UCLA Olga Radko Endowed Math Circle

ORMC Meetings Archive • Fall 2007–Spring 2024

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For the current schedule, visit the Circle Calendar

2007–2008 2008–2009 2009–2010 2010–2011 2011–2012 2012–2013 2013–2014 2014–2015 2015–2016 2016–2017 2017–2018 2018–2019 2019–2020 2020–2021 2021–2022 2022–2023 2023–2024
1. Introduction to the Math circle.
2. Lecture: Mathematics of Color (Olga Radko) (Boelter Hall 3400)
3. Presenting solutions of the Welcome problem set
Handouts: Solutions: Geometry | Solutions: Algebra | Welcome problem set: Geometry | Welcome problem set: Algebra
Group A (problem-solving techniques): MS6627
Group B (geometry): MS6221
Handouts: Group A (Parity) | Group B (Geometry of triangles)
Handouts: Group B (Geometry of Triangles) | Group A (Parity)
Group A: MS6201
Group B: MS6221
Handouts: Angles and Circles (group B) | Triangles (group B) | Divisibility (group A)
Group A (MS6201): Problems on Logic
Group B (MS6221): Point Mass geometry and barycentric coordinates;
Handouts: Point Mass Geometry | Logic

Pigeonhole principle:

We will motivate, prove and apply the Pigeonhole principle in a series of problems

Knots and Invariants:

In mathematics, a knot is a closed piece of string in three-space. Two knots are equivalent if they can be deformed into each other. We will explore several ways of showing that certain knots are not equivalent, using invariants.

