VMO 2005 — Division A

Let $x,y$ be real numbers satisfying
$$x-3\sqrt{x+1}=3\sqrt{y+2}-y.$$
Determine the maximum and minimum of $P=x+y$.

Let $(O)$ be a fixed circle of radius $R$. Fixed points $A,B$ on $(O)$ satisfy that $A,B,O$ are not collinear. Let $C$ vary on $(O)$, with $C\ne A,B$. Construct circles $(O_1),(O_2)$ passing through $A,B$ and tangent at $C$ to $BC,AC$, respectively. The circles meet again at $D\ne C$. Prove that:
(a) $CD\le R$;
(b) line $CD$ always passes through a point independent of $C$.

Let $A_1A_2A_3A_4A_5A_6A_7A_8$ be a convex octagon such that no three diagonals are concurrent. The intersection of any two diagonals is called a “node.” Consider all convex quadrilaterals formed by four vertices of the octagon, called “subquadrilaterals.” Determine the smallest positive integer $n$ such that:
We can color $n$ “nodes” so that for all distinct $i,k\in\{1,2,3,4,5,6,7,8\}$, the numbers $s(i,k)$ are equal, where $s(i,k)$ denotes the number of “subquadrilaterals” having $A_i,A_k$ as vertices and whose diagonal intersection is a node.

Find all functions $f:\mathbb R\to\mathbb R$ satisfying
$$f(f(x-y))=f(x)f(y)-f(x)+f(y)-xy$$
for all $x,y\in\mathbb R$.

Find all triples $(x,y,n)$ of positive integers satisfying
$$\frac{x!+y!}{n!}=3^n.$$

Let $\{x_n\}$ be the real sequence defined by
$$x_1=a,\qquad x_{n+1}=3x_n^3-7x_n^2+5x_n\quad(n\in\mathbb N).$$
Determine all real $a$ for which the sequence has a finite limit as $n\to+\infty$, and determine the limit in each case.