I recently watched a movie (A Brilliant Young Mind) in which this problem appeared:

Let the vertices of a regular $72$-gon be colored red, blue, and green in equal parts. Show that there are $4$ vertices of each color such that the resulting monochromatic quadrilaterals formed are congruent to each other.

I don't know the solution to this problem, nor do I even know if the problem is actually true (it is from a movie after all). But I would love to see a proof, if one exists, or otherwise a counter-example.