2. Given that $\triangle A B C$ is an equilateral triangle, $M$ and $N$ are the midpoints of sides $B C$ and $B A$ respectively, $O$ is the circumcenter of $\triangle B M N$, and there is a point $D$ on side $A C$ such that the area of $\triangle A O D$ is $\frac{1}{n}$ of the area of $\triangle A B C$ $\left(n>1, n \in \mathbf{N}_{+}\right)$. Then the value of $\frac{A D}{D C}$ is $\qquad$
2. $\frac{3}{2 n-3}$.

Since the distance from point $O$ to $AC$ is equal to $\frac{2}{3}$ of the distance from point $B$ to $AC$, we have:
$$
\begin{array}{l}
\frac{S_{\triangle A O D}}{S_{\triangle A B C}}=\frac{\frac{1}{2} A D \cdot \frac{2}{3}}{\frac{1}{2} A C \cdot 1}=\frac{1}{n} \Rightarrow \frac{A D}{A C}=\frac{3}{2 n} \\
\Rightarrow \frac{A D}{D C}=\frac{3}{2 n-3} .
\end{array}
$$