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.

We use the first part as motivation to define metrics in general. We talk about a few different examples and introduce the notion of convergence of a series.

We take a look at a measure of income inequality in a given population.

Students will learn what an error of measurement is and how the said error propagates through computations that use the result of the measurement. In the process, the students will derive the sum, product, and quotient rules for the derivative using an engineering approach instead of taking limits.

We use the error formulas derived in the first part, throw in the concept of limits, and see some applications for measurement error.

We'll define the most common lattice and get some results relating regions to lattice points. We'll also see an application to Polya's Orchard problem.

We will develop and study different types of binary codes that detect when a user has made an error. These will include ISBN, repeating codes, Hamming's square code, and Hamming's [7,4]-code. We will also be able to compare the efficiency of these codes.

Sequences are a fundamental part of advanced mathematics. We continue our study of metric spaces from last quarter by using them to define and study sequences.