Find the largest positive integer $n(n \geqslant 3)$, such that there exists a convex $n$-gon, where the tangent values of all its interior angles are integers.
(Proposed by the Problem Committee)
On the one hand, since each interior angle of a regular octagon is $135^{\circ}$, and its tangent value is -1, $n=8$ satisfies the condition.

On the other hand, if $n \geqslant 9$, let the exterior angles of the $n$-sided polygon be $\angle A_{1}, \angle A_{2}, \cdots, \angle A_{n}\left(0<\angle A_{1} \leqslant \angle A_{2}\right.$ $\left.\leqslant \cdots \leqslant \angle A_{n}\right)$. Then
$$
\begin{array}{l}
\angle A_{1}+\angle A_{2}+\cdots+\angle A_{n}=2 \pi \\
\Rightarrow 0<\angle A_{1} \leqslant \frac{2 \pi}{n}<\frac{\pi}{4} \Rightarrow 0<\tan A_{1}<1 .
\end{array}
$$

Thus, the tangent value of the corresponding interior angle of $\angle A_{1}$ is not an integer. Therefore, when $n \geqslant 9$, there is no polygon that satisfies the condition. In summary, the maximum value of $n$ is 8.