SE-SKOLORNAS 2018 Problem 5

In a triangle $ABC$ , two lines are drawn that together trisect the angle at $A$ . These intersect the side $BC$ at points $P$ and $Q$ so that $P$ is closer to $B$ and $Q$ is closer to $C$ . Determine the smallest constant $k$ such that $| P Q | \le k (| BP | + | QC |)$ , for all such triangles. Determine if there are triangles for which equality applies.