We'll do a variety of Olympiad problems to warm up for the year of problem-solving.
The problems will range in level, from practice problems through BAMO up to Putnam, so there should be something for everyone.
We'll focus on writing clear proofs as we do problems from books and from a variety of actual Olympiads.
This week, we will solve competition problems with invariants.
When there is some repeated process, rather than studying what does change, we may want to look at what stays the same.
This allows us to make connections between the starting and ending positions, and we can rule out many possibilities this way.
We'll look at a few problems involving polynomials and the tricks to solve them.
We consider some more techniques for working with polynomials, such as Vieta's Formulas and tools for working with irreducible polynomials.
We will do competition problems using inequalities such as AM-GM and Cauchy-Schwarz.
We will go over the basics of generating functions, and see how they can be used in olympiad problems about counting.