OMM 2022 Problem 6

Find all integers $n\geq 3$ such that there exists a convex $n$ -gon $A_1A_2\dots A_n$ which satisfies the following conditions:

• All interior angles of the polygon are equal
• Not all sides of the polygon are equal
• There exists a triangle $T$ and a point $O$ inside the polygon such that the $n$ triangles $OA_1A_2,\ OA_2A_3,\ \dots,\ OA_{n-1}A_n,\ OA_nA_1$ are all similar to $T$ , not necessarily in the same vertex order.