PMO 2011

Find all nonempty finite sets $X$ of real numbers such that for all $x\in X$ , $x+|x| \in X$ .

In triangle $ABC$ , let $X$ and $Y$ be the midpoints of $AB$ and $AC$ , respectively. On segment $BC$ , there is a point $D$ , different from its midpoint, such that $\angle{XDY}=\angle{BAC}$ . Prove that $AD\perp BC$ .

The $2011$ th prime number is $17483$ and the next prime is $17489$ . Does there exist a sequence of $2011^{2011}$ consecutive positive integers that contain exactly $2011$ prime numbers?

Find all (if there is one) functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x\in\mathbb{R}$ , $$f(f(x))+xf(x)=1.$$

The chromatic number $\chi$ of an (infinite) plane is the smallest number of colors with which we can color the points on the plane in such a way that no two points of the same color are one unit apart. Prove that $4 \leq \chi \leq 7$ .