SMO 2012 Open Round 2

The incircle with centre $I$ of triangle $ABC$ touches the sides $BC,CA,AB$ at $D,E,F$, respectively. The line $ID$ intersects segment $EF$ at $K$. Prove that $A,K,M$ are collinear, where $M$ is the midpoint of $BC$.

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

For each $i=1,2,\ldots,N$, let $a_i,b_i,c_i$ be integers such that at least one is odd. Show that there exist integers $x,y,z$ such that $xa_i+yb_i+zc_i$ is odd for at least $4N/7$ different values of $i$.

Let $p$ be an odd prime. Prove that
$$1^{p-2}+2^{p-2}+3^{p-2}+\cdots+\left(\frac{p-1}{2}\right)^{p-2}\equiv\frac{2-2^p}{p}\pmod p.$$

There are $2012$ distinct points in the plane, each to be coloured using one of $n$ colours, so that the numbers of points of the respective colours are distinct. A set of $n$ points is called multi-coloured if all their colours are distinct. Determine $n$ that maximizes the number of multi-coloured sets.