SMO 2012

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

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

For each $i=1,2,..N$ , let $a_i,b_i,c_i$ be integers such that at least one of them is odd. Show that one can find integers $x,y,z$ such that $xa_i+yb_i+zc_i$ is odd for at least $\frac{4}{7}N$ different values of $i$ .

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

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