(3) Let $m, n$ be given integers, $4<m<n, A_{1} A_{2} \cdots A_{2 n+1}$ be a regular $(2 n+1)$-gon, $P=\left\{A_{1}, A_{2}, \cdots, A_{2 n+1}\right\}$. Find the number of convex $m$-gons with vertices in $P$ and exactly two acute internal angles.
First, we prove a lemma: A convex $m$-gon with vertices in $P$ has at most two acute angles, and if there are two acute angles, they must be adjacent.

Indeed, let this convex $m$-gon be $P_{1} P_{2} \cdots P_{m}$. We only consider the case where there is at least one acute angle. Assume $\angle P_{m} P_{1} P_{2} < \frac{\pi}{2}$, and for $3 \leqslant j \leqslant m-1$,
$$
\angle P_{j-1} P_{j} P_{j+1} > \frac{\pi}{2}, 3 \leqslant j \leqslant m-1
$$
Furthermore, $\angle P_{1} P_{2} P_{3} + \angle P_{m-1} P_{m} P_{1} > \pi$, so at most one of them can be an acute angle. This proves the lemma.

By the lemma, if a convex $m$-gon has exactly two acute angles, then the corresponding vertices are adjacent.

In a convex $m$-gon, let vertices $A_{i}$ and $A_{j}$ be two adjacent vertices, and the interior angles at these two vertices are acute. Suppose the minor arc $A_{i} A_{j}$ contains $r$ edges of $P$ $(1 \leqslant r \leqslant n)$, and there are exactly $2n+1$ pairs of $(i, j)$ when $r$ is fixed.
(1) If the remaining $m-2$ vertices of the convex $m$-gon are all on the minor arc $A_{i} A_{j}$, and the minor arc $A_{i} A_{j}$ contains $r-1$ points of $P$, the number of ways to choose these $m-2$ vertices is $\mathrm{C}_{r-1}^{m-2}$.
(2) If the remaining $m-2$ vertices of the convex $m$-gon are all on the major arc $A_{i} A_{j}$, take the antipodal points $B_{i}$ and $B_{j}$ of $A_{i}$ and $A_{j}$, respectively. Since the interior angles at vertices $A_{i}$ and $A_{j}$ are acute, the remaining $m-2$ vertices are all on the minor arc $B_{i} B_{j}$, and the minor arc $B_{i} B_{j}$ contains exactly $r$ points of $P$. The number of ways to choose these $m-2$ vertices is $\mathrm{C}_{r}^{m-2}$.
Therefore, the number of convex $m$-gons that satisfy the given conditions is
$$
\begin{array}{l}
(2 n+1) \sum_{r=1}^{n}\left(\mathrm{C}_{r-1}^{m-2}+\mathrm{C}_{r}^{m-2}\right)=(2 n+1)\left(\sum_{r=1}^{n} \mathrm{C}_{r-1}^{m-2}+\sum_{r=1}^{n} \mathrm{C}_{r}^{m-2}\right)= \\
(2 n+1)\left(\sum_{r=1}^{n}\left(\mathrm{C}_{r}^{m-1}-\mathrm{C}_{r-1}^{m-1}\right)+\sum_{r=1}^{n}\left(\mathrm{C}_{r+1}^{m-1}-\mathrm{C}_{r}^{m-1}\right)\right)= \\
(2 n+1)\left(\mathrm{C}_{n+1}^{m-1}+\mathrm{C}_{n}^{m-1}\right)
\end{array}
$$