SMO 2011 Open Round 2

In the acute-angled non-isosceles triangle $ABC$, $O$ is its circumcentre, $H$ its orthocentre, and $AB\gt AC$. Let $Q$ be a point on $AC$ such that the extension of $HQ$ meets the extension of $BC$ at $P$. 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 coloured red on a $9\times9$ board, show that there is a $2\times2$ block on the board in which at least three squares are coloured red.

Let $x,y,z\gt 0$ satisfy
$$\frac1x+\frac1y+\frac1z\lt \frac1{xyz}.$$
Show that
$$\frac{2x}{\sqrt{1+x^2}}+\frac{2y}{\sqrt{1+y^2}}+\frac{2z}{\sqrt{1+z^2}}\lt 3.$$

Find all polynomials $P(x)$ with real coefficients such that, for every real $a$,
$$P(a)\in\mathbb Z\quad\Longrightarrow\quad a\in\mathbb Z.$$

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