3. In $\triangle A B C$, $\angle A=60^{\circ}, A B>A C, O$ is the circumcenter, the two altitudes $B E$ and $C F$ intersect at point $H$, points $M$ and $N$ lie on segments $B H$ and $H F$ respectively, and satisfy $B M=C N$. Find the value of $\frac{M H+N H}{O H}$.
(2002, National High School Mathematics Competition)
From the known conditions, we have
$$
\begin{array}{l}
M H+N H=B H-C H \\
=2 R(\cos B-\cos C)=2 \sqrt{3} \sin \frac{C-B}{2},
\end{array}
$$

and \( |\overrightarrow{A H}|=R \) (where \( R \) is the radius of the circumcircle of \( \triangle A B C \)).
From \( \overrightarrow{O H}=\overrightarrow{O A}+\overrightarrow{A H} \), we get
$$
O H=2 R \sin \frac{\angle O A H}{2}=2 R \sin \frac{C-B}{2} .
$$

Therefore, \(\frac{M H+N H}{O H}=\sqrt{3}\).