In our final chapter, we present a collection of theorems and problems from various branches of mathematics and their proofs and solutions. We begin by discussing some set theoretic results concerning infinite sets, including the Cantor-Schröder-Bernstein theorem. In the next two sections we present proofs of the Cauchy-Schwarz inequality and the AM-GM inequality for sets of size*n*. We then use origami to solve the classical problems of trisecting angles and doubling cubes, followed by a proof that the Peaucellier-Lipkin linkage draws a straight line. We then look at several gems from the theory of functional equations and inequalities. In