2015 IMO Shortlist G4

Let $A B C$ be an acute triangle, and let $M$ be the midpoint of $A C$. A circle $\omega$ passing through $B$ and $M$ meets the sides $A B$ and $B C$ again at $P$ and $Q$, respectively. Let $T$ be the point such that the quadrilateral $B P T Q$ is a parallelogram. Suppose that $T$ lies on the circumcircle of the triangle $A B C$. Determine all possible values of $B T / B M$. (Russia)

设$A B C$ 为锐角三角形，$M$ 为$A C$ 的中点。穿过$B$和$M$的圆$\omega$再次分别在$P$和$Q$处与边$A B$和$B C$相交。令 $T$ 为四边形 $B P T Q$ 为平行四边形的点。假设 $T$ 位于三角形 $A B C$ 的外接圆上。确定 $B T / B M$ 的所有可能值。 (俄罗斯)