As $OQ$ is the reflection of $OP$ in $OM$, $\angle QOM = \angle POM$.

Similarly, $\angle RON = \angle PON$.

Hence reflex $\angle QOR = 2 \times \angle MON = 260^{\circ}$

Therefore $\angle QOR =360^{\circ} - 260^{\circ} = 100^{\circ}$.

This problem is taken from the UKMT Mathematical Challenges.

You can find more short problems, arranged by curriculum topic, in our short problems collection.