Find the product $ \cos a \cdot \cos 2a\cdot \cos 3a \cdots \cos 1006a$ where $a=\frac{2\pi}{2013}$.

Sequence $x_1 , x_2 , ..., $ with $x_1=20$ ; $x_2=12$ for all $n\geq 1$ such that $x_{n+2}=x_n+x_{n+1}+2\sqrt{x_{n}*x_{n+1}+121} $then prove that $x_{2013}$ is an integer number.

If a,b,c positive numbers and such that $a+\sqrt{b+\sqrt{c}}=c+\sqrt{b+\sqrt{a}}$. Prove that if $a\neq c$ then $40ac<1$.

Let $ ABCD$ be a convex quadrilateral such that the sides $ AB, AD, BC$ satisfy $ AB = AD + BC.$ There exists a point $ P$ inside the quadrilateral at a distance $ h$ from the line $ CD$ such that $ AP = h + AD$ and $ BP = h + BC.$ Show that: \[ \frac {1}{\sqrt {h}} \geq \frac {1}{\sqrt {AD}} + \frac {1}{\sqrt {BC}} \]