CR-MO 2013

Find all functions $f: R\to R$ such that for all real numbers $x, y$ is satisfied that $$f (x + y) = (f (x))^{ 2013} + f (y).$$

Find all functions $f:R -\{0,2\} \to R$ that satisfy for all $x \ne 0,2$ $$f(x) \cdot \left(f\left(\sqrt[3]{\frac{2+x}{2-x}}\right) \right)^2=\frac{x^3}{4}$$

Consider the triangle $ABC$ . Let $P, Q$ inside the angle $A$ such that $\angle BAP=\angle CAQ$ and $PBQC$ is a parallelogram. Show that $\angle ABP=\angle ACP.$

Let $ABCD$ be a rectangle with center $O$ such that $\angle DAC = 60^o$ . Bisector of $\angle DAC$ cuts a $DC$ at $S$ , $OS$ and $AD$ intersect at $L$ , $BL$ and $AC$ intersect at $M$ . Prove that $SM \parallel CL$ .

Find all triples $(a, b, p)$ of positive integers, where $p$ is a prime number, such that $a^p - b^p = 2013$ .