VMO 2008

Determine the number of solutions of the simultaneous equations $x^2 + y^3 = 29$ and $\log_3 x \cdot \log_2 y = 1.$

Given a triangle with acute angle $\angle BEC,$ let $E$ be the midpoint of $AB.$ Point $M$ is chosen on the opposite ray of $EC$ such that $\angle BME = \angle ECA.$ Denote by $\theta$ the measure of angle $\angle BEC.$ Evaluate $\frac{MC}{AB}$ in terms of $\theta.$

Let $m = 2007^{2008}$ , how many natural numbers n are there such that $n < m$ and $n(2n + 1)(5n + 2)$ is divisible by $m$ (which means that $m \mid n(2n + 1)(5n + 2)$ ) ?

he sequence of real number $(x_n)$ is defined by $x_1 = 0,$ $x_2 = 2$ and $x_{n+2} = 2^{-x_n} + \frac{1}{2}$ $\forall n = 1,2,3 \ldots$ Prove that the sequence has a limit as $n$ approaches $+\infty.$ Determine the limit.

What is the total number of natural numbes divisible by 9 the number of digits of which does not exceed 2008 and at least two of the digits are 9s?

Let $x, y, z$ be distinct non-negative real numbers. Prove that

$$\frac{1}{(x-y)^2} + \frac{1}{(y-z)^2} + \frac{1}{(z-x)^2} \geq \frac{4}{xy + yz + zx}.$$

When does the equality hold?

Let $AD$ is centroid of $ABC$ triangle. Let $(d)$ is the perpendicular line with $AD$ . Let $M$ is a point on $(d)$ . Let $E, F$ are midpoints of $MB, MC$ respectively. The line through point $E$ and perpendicular with $(d)$ meet $AB$ at $P$ . The line through point $F$ and perpendicular with $(d)$ meet $AC$ at $Q$ . Let $(d')$ is a line through point $M$ and perpendicular with $PQ$ . Prove $(d')$ always pass a fixed point.

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