KZ-MO 2003

Find all natural numbers $n$ ,such that there exist $x_1,x_2,\dots,x_{n+1}\in\mathbb{N}$ ,such that $\frac{1}{x_1^2}+\frac{1}{x_2^2}+\dots+\frac{1}{x_n^2}=\frac{n+1}{x_{n+1}^2}$ .

Two square sheets have areas equal to $2003$ . Each of the sheets is arbitrarily divided into $2003$ nonoverlapping polygons, besides, each of the polygons has an unitary area. Afterward, one overlays two sheets, and it is asked to prove that the obtained double layer can be punctured $2003$ times, so that each of the $4006$ polygons gets punctured precisely once.

Let the inscribed circle $\omega$ of triangle $ABC$ touch the side $BC$ at the point $A '$ . Let $AA '$ intersect $\omega$ at $P \neq A$ . Let $CP$ and $BP$ intersect $\omega$ , respectively, at points $N$ and $M$ other than $P$ . Prove that $AA ', BN$ and $CM$ intersect at one point.

Let the point $B$ lie on the circle $S_1$ and let the point $A$ , other than the point $B$ , lie on the tangent to the circle $S_1$ passing through the point $B$ . Let a point $C$ be chosen outside the circle $S_1$ , so that the segment $AC$ intersects $S_1$ at two different points. Let the circle $S_2$ touch the line $AC$ at the point $C$ and the circle $S_1$ at the point $D$ , on the opposite side from the point $B$ with respect to the line $AC$ . Prove that the center of the circumcircle of triangle $BCD$ lies on the circumcircle of triangle $ABC$ .

Determine all functions $f: \mathbb R \to \mathbb R$ with the property $$f(f(x)+y)=2x+f(f(y)-x), \forall x,y \in \mathbb R.$$