3. As shown in Figure 1, in $\triangle A B C$, $O$ is the midpoint of side $B C$, and a line through $O$ intersects lines $A B$ and $A C$ at two distinct points $M$ and $N$ respectively. If
$$
\begin{array}{l}
\overrightarrow{A B}=m \overrightarrow{A M}, \\
\overrightarrow{A C}=n \overrightarrow{A N},
\end{array}
$$

then $m+n=$
3. 2 .

Solution 1 Notice that,
$$
\overrightarrow{A O}=\frac{1}{2}(\overrightarrow{A B}+\overrightarrow{A C})=\frac{m}{2} \overrightarrow{A M}+\frac{n}{2} \overrightarrow{A N} .
$$

Since points $M, O, N$ are collinear, we have,
$$
\frac{m}{2}+\frac{n}{2}=1 \Rightarrow m+n=2 \text {. }
$$

Solution 2 Since points $M, O, N$ are on the sides or their extensions of $\triangle A B C$, and the three points are collinear, by Menelaus' theorem we get
$$
\begin{array}{l}
\frac{A M}{M B} \cdot \frac{B O}{O C} \cdot \frac{C N}{N A}=1 . \\
\text { Hence } \frac{n-1}{1-m}=1 \Rightarrow m+n=2 .
\end{array}
$$