2013 IMO Shortlist G4

Let $A B C$ be a triangle with $\angle B>\angle C$. Let $P$ and $Q$ be two different points on line $A C$ such that $\angle P B A=\angle Q B A=\angle A C B$ and $A$ is located between $P$ and $C$. Suppose that there exists an interior point $D$ of segment $B Q$ for which $P D=P B$. Let the ray $A D$ intersect the circle $A B C$ at $R \neq A$. Prove that $Q B=Q R$. (Georgia)

设$A B C$ 是一个三角形，$\angle B>\angle C$。令$P$和$Q$为$A C$线上的两个不同点，使得$\angle P B A=\angle Q B A=\angle A C B$和$A$位于$P$和$C$之间。假设线段$B Q$ 存在一个内点$D$，其中$P D=P B$。让射线$A D$ 与圆$A B C$ 相交于$R \neq A$。证明$Q B=Q R$。 (乔治亚州)