VMO 2004 — Division A

Solve the system
$$\begin{cases}x^3+x(y-z)^2=2,\\ y^3+y(z-x)^2=30,\\ z^3+z(x-y)^2=16.\end{cases}$$

Let $ABC$ be a triangle in the plane, and let $D$ be the intersection of segment $AB$ with the internal bisector of $\angle ACB$. A variable circle $(O)$ always passes through $C,D$ and is not tangent to either line $BC$ or $CA$. It meets $BC,CA$ again at $M,N$, respectively.
(a) Prove that there exists a circle $(S)$ tangent to lines $DM,DN$ at $M,N$, respectively.
(b) Circle $(S)$ meets lines $BC,AC$ at $P,Q$, respectively. Prove that the lengths $MP,NQ$ are constant as $(O)$ varies.

Let $A$ be the set of the first 16 positive integers, $A=\{1,2,\ldots,16\}$. Determine the smallest positive integer $k$ such that every $k$-element subset of $A$ contains two elements $a,b$ for which $a^2+b^2$ is prime.

Consider the real sequence $\{x_n\}_{n=1}^{+\infty}$ defined by
$$x_1=1,\qquad x_{n+1}=\frac{(2+2\cos2\alpha)x_n+\cos^2\alpha}{(2-2\cos2\alpha)x_n+2-\cos2\alpha}\quad(n=1,2,\ldots),$$
where $\alpha$ is a real parameter. Determine all possible values of $\alpha$ such that the sequence $\{y_n\}_{n=1}^{+\infty}$ with
$$y_n=\sum_{k=1}^{n}\frac1{1+2x_n}$$
has a finite limit as $n$ tends to infinity, and determine those limits.

Positive real numbers $x,y,z$ satisfy
$$(x+y+z)^3=32xyz.$$
Determine the maximum and minimum of
$$P=\frac{x^4+y^4+z^4}{(x+y+z)^4}.$$

For each positive integer $n$, let $S(n)$ be the sum of all its decimal digits. Among all positive integers $m$ divisible by $2003$, find the smallest possible value of $S(m)$.