VMO 2005

Let $x,y$ be real numbers satisfying the condition: $$x-3\sqrt {x+1}=3\sqrt{y+2} -y$$ Find the greatest value and the smallest value of: $$P=x+y$$

Let $(O)$ be a fixed circle with the radius $R$ . Let $A$ and $B$ be fixed points in $(O)$ such that $A,B,O$ are not collinear. Consider a variable point $C$ lying on $(O)$ ( $C\neq A,B$ ). Construct two circles $(O_1),(O_2)$ passing through $A,B$ and tangent to $BC,AC$ at $C$ , respectively. The circle $(O_1)$ intersects the circle $(O_2)$ in $D$ ( $D\neq C$ ). Prove that: a) $$CD\leq R$$ b) The line $CD$ passes through a point independent of $C$ (i.e. there exists a fixed point on the line $CD$ when $C$ lies on $(O)$ ).

Let $A_1A_2A_3A_4A_5A_6A_7A_8$ be convex 8-gon (no three diagonals concruent). The intersection of arbitrary two diagonals will be called "button".Consider the convex quadrilaterals formed by four vertices of $A_1A_2A_3A_4A_5A_6A_7A_8$ and such convex quadrilaterals will be called "sub quadrilaterals".Find the smallest $n$ satisfying: We can color n "button" such that for all $i,k \in\{1,2,3,4,5,6,7,8\},i\neq k,s(i,k)$ are the same where $s(i,k)$ denote the number of the "sub quadrilaterals" has $A_i,A_k$ be the vertices and the intersection of two its diagonals is "button".

Find all function $f: \mathbb R\to \mathbb R$ satisfying the condition: $$f(f(x - y)) = f(x)\cdot f(y) - f(x) + f(y) - xy$$

Find all triples of natural $(x,y,n)$ satisfying the condition: $$\frac {x! + y!}{n!} = 3^n$$

Define $0! = 1$

Let $\{x_n\}$ be a real sequence defined by: $$x_1=a,x_{n+1}=3x_n^3-7x_n^2+5x_n$$ For all $n=1,2,3...$ and a is a real number. Find all $a$ such that $\{x_n\}$ has finite limit when $n\to +\infty$ and find the finite limit in that cases.