VMO 2010

Solve the system equations

$$\left\{\begin{array}{cc}x^{4}-y^{4}=240x^{3}-2y^{3}=3(x^{2}-4y^{2})-4(x-8y)\end{array}\right.$$

Let $\{a_{n}\}$ be a sequence which satisfy $a_{1}=5$ and $a_{n=}\sqrt[n]{a_{n-1}^{n-1}+2^{n-1}+2.3^{n-1}} \forall n\geq2$ (a) Find the general fomular for $a_{n}$ (b) Prove that $\{a_{n}\}$ is decreasing sequences

In plane,let a circle $(O)$ and two fixed points $B,C$ lies in $(O)$ such that $BC$ not is the diameter.Consider a point $A$ varies in $(O)$ such that $A\neq B,C$ and $AB\neq AC$ .Call $D$ and $E$ respective is intersect of $BC$ and internal and external bisector of $\widehat{BAC}$ , $I$ is midpoint of $DE$ .The line that pass through orthocenter of $\triangle ABC$ and perpendicular with $AI$ intersects $AD,AE$ respective at $M,N$ .

1/Prove that $MN$ pass through a fixed point

2/Determint the place of $A$ such that $S_{AMN}$ has maxium value

Prove that for each positive integer n,the equation $x^{2}+15y^{2}=4^{n}$ has at least $n$ integer solution $(x,y)$

Let a positive integer $n$ .Consider square table $3*3$ .One use $n$ colors to color all cell of table such that

each cell is colored by exactly one color.

Two colored table is same if we can receive them from other by a rotation through center of $3*3$ table

How many way to color this square table satifies above conditions.