Question 1
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 $\angle ODQ=90^\circ$.