CN-LEAGUE 2006

An ellipse with foci $B_0,B_1$ intersects $AB_i$ at $C_i$ $(i=0,1)$ . Let $P_0$ be a point on ray $AB_0$ . $Q_0$ is a point on ray $C_1B_0$ such that $B_0P_0=B_0Q_0$ ; $P_1$ is on ray $B_1A$ such that $C_1Q_0=C_1P_1$ ; $Q_1$ is on ray $B_1C_0$ such that $B_1P_1=B_1Q_1$ ; $P_2$ is on ray $AB_0$ such that $C_0Q_1=C_0Q_2$ . Prove that $P_0=P_2$ and that the four points $P_0,Q_0,Q_1,P_1$ are concyclic.

Let $x,y$ be real numbers. Define a sequence $\{a_n \}$ through the recursive formula $$a_0=x,a_1=y,a_{n+1}=\frac{a_na_{n-1}+1}{a_n+a_{n-1}},$$ Find $a_n$ .

Solve the system of equations in real numbers: $$\begin{cases} x-y+z-w=2 x^2-y^2+z^2-w^2=6 x^3-y^3+z^3-w^3=20 x^4-y^4+z^4-w^4=66 \end{cases}$$