MD-MO 2008

Let $f_m: \mathbb R \to \mathbb R$ , $f_m(x)=(m^2+m+1)x^2-2(m^2+1)x+m^2-m+1,$ where $m \in \mathbb R$ .

1. Find the fixed common point of all this parabolas.

2. Find $m$ such that the distance from that fixed point to $Oy$ is minimal.

Find $f(x): (0,+\infty) \to \mathbb R$ such that $$f(x)\cdot f(y) + f(\frac{2008}{x})\cdot f(\frac{2008}{y})=2f(x\cdot y)$$ and $f(2008)=1$ for $\forall x \in (0,+\infty)$ .

From the vertex $A$ of the equilateral triangle $ABC$ a line is drown that intercepts the segment $[BC]$ in the point $E$ . The point $M \in (AE$ is such that $M$ external to $ABC$ , $\angle AMB = 20 ^\circ$ and $\angle AMC = 30 ^ \circ$ . What is the measure of the angle $\angle MAB$ ?

Let $n$ be a positive integer. Find all $x_1,x_2,\ldots,x_n$ that satisfy the relation: $$\sqrt{x_1-1}+2\cdot \sqrt{x_2-4}+3\cdot \sqrt{x_3-9}+\cdots+n\cdot\sqrt{x_n-n^2}=\frac{1}{2}(x_1+x_2+x_3+\cdots+x_n).$$