbb{N}_0$ denote the set of all nonnegative integers. Determine all functions $f:\mathbb{N}_0\to\mathbb{N}_0$ with the following two properties:

• $0\le f(x)\le x^2$ for all $x\in\mathbb{N}_0$
• $x-y$ divides $f(x)-f(y)$ for all $x,y\in\mathbb{N}_0$ with $x>y$

In triangle $ABC$ , the incircle touches $BC$ in $D$ , $CA$ in $E$ and $AB$ in $F$ . The bisector of $\angle BAC$ intersects $BC$ in $G$ . The lines $BE$ and $CF$ intersect in $J$ . The line through $J$ perpendicular to $EF$ intersects $BC$ in $K$ . Prove that $\frac{GK}{DK}=\frac{AE}{CE}+\frac{AF}{BF}$