SAUDI ARABIAN MATHEMATICAL COMPETITIONS

Denote $N$ as the midpoint of $AB$. Because $A$, $B$, $C$, $D$ belong to the same circle, then we have $$△IAB\sim △IDC.$$ Since $M$ is the midpoint of $CD$ and $N$ is the midpoint of $AB$ then $IM$, $IN$ are isogonal conjugate with respect to the angle $\angle CID$.

In the other hand, $O$ is the intersection of two tangent lines of $(O_2)$ at $A$, $B$, then $IO$ is the symmedian of triangle $IAB$. It means $IO$, $IN$ are isogonal conjugate with respect to $\angle AIB$.

Hence, $I$, $M$, $O$ are collinear.