VMO 2009

**Problem 1.**Find all $(x,y)$ such that: $$\{\begin{matrix} \dfrac {1}{\sqrt {1 + 2x^2}} + \dfrac {1}{\sqrt {1 + 2y^2}} & = & \dfrac {2}{\sqrt {1 + 2xy}} \sqrt {x(1 - 2x)} + \sqrt {y(1 - 2y)} & = & \dfrac {2}{9} \end{matrix}$$

The sequence $\{x_n\}$ is defined by $$\left\{ \begin{array}{l}x_1 = \frac{1}{2} x_n = \frac{{\sqrt {x_{n - 1} ^2 + 4x_{n - 1} } + x_{n - 1} }}{2} \end{array} \right.$$ Prove that the sequence $\{y_n\}$ , where $y_n=\sum_{i=1}^{n}\frac{1}{{{x}_{i}}^{2}}$ , has a finite limit and find that limit.

Let $A$ , $B$ be two fixed points and $C$ is a variable point on the plane such that $\angle ACB=\alpha$ (constant) ( $0^{\circ}\le \alpha\le 180^{\circ}$ ). Let $D$ , $E$ , $F$ be the projections of the incenter $I$ of triangle $ABC$ to its sides $BC$ , $CA$ , $AB$ , respectively. Denoted by $M$ , $N$ the intersections of $AI$ , $BI$ with $EF$ , respectively. Prove that the length of the segment $MN$ is constant and the circumcircle of triangle $DMN$ always passes through a fixed point.

Let $a$ , $b$ , $c$ be three real numbers. For each positive integer number $n$ , $a^n + b^n + c^n$ is an integer number. Prove that there exist three integers $p$ , $q$ , $r$ such that $a$ , $b$ , $c$ are the roots of the equation $x^3 + px^2 + qx + r = 0$ .

Let $S =\{1,2,3, \ldots, 2n\}$ ( $n \in \mathbb{Z}^+$ ). Ddetermine the number of subsets $T$ of $S$ such that there are no 2 element in $T$ $a,b$ such that $|a-b|=\{1,n\}$