VMO 2008 — National examination

Determine the number of real solutions $(x,y)$ of the system
$$\begin{cases}x^2+y^3=29,\\[3pt]\log_3x\cdot\log_2y=1.\end{cases}$$

In triangle $ABC$, let $E$ be the midpoint of $AB$ and suppose $\angle BEC$ is acute. On ray $EC$, take $M$ such that $\angle BME=\angle ECA$. If $\alpha=\angle BEC$, express $MC/AB$ in terms of $\alpha$.

Let $m=2007^{2008}$. How many natural numbers $n$ satisfy $n\lt m$ and $m\mid n(2n+1)(5n+2)$?

Let the real sequence $(x_n)$ be defined by $x_1=0$, $x_2=2$ and
$$x_{n+2}=2^{-x_n}+\frac12\qquad(n=1,2,3,\ldots).$$
Prove that it converges to a finite limit as $n\to+\infty$, and find the limit.

How many natural numbers are divisible by $9$, have at most $2008$ decimal digits, and contain at least two digits equal to $9$?

Let $x,y,z$ be pairwise distinct nonnegative real numbers. Prove
$$(xy+yz+zx)\left(\frac1{(x-y)^2}+\frac1{(y-z)^2}+\frac1{(z-x)^2}\right)\ge4.$$
When does equality occur?

Let $ABC$ be a triangle with median $AD$. Let a line $d$ be perpendicular to $AD$, and let $M$ vary on $d$. Let $E,F$ be the midpoints of $MB,MC$, respectively. The line through $E$ perpendicular to $d$ meets $AB$ at $P$, and the line through $F$ perpendicular to $d$ meets $AC$ at $Q$. Prove that the line through $M$ perpendicular to $PQ$ always passes through a fixed point as $M$ varies on $d$.