JBMO-SL 2013 G5

A circle passing through the midpoint $M$ of the side $B C$ and the vertex $A$ of a triangle $A B C$ intersects the sides $A B$ and $A C$ for the second time at the points $P$ and $Q$, respectively. Show that if $\angle B A C=60^{\circ}$ then

$$A P+A Q+P Q<A B+A C+\frac{1}{2} B C$$