#topology

Let $C$ be any simple closed curve in the plane. Prove or disprove that $C$ always contains four points that are the vertices of a square.

Prove that every closed (compact, without boundary) three-dimensional manifold in which every loop can be continuously shrunk to a point is homeomorphic to the three-dimensional sphere.

A convex polyhedron has $V$ vertices, $E$ edges and $F$ faces. Prove that $V-E+F=2$.