UZ-MO 2014

Find all functions $f:R\rightarrow R$ such that $$f(x^3)+f(y^3)=(x+y)(f(x^2)+f(y^2)-f(xy))$$ for all $x,y\in R$ .

For all $x,y,z\in \mathbb{R}\backslash \{1\}$ , such that $xyz=1$ , prove that $$\frac{x^2}{(x-1)^2}+\frac{y^2}{(y-1)^2}+\frac{z^2}{(z-1)^2}\ge 1$$

A circle passes through the points $A,C$ of triangle $ABC$ intersects with the sides $AB,BC$ at points $D,E$ respectively. Let $\frac{BD}{CE}=\frac{3}{2}$ , $BE=4$ , $AD=5$ and $AC=2\sqrt{7}$ . Find the angle $\angle BDC$ .

Let $PA_1A_2...A_{12}$ be the regular pyramid, $A_1A_2...A_{12}$ is regular polygon, $S$ is area of the triangle $PA_1A_5$ and angle between of the planes $A_1A_2...A_{12}$ and $PA_1A_5$ is equal to $\alpha$ . Find the volume of the pyramid.