Handouts: Knots and Invariants | Pigeonhole principle
We will continue with the topics that we have started earlier;
Please solve the problems from the problem sets distributed last time at home.
Handouts: Point Mass Geometry (II) | Point Mass Geometry (I)
Group A: We will solve a series of problems which can be easily attacked using trees, even if trees are not mentioned in the formulation of the problem.
Group B: Students will first present solutions of problems on barycentric coordinates. Then, after reviewing the definition of the center of mass via vectors, we will prove (and apply) the theorems by Cheva and Menelaus.
Handouts: Trees | Theorems by Cheva and Menelaus
Group A: We will take the first steps in learning the method of mathematical induction;
Group B: Any three non-collinear points always lie on some common circle, but in general it is rare for four points to have that property - unless one is trying to solve an Olympiad problem. Then, the observation that certain quadrilaterals are cyclic often turns out to be the key to the solution. We will survey some facts about cyclic quadrilaterals, and use them to solve actual Olympiad problems.
Handouts: Cyclic quadrilaterals | Induction
Graphs: We will learn about simple properties of graphs through problem-solving.
Inversion: Inversion with respect to a circle is a spectacular geometric transformation of the plane that allows to solve some seemingly unapproachable problems. E.g., the following two problems posed by an ancient Greek mathematician Appolonius:
  • Using a compass and a ruler, construct a circle tangent to three given circles;
  • Given two fixed points on the plane, find the set of all points whose distance to the first fixed point is k times the distance to the other point.
  • Handouts: Inversion: workbook | Inversion: problem set
    Group A: We will practice constructing inductive proves, and will also solve several problems related to graphs;
    Group B: in MS 6627 We will discuss basic mathematical aspects of computer games and movie special effects. Specifically, we will cover the mathematical representations used for representing geometry of digital characters and objects, clothing, fluids like water and smoke and a few other examples.
    We will have a small party after the end of the session!!!
    Group A: This is the first of a series of meetings devoted to combinatorics. We will see how permutations and combinations arise naturally in a variety of problems.
    Group B: We will review the basics of modular arithmetic and Euclidean algorithm in preparation for Unique factorization in exotic number systems presented by Jared Weinstein in a series of two future meetings.
    Handouts: Modular Arithmetic
    Group A: We will continue solving problems involving permutations and permutations with repetitions;
    Group B: We will solve many geometry problems from AMC 10/12 exam. Note that the review session will last 3 hours, 1 - 4 p.m. in MS 6627 . The session will include a practice AMC test.
    Group A: Please read Chapter 1 of Introduction to Counting and Probability by D. Patrick in preparation for the session;
    Group B: We will learn about quadratic residues. It is enough to know the basics of modular arithmetic (to the extent we covered on Jan. 11th meeting).
    Handouts: Quadratic residues
    Group A: As homework for next Sunday, please read Sections 1.4-1.5 and 2.1-2.3 (as far as page 37) and do the following exercises:
    1.5.1(a)-(d), 1.5.3(a)-(c)
    2.2.2, 2.2.3
    2.3.1, 2.3.4
    Challenge: 1.34, 1.36, 2.28, 2.32
    Group B: We will continue learning about quadratic residues and unique factorization for Gaussian integers and some other number systems.
    Handouts: Unique factorization in exotic number systems
    Group B: We will learn to solve linear Diophantine equations in two variables.
    Handouts: Handout
    Group A: Please read sections 5.1-5.4 and solve 5.2.1, 5.2.2, 5.2.3 5.3.1, 5.3.3, 5.3.4 5.4.1 Challenge: 5.3.5, 5.18, 5.21
    Group A homework: Read Chapter 6.
    5.4.3 Review 5.11, 5.13, 5.15 Challenge: 5.26, 6.14, 6.16, 6.19
    Group B: this will be a problem-solving session devoted to AIME preparation. If you are not participating in AIME this year, I recommend attending, just to do the problem-solving.
    Handouts: Non-Linear Diophantine equations
    Group A: Please work on problems 8.2.1, 8.2.2, 8.2.3, 8.2.5 for homework.
    Handouts: Image processing slides | Image processing worksheets
    Group A homework: 7.2.2, 7.2.3 7.3.1, 7.3.3, 7.3.4 7.4.1, 7.4.3, 7.4.6 Challenge: 7.22, 7.24
    Handouts: Public Key Cryptography | Probability and Independence
    Group A: We'll be covering material related to chapters 8-9. Please read sections 8.2-8.4, and if you have time also take a look at 8.5, and perhaps a problem or two from chapter 9, which consists of a number of challenging problems. Since we've spent a few weeks going over the basics, this session will be about reviewing what we've learned and attempting some of these more challenging problems. I've included a number of problems below, with suggested numbers of problems per section of chapter 8. Here are some problems from the book: 8.2.2, 8.2.3, 8.3.1, 8.3.2, 8.3.4, 8.4.1, 8.4.3 From section 8.5: (If you get there) 8.5.1 Review: (If you'd like more practice) 8.17, 8.24 Challenge: (Try to solve at least one): 8.37, 9.8, 9.9, 9.10
    Group B: Secret sharing refers to any method for distributing a secret amongst a group of participants, each of which is allocated a share of the secret. The secret can be reconstructed only when the shares are combined together; individual shares are of no use on their own. Secret sharing is an important primitive in several protocols for "secure multiparty computation (MPC)" (We will explore how mathematics and basic number theory can be used in cryptography to develop Secret Sharing schemes and Secure Multi-party Computation schemes.
    Group A: Please read sections 8.5-8.6, and 9.2, and work on the following homework problems: 8.5.3, 8.5.4 Review problems: 8.22, 8.27 Challenge: 9.11, 9.12
    Group A: Here are some HW problems: 10.2.2, 10.2.3 10.3.2, 10.3.4 11.2.1, 11.2.2 Challenge: 10.17, 10.21 Group B: What are finite fields? We'll answer this question and you'll see that you are already familiar with many of them, in a different guise. There are some nice applications that are easy to understand; we'll look at a few.
    Group A: TBA
    Group B: We will look at the Dirichlet product of arithmetic functions and the Mobius inversion formula. Special attention will be given to multiplicative functions.
    Group B: Please go over the proof of the fact that Euler's function phi(n) is multiplicative in preparation for the meeting. We will discover some properties of convolution; prove Mobius inversion formula (via convolution), and use all of these in problem solving.
    Handouts: Sum-functions and convolution