3.82 Let point $O$ be an internal point of a convex $n$-sided polygon $A_{1} A_{2} \cdots A_{n}$. Connect $O A_{1}, O A_{2}, \cdots, O A_{n}$. Label the natural numbers from 1 to $n$ in some order on the $n$ sides of the $n$-sided polygon, and do the same for the line segments $O A_{1}, O A_{2}, \cdots, O A_{n}$.
(1) Give a labeling method for $n=9$ such that the sum of the integers labeled on the 3 sides of each triangle $\triangle A_{i} O A_{i+1}$ (i $\left.=1,2, \cdots, 9, A_{10}=A_{1}\right)$ is equal.
(2) Prove that for $n=10$, there does not exist a labeling method that satisfies the requirement in (1).
［Solution] The labeling method in the right figure satisfies the requirements of the problem, where the sum of the numbers labeled on the 3 sides of each triangle is 15.

When $n=10$, the total sum of the numbers labeled on the 30 sides of the 10 triangles is
$$
\begin{aligned}
S & =3(1+2+\cdots+10)=3 \times \frac{1}{2} \times 10 \times 11 \\
& =165 .
\end{aligned}
$$

The sum of the numbers labeled on the 3 sides of each triangle is 16.5, which is not an integer, so it is impossible to achieve.