This is a problem that I have spent a good 2 hours on but can seem to come up with any rigorous solution. If someone could provide one that would be great!
If we color 21 vertices of a 50-gon, how do we go about proving that will always exist 3 vertices such that if you connect them you get an isosceles triangle?
Join alternate vertices of the $50$-gons to produce two $25$-gons. One of the two $25$-gons must have at-least $11$ colored vertices. Further partition this polygon into $5$ - regular pentagons by joining every fifth vertex. One of the pentagons must contain at least three colored vertices. But then those three vertices necessarily form an isosceles triangle.