Post Number:#2 by Thanic Nur Samin » Wed Jan 11, 2017 6:46 pm
Cute problem.
Let $E$ be the midpoint of $AB$. Let $Y$ be the intersection of $AD$ and $EM$. Clearly, $\angle DXM=\angle DYM=90^{\circ}$, and so $DYXM$ is cyclic.
Again, $\angle ADB=\angle AXB=90^{\circ}$. So, $ABDX$ is cyclic.
Now, $\angle XME=\angle XMY=\angle XDY=\angle XDA=\angle XBA=\angle XBE$. So, $BMXE$ is cyclic.
Also, since $\triangle AXB$ is a right traingle, $E$ is the circumcenter. So, $EB=EX$. But since $BMXE$ is cyclic, this implies $\angle XME=\angle EMB=\angle MBC$. So, $\angle XMB=\angle XME+\angle EMB=2\angle MBC$.
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