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I found the following five types.

1. $(m,n)=(kn,n)$ for $n\ge3, k\ge1$.

2. $(m,n)=(hk,2k)$ where $h\ge1$ is an odd number and $k\ge2$.

3. $(m,n)=(m,3)$ where $m\ge4$ and $m\not\equiv0$ (mod $3$).

4. $(m,n)=(m,4)$ where $m\ge3$ is an odd number.

5. $(m,n)=(3,3)$.

I'm going to write that a regular $n$-gon can be inscribed inside a regular $m$-gon for each of these sets.

1. You can get a regular $n$-gon by choosing every $k$-th point of a regular $(m=)kn$-gon.

2. You can get a regular $2k$-gon by cutting every corner of a regular $k$-gon in an 'appropriate' way. Then, you get $(m,n)=(k,2k)$. Letting $k=hk^{\prime}$, you can get $(m,n)=(hk^{\prime},2hk^{\prime})$. Then, you can get $(m,n)=(hk^{\prime}, 2k^{\prime})$ by choosing every $h$-th point of a regular $(n=)2hk^{\prime}$-gon. Note that this type excludes the $h=2l$ cases, which are special cases of 1.

3. Take a point $P_0$ on an edge of a regular $m$-gon. Next, you go counterclockwise on edges then take a point $P_1$ such that $P_0P_1=x$. Again, take $P_2, P_3$ in the same way as above. Then, there exists a value $x$ such that $P_3=P_0$ if you make $x$ larger gradually. Now you get an equilateral triangle $P_0P_1P_2$. We need some more details in the case that $x$ doesn't increase. Note that points move continuously. Note type excludes the $m=3k$ cases, which are special cases of 1.

4. Take an axis of symmetry $AB$ of a regular $m$-gon. Next, take a point $P_0$ near a point $A$ and take a longitudinal rectangle $P_0P_1P_2P_3$ such that $P_0P_1\parallel AB$ (sorry for no figures). You can get a square when you get $P_0=P_1$ if you move $P_0$. Note that this type excludes the $m=2k$ cases, which are included in 1 or 2.

5. This type is obvious.

By the way, are these all sets? Though I've tried to find another type, I neither can find any set nor can prove that these are all possible sets. Then, here are my questions.

Question 1: Does there exist another set $(m,m)$ such that a regular $n$-gon is inscribed inside a regular $m$-gon ?

Question 2: If the answer for question 1 is yes, then could you show me how to prove that? If the answer for question 2 is no, then could you show me another set?

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