SMO 2011

In the acute-angled non-isosceles triangle $ABC$ , $O$ is its circumcenter, $H$ is its orthocenter and $AB>AC$ . Let $Q$ be a point on $AC$ such that the extension of $HQ$ meets the extension of $BC$ at the point $P$ . Suppose $BD=DP$ , where $D$ is the foot of the perpendicular from $A$ onto $BC$ . Prove that $\angle ODQ=90^{\circ}$ .

If 46 squares are colored red in a $9\times 9$ board, show that there is a $2\times 2$ block on the board in which at least 3 of the squares are colored red.

Let $x,y,z>0$ such that $\frac1x+\frac1y+\frac1z<\frac{1}{xyz}$ . Show that $$\frac{2x}{\sqrt{1+x^2}}+\frac{2y}{\sqrt{1+y^2}}+\frac{2z}{\sqrt{1+z^2}}<3.$$

Find all polynomials $P(x)$ with real coefficients such that $$P(a)\in\mathbb{Z}\ \ \ \text{implies that}\ \ \ a\in\mathbb{Z}.$$

Find all pairs of positive integers $(m,n)$ such that $$m+n-\frac{3mn}{m+n}=\frac{2011}{3}.$$