ORMC Meetings • 2023-2024 Academic Year

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.

We'll continue thinking about generating functions, showing some more advanced techniques for particular competition problems.

We'll take a look at a variety of different combinatorics problems, from olympiad training books and actual competitions.

I've selected the easier half of this year's Putnam problems, and broken them into approachable pieces with some hints.

Inspired by a problem from last week's worksheet on the 2023 Putnam, this week we will be diving into uses of expected values in probability problems.

We'll take a look at some interesting number theory problems from problem books and competitions.

This week, we will talk about combinatorial geometry.

We'll start with the classic Lazy Caterer's Problem and build up to harder problems about counting geometrical objects.

We use the complex plane interpretation to solve geometry problems using complex algebra.

We'll take a look at a particular kind of coordinate system based on triangles.

We'll then use these barycentric coordinates to prove theorems and solve competition problems based on triangles.

Many combinatorial problems have the same solution - the Catalan numbers.

We will use some of these problems to practice bijective combinatorial proofs, and then apply those techniques to competition problems.

We'll prove some formulas for summing binomial coefficients, and use them to solve competition problems.

We return to number theory, and solve some problems with factorization and gcds.

We'll pick up last week's theme of factorization, but instead of focusing on gcds, we focus on direct applications of unique factorization into primes.

We will study multiplicative functions such as the sum-of-divisors function and Euler's totient function, and use them to solve Olympiad number theory problems.

We learn about vector spaces, and use dimensions of vector spaces to solve competition problems.

We pick up the linear algebra theme from last week. This time, will look at matrices, and the functions they define between vector spaces.