OME 2009

Find all the finite sequences with $n$ consecutive natural numbers $a_1, a_2,..., a_n$ , with $n\geq3$ such that $a_1+ a_2+...+ a_n=2009$ .

Let $ABC$ be an acute triangle with the incircle $C(I,r)$ and the circumcircle $C(O,R)$ . Denote $D\in BC$ for which $AD\perp BC$ and $AD = h_a$ . Prove that $DI^2 = (2R - h_a)(h_a - 2r)$ .

Some edges are painted in red. We say that a coloring of this kind is good, if for each vertex of the polyhedron, there exists an edge which concurs in that vertex and is not painted red. Moreover, we say that a coloring where some of the edges of a regular polyhedron is completely good, if in addition to being good, no face of the polyhedron has all its edges painted red. What regular polyhedrons is equal the maximum number of edges that can be painted in a good color and a completely good? Explain your answer.

Find all the integer pairs $(x,y)$ such that: $$x^2-y^4=2009$$

Let, $a,b,c$ real positive numbers with $abc = 1$ Prove:

$(\frac {a}{1 + ab})^2 + (\frac {b}{1 + bc})^2 + (\frac {c}{1 + ca})^2\geq \frac {3}{4}$

Thanks!

Inside a circle of center $O$ and radius $r$ , take two points $A$ and $B$ symmetrical about $O$ . We consider a variable point $P$ on the circle and draw the chord $\overline{PP'}\perp \overline{AP}$ . Let $C$ is the symmetric of $B$ about $\overline{PP'}$ ( $\overline{PP}'$ is the axis of symmetry) . Find the locus of point $Q = \overline{PP'}\cap\overline{AC}$ when we change $P$ in the circle.