UCLA Olga Radko Endowed Math Circle

ORMC Meetings Archive • Fall 2007–Summer 2021

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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
Fall 2020 quarter // Filter groups by:
10/11/2020

We warm-up for the new academic year by solving problems from some mathematical competitions.

Handouts: Handout

We warm-up for the new academic year by solving problems from some older mathematical competitions.

Handouts: Handout

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

Handouts: Handout | Solutions

We will start the year with a wide variety of competition style problems.

Handouts: Competition Problems

Prove the existence of e, and see it arise in various natural situations.

Handouts: eulers_number_handout

For our first meeting, we're going to cover logic problems and do some fun puzzles.

The main goal of the first lesson is to engage students and show them several mathematical surprises. We will also look at an optical illusion

The main goal of the first lesson is to engage students and show them several mathematical surprises. We will also look at an optical illusion

We will talk about some sharing problems, and start to think about how we can make careful arguments to justify our answers.

We will go over a simple version of "I cut, you choose" and some brain teasers!

Handouts: Handout: I_Cut_You_Choose

We will go over a simple version of "I cut, you choose" and some brain teasers!

Handouts: Handout: I_Cut_You_Choose

For our first lesson, we will introduce pooled testing in the context of coronavirus.

Handouts: Key | Handout

For our first lesson, we will introduce pooled testing in the context of coronavirus.

Handouts: Key | Handout

First meeting of the year for intermediate 2A. We will be doing some fun problem solving!

Handouts: Intermediate 2A Problem Solving

This year we will be starting off with a great worksheet filled with a variety of problems, and we will go through it in teams in competition. cant wait!

Handouts: Solutions | Worksheet

We quickly review mathematical induction and then solve a few related problems of various levels of difficulty, from easy to very hard.

Handouts: Handout

Many nontrivial Olympiad problems can be solved by angle chasing. This class we covered inscribed angles, cyclic quadrilaterals, and properties of the orthocenter.

Handouts: Homework | Lecture Notes
10/18/2020

We start our exploration of quadratic equations with Vieta's Theorem. As usual, we also do some geometry.

Handouts: Handout

Some low degree polynomial analysis starting with quadratics including Vieta's formulas.

Handouts: Handout

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

Handouts: Handout | Solutions

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

Handouts: Gaussian Integers I

Does e really exist? Let's try to fill in missing details of our previous claims.

Handouts: Euler's Number Worksheet 2: The Least Upper Bound
Handouts: Week 2 slides

Lesson 2 - Sharing Problems and Birthday Cakes

We will return to the fair share problems that we introduced in week 1, by considering what happens when we have to divide among three people. We will also think about the geometry of plane figures and how many places lines will intersect with different figures.

We will be developing logic skills by looking at problems with different colored hats. The goal of this handout is to learn how to make an assumption, test the assumption, and readjust the original assumption if necessary.

Handouts: Handout: Wizards_Hats | Solutions: Wizards_Hats

We will be developing logic skills by looking at problems with different colored hats. The goal of this handout is to learn how to make an assumption, test the assumption, and readjust the original assumption if necessary.

Handouts: Handout: Wizards_Hats | Solutions: Wizards_Hats

We will be introduced to modular math. We completed pages 1-6 of the handout.

Handouts: Handout

We will be introduced to modular math. We completed pages 1-6 of the handout.

Handouts: Handout

We introduce permutations and the 15 puzzle problem!

Handouts: Handout

This week we will be learning about a very important topic to the theory of Abstract Algebra: Permutations.

Handouts: Worksheet | Solutions

Today we will approach problems involving area computations, that should hopefully cover all necessary techniques for Olympiad problems.

Handouts: Handout

Directed angles, incenter/excenter lemma, problems.

Handouts: Homework | Lecture Notes
10/25/2020

We start the long handout with a long division.

Handouts: Handout | HW

We finish last weeks handout with the convex property and start cubic equations.

Handouts: Cubics | Convex

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

Handouts: Handout | Solutions

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

Handouts: Gaussian Integers II
Handouts: Hats

Slides for this week: https://docs.google.com/presentation/d/1nhl_yW5xpqtiZZOn1Dv4G5eVvvJ1qe6tlebHdXeK6x8/edit?usp=sharing

More optical illusions and crossing problems.

More optical illusions and crossing problems.

We will continue with our exploration of logic puzzles involving hats and if we have additional time, work through some math kangaroo problems.

Handouts: Math_Kangaroo_Problems | Handout_Solutions_No_Challenge_Problem

We will continue with our exploration of logic puzzles involving hats and if we have additional time, work through some math kangaroo problems.

Handouts: Math_Kangaroo_Problems | Solutions: No_Challenge_Problem

We will be continuing the second half of the handout on modular math. We will also learn modular math applications, such as checksums.

