3. Let $m, n (4 < m < n)$ be given integers, $A_{1} A_{2} \cdots A_{2 n+1}$ be a regular $(2 n+1)$-gon, and $P=\left\{A_{1}, A_{2}, \cdots, A_{2 n+1}\right\}$. Find the number of convex $m$-gons with vertices in $P$ and exactly two acute interior angles.
（Leng Gangsong, problem contributor）
3. First, prove a lemma.

Lemma: A convex $m$-gon with vertices in $P$ has at most two acute angles; and if there are two acute angles, these two angles must be adjacent.

Proof of the lemma: Let the convex $m$-gon be $P_{1} P_{2} \cdots P_{m}$. We only consider the case where there is at least one acute angle. Without loss of generality, assume $\angle P_{m} P_{1} P_{2} < \frac{\pi}{2}$. If $\angle P_{1} P_{2} P_{3} < \frac{\pi}{2}$, then $\angle P_{2} P_{3} P_{4} > \frac{\pi}{2}$, and so on, we have $\angle P_{j-1} P_{j} P_{j+1} > \frac{\pi}{2} (3 \leqslant j \leqslant m-1)$.
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Moreover, $\angle P_{j-1} P_{j} P_{j+1} > \frac{\pi}{2} (3 \leqslant j \leqslant m-1)$. And $\angle P_{1} P_{2} P_{3} + \angle P_{m-1} P_{m} P_{1} > \pi$, so at most one of them is an acute angle.
Returning to the original problem.
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 $A_{i}$ and $A_{j}$ be two adjacent vertices, and the internal angles at these two vertices are acute. Suppose the minor arc $\overparen{A_{i} A_{j}}$ 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 $\overparen{A_{i} A_{j}}$, and there are $r-1$ points of $P$ on $\overparen{A_{i} A_{j}}$, then 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 $\overparen{A_{i} A_{j}}$, take the antipodal points $B_{i}$ and $B_{j}$ of $A_{i}$ and $A_{j}$, respectively. Since the internal angles at $A_{i}$ and $A_{j}$ are acute, the remaining $m-2$ vertices are all on the minor arc $\overparen{B_{i} B_{j}}$, and there are exactly $r$ points of $P$ on $\overparen{B_{i} B_{j}}$. Thus, 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 problem is
\[
\begin{array}{l}
(2 n+1) \sum_{r=1}^{n}\left(\mathrm{C}_{r-1}^{m-2}+\mathrm{C}_{r}^{m-2}\right) \\
=(2 n+1)\left(\sum_{r=1}^{n} \mathrm{C}_{r-1}^{m-2}+\sum_{r=1}^{n} \mathrm{C}_{r}^{m-2}\right) \\
=(2 n+1)\left[\sum_{r=1}^{n}\left(\mathrm{C}_{r}^{m-1}-\mathrm{C}_{r-1}^{m-1}\right)+\right. \\
\left.\sum_{r=1}^{n}\left(\mathrm{C}_{r+1}^{m-1}-\mathrm{C}_{r}^{m-1}\right)\right] \\
=(2 n+1)\left(\mathrm{C}_{n+1}^{m-1}+\mathrm{C}_{n}^{m-1}\right) .
\end{array}
\]