TH-TMO 2023 Problem 10

To celebrate the 20th Thailand Mathematical Olympiad (TMO), Ratchasima Witthayalai School put up flags around the Thao Suranari Monument so that [list=i] [] Each flag is painted in exactly one color, and at least $2$ distinct colors are used. [] The number of flags are odd. [] Every flags are on a regular polygon such that each vertex has one flag. [] Every flags with the same color are on a regular polygon. [/list] Prove that there are at least $3$ colors with the same amount of flags.