$\left[\begin{array}{l}{[\text { Similar auxiliary triangles }} \\ {[\text { Properties and characteristics of a parallelogram }]}\end{array}\right]$

Points $M$ and $N$ are the midpoints of sides $B C$ and $C D$ of parallelogram $A B C D$, respectively. Segments $A M$ and $B N$ intersect at point $O$.

Find the ratio $M O: O A$.
Let the extensions of segments $B N$ and $A D$ intersect at point $E$. Triangle $D N E$ is congruent to triangle $C N B$ by a side and the angles adjacent to it, therefore $D E = B C = 2 B M, A E = 4 a$.

Triangle $B O M$ is similar to triangle $E O A$, hence $M O: O A = 1: 4$.

## Answer

$1: 4$.