Bakayev E.B.

In triangle $A B C \quad \angle A=60^{\circ}$, points $M$ and $N$ on sides $A B$ and $A C$ respectively are such that the circumcenter of triangle $A B C$ bisects the segment $M N$. Find the ratio $A N: M B$.

#
Let $P, Q$ be the projections of point $N$ and the center $O$ of the circumcircle onto $AB$ (see figure). Then, by the condition, $MQ = QP$. On the other hand, $Q$ is the midpoint of $AB$, hence $BM = AP = AN / 2$ (since in the right triangle $APN$, $\angle A = 60^\circ$).

![](https://cdn.mathpix.com/cropped/2024_05_06_82cd6cdb4fc29ee56532g-83.jpg?height=666&width=934&top_left_y=-1&top_left_x=562)

## Answer

$2: 1$.