Franklin 5.

An inscribed $n$-gon ( $n>3$ ) is divided by non-intersecting (in interior points) diagonals into triangles. Each of the resulting triangles is similar to at least one of the others. For which $n$ is the described situation possible?
For even $n$, a regular $n$-sided polygon can be divided into two equal polygons by a diagonal passing through its center, and then these two polygons can be cut in the same way. In addition, equal triangles with vertices on the circumscribed circle can be constructed on three sides of a regular $2k$-sided polygon. Therefore, the desired situation is also possible for odd $n>5$. It remains to prove that it is impossible for $n=5$.

If the center of the circumscribed circle around the pentagon does not lie on any of the drawn diagonals, then the triangle containing it is acute, and the others are obtuse, which means that the described situation cannot occur. If the center lies on a diagonal, then the two triangles adjacent to this diagonal are right, and the third is obtuse. Therefore, the specified situation is also impossible.

## Answer

For $n=4$ and for $n>5$.