Find all nonegative integers $ x,y,z$ such that $ 12^x+y^4=2008^z$

Triangle $ \triangle ABC$ is given. Points $ D$ i $ E$ are on line $ AB$ such that $ D - A - B - E, AD = AC$ and $ BE = BC$. Bisector of internal angles at $ A$ and $ B$ intersect $ BC,AC$ at $ P$ and $ Q$, and circumcircle of $ ABC$ at $ M$ and $ N$. Line which connects $ A$ with center of circumcircle of $ BME$ and line which connects $ B$ and center of circumcircle of $ AND$ intersect at $ X$. Prove that $ CX \perp PQ$.

Let $ a$, $ b$, $ c$ be positive real numbers such that $ a + b + c = 1$. Prove inequality: \[ \frac{1}{bc + a + \frac{1}{a}} + \frac{1}{ac + b + \frac{1}{b}} + \frac{1}{ab + c + \frac{1}{c}} \leqslant \frac{27}{31}.\]

Each point of a plane is painted in one of three colors. Show that there exists a triangle such that: $ (i)$ all three vertices of the triangle are of the same color; $ (ii)$ the radius of the circumcircle of the triangle is $ 2008$; $ (iii)$ one angle of the triangle is either two or three times greater than one of the other two angles.

The sequence $ (a_n)_{n\ge 1}$ is defined by $ a_1 = 3$, $ a_2 = 11$ and $ a_n = 4a_{n-1}-a_{n-2}$, for $ n \ge 3$. Prove that each term of this sequence is of the form $ a^2 + 2b^2$ for some natural numbers $ a$ and $ b$.

In a convex pentagon $ ABCDE$, let $ \angle EAB = \angle ABC = 120^{\circ}$, $ \angle ADB = 30^{\circ}$ and $ \angle CDE = 60^{\circ}$. Let $ AB = 1$. Prove that the area of the pentagon is less than $ \sqrt {3}$.