Problem 2. (4 points) In triangle $ABC$, point $M$ is the midpoint of side $AC$, and point $P$ lies on side $BC$. Segment $AP$ intersects $BM$ at point $O$. If $BO = BP$, find the ratio $OM: PC$.

Answer: 1:2.
Solution. Let's construct the midsegment $M N$ in triangle $A C P$ (parallel to $A P$). By Thales' theorem, $O M / P N = B O / B P$, therefore $O M = P N = 1 / 2 P C$.