Q:

Getting four points on a closed non-intersecting curve such that they form a square.

Prove that in any closed non-intersecting curve there exist four points on the curve such that they make a square.
I have no idea from where to start.

A:

This is known as the inscribed square problem, and it remains an unsolved problem. However, it is true if the curve is convex or piecewise smooth.
You can read more about the problem at the link to Wikipedia, above, where you'll also find references for further research into the problem and attempts to solve it.

