#geometry

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.

A rigid planar shape must be moved around a right-angled corner in a hallway of width 1. What is the largest possible area of such a shape? Gerver's sofa has area about 2.2195. Prove that no shape with larger area can make the turn.

Using only a compass and an unmarked straightedge, can one construct in finitely many steps a square with exactly the same area as a given circle? Prove your answer.

A regular tetrahedron and a cube have the same volume. Can the tetrahedron be cut into finitely many polyhedral pieces that can be reassembled to form the cube? Prove your answer.

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

A simple polygon has all of its vertices at points with integer coordinates. It has $i$ integer points in its interior and $b$ integer points on its boundary. Find a formula for its area in terms of $i$ and $b$, and prove it.