For any given $n \geqslant 3(n \in$ $\left.\mathbf{Z}_{+}\right)$, take $n$ points $A_{1}, A_{2}, \cdots$, $A_{n}$ on the circumference of $\odot O$. Among the $\mathrm{C}_{n}^{2}$ angles $\angle A_{i} O A_{j}(1 \leqslant i<j \leqslant n)$, there are $S(n)$ angles that are not greater than $120^{\circ}$. Find the minimum value of $S(n)$.
If $\angle A_{i} O A_{j}>120^{\circ}(1 \leqslant i<j \leqslant n)$, then a line is drawn between points $A_{i}$ and $A_{j}$; otherwise, no line is drawn. Thus, a graph $G$ is obtained. It is easy to see that $G$ contains no triangles.

Let the number of edges in $G$ be $m$, and any three points on the circle are not collinear.
(1) When $n$ is an odd number greater than 1, the proof is the same as in problem 2 (1).

Thus, among these $\mathrm{C}_{n}^{2}$ angles, the number of angles not greater than $120^{\circ}$ is at least $\frac{(n-1)^{2}}{4}$.

Therefore, when $n$ is odd, the minimum value of $S(n)$ is $\frac{(n-1)^{2}}{4}$.
(2) When $n$ is an even number greater than 2, the proof is the same as in problem 2 (2).

Thus, among these $\mathrm{C}_{n}^{2}$ angles, the number of angles not greater than $120^{\circ}$ is at least $\frac{n^{2}}{4}-\frac{n}{2}$.

Therefore, when $n$ is even, the minimum value of $S(n)$ is $\frac{n^{2}}{4}-\frac{n}{2}$.