Question 1
In a scalene triangle $ABC$ with centroid $G$ and circumcircle $\omega$ centred at $O$, the extension of $AG$ meets $\omega$ at $M$, lines $AB$ and $CM$ intersect at $P$, and lines $AC$ and $BM$ intersect at $Q$. Suppose the circumcentre $S$ of the triangle $APQ$ lies on $\omega$ and $A,O,S$ are collinear. Prove that $\angle AGO=90^\circ$.
