Question 12
Pyramid $OABCD$ has square base $ABCD,$ congruent edges $\overline{OA},$ $\overline{OB},$ $\overline{OC},$ and $\overline{OD},$ and $\angle AOB=45^\circ.$ Let $\theta$ be the measure of the dihedral angle formed by faces $OAB$ and $OBC.$ Given that $\cos\theta=m+\sqrt n,$ where $m$ and $n$ are integers, find $m+n.$