On the median $A A_{1}$ of triangle $A B C$, a point $M$ is taken such that $A M: M A_{1}=1: 3$. In what ratio does the line $B M$ divide the side $A C$?
Let's draw the midline $A_{1} K$ of triangle $A C B_{1}$, parallel to $B B_{1}$. Then $B_{1} K=K C$.

By Thales' theorem, $A B_{1}=1 / 3 B_{1} K=1 / 6 B_{1} C$.

## Answer

$1: 6$, measured from point $A$.