ORMC Meetings • 2020-2021 Academic Year

We kick off the academic year with some competition-style problems gathered from AMC, AIME, and Olympiads.

We will introduce Gaussian integers in order to decide which prime numbers can be written as a sum of two squares.

We will wrap up the discussion of Gaussian integers and prove which prime numbers are the sum of two squares.

We will study the limitations of polynomials as prime generating functions.

We will study recursive formulas for generating primes.

We will take our first peak into algorithms, with the goal of discussing the P vs NP problem the following week.

We continue our unit on algorithms by discussing the most famous open problem in the field, P vs NP.

The Fundamental Theorem of Algebra... Everyone's heard of it. If it's so fundamental then why haven't we seen a proof?! Look no further, we consider complex polynomials and graphing techniques to prove that every complex polynomial has a root.

To start off the new year we are going to split the class into two competing groups. There will be a variety of problems to work on and it will be up to the teams to organize how they feel is best.

We start our unit on graphs with a dive into planar graphs and the Euler characteristic.

We continue our unit on graph theory with a handout on graph colorings. We will define what it means to color a graph, connect this to coloring maps (geographically speaking), and prove some bounds on how many colors are needed.

We continue with a proof of the 5-color theorem and some Ramsey theory.

We look at continued fractions and their relation to rational and irrational numbers.

We introduce the concept of metrics using a motivating example: the taxicab metric. This worksheet is especially useful for all those of you who are part time students and part time taxi drivers in New York City.