Handouts: Handout | Homework

We will be continuing the second half of the handout on modular math. We will also learn modular math applications, such as checksums.

Handouts: Handout | Homework

This week we continue developing some more language and notation to be able to prove whether or not the 15 puzzle has a solution...

Handouts: Permutations Part 2 - handout

Hello! This week we are continuing on the topic of permutations.

Handouts: Worksheet | Solutions

Today we will discuss various standard techniques in solving diophantine equations (i.e. equations with integers). The material is for 2 weeks.

Handouts: Handout

More advanced angle chasing problems

Handouts: Homework | Lecture Notes
11/1/2020
Handouts: Handout | Homework

We continue the cubic handout from last week.

Handouts: Handout

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

Handouts: Prime Generating Functions I

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

Handouts: Prime Generating Functions I
Handouts: Worksheet

https://docs.google.com/presentation/d/1EemxMNiCYWj3wna_inEKic_Z_X2BXtopN7d2y1A7Vw0/edit?usp=sharing

We will review the homeworks from lesson 3, and introduce Roman numerals as a new topic. Remember to bring your craft sticks to class!

This week, we will finish up the wizards and hats packet by completing the challenge problem. We will then work through some Halloween math problems.

Handouts: Solution: Challenge Problem | Handout: Halloween Math

This week, we will finish up the wizards and hats packet by completing the challenge problem. We will then work through some Halloween math problems.

Handouts: Handout: Halloween_Math | Solution: Challenge Problem

We will finish up our review on modular math, learning how to subtract and divide in modular math as well as learning how to prove divisibility rules with modular math.

Handouts: Handout | Divisibility Rules | Answer Key

We will finish up our review on modular math, learning how to subtract and divide in modular math as well as learning how to prove divisibility rules with modular math.

Handouts: Answer Key | Handout | Divisibility Rules
Handouts: Handout

Today is the last day of this topic and we finally be able to combine taxicab distance and permutations to solve the 15 puzzle.

Handouts: Worksheet

The handout is the same as last time (see Part I)

We will move on to problems 4, 5, 6... and so on today.

Power of a Point and Radical Axis

11/8/2020
Handouts: Homework | Handout (same as two previous weeks)

We have a more in depth coverage of complex numbers.

Handouts: Handout

We will study recursive formulas for generating primes.

Handouts: Handout

We will study recursive formulas for generating primes.

Handouts: Prime Generating Functions II
Handouts: Worksheet

https://docs.google.com/presentation/d/1PGTLoVpI4mN1a6ucCoe-fMBd2I32JwJKkpjp5LrVYts/edit?usp=sharing

We will look at the rules for making Roman numerals and we will learn how to add and subtract with Roman numerals.

We will look at the rules for making Roman numerals and we will learn how to add and subtract with Roman numerals.

This week, we finish up the Halloween Math packet and begin our exploration of Venn Diagrams.

Handouts: Handout: Venn Diagrams | Solutions: Halloween Math

This week, we finish up the Halloween Math packet and begin our exploration of Venn Diagrams.

Handouts: Handout: Venn Diagrams | Solutions: Halloween Math

We will be exploring an ancient Egyptian technique of splitting fractions into their unit parts. We completed pages 1-16.

Handouts: Handout | Key

We will be exploring an ancient Egyptian technique of splitting fractions into their unit parts. We completed pages 1-16.

Handouts: Handout | Key
Handouts: Handout

Today we are going to test the limits of zoom by going through an awesome geometry worksheet that focuses on the use of compass and straight edge.

Handouts: Worksheet

Today we are doing problems involving mathematical games, and talk about the method of retrograde analysis.

Handouts: Handout

Harder problems with Power of a Point

Handouts: Homework
11/15/2020

We get back to small handouts for one lesson. This one is about functions.

Handouts: Handout

We discuss e and one of its definitions coming off Euler's equation.

Handouts: handout

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

Handouts: Handout

We introduce some classic algorithms and the notion of runtime.

Handouts: Intro to Algorithms
Handouts: CryptographyHandout

https://docs.google.com/presentation/d/1csp0eYKJrFWvBkwCMrCf_zZYoP9U85WB3K1AErb-lBE/edit?usp=sharing

This week, we will be continuing our exploration of Venn Diagrams and we if have additional time, work through some Math Kangaroo problems.

Handouts: Venn Diagrams Solutions

This week, we will be continuing our exploration of Venn Diagrams and we if have additional time, work through some Math Kangaroo problems.

Handouts: Venn Diagram Solutions

We will continue our exploration of Egyptian fractions (pages 17-24). Homework is pages 25-26.

Handouts: Handout: part 1

We will continue our exploration of Egyptian fractions (pages 17-24). Homework is pages 25-26.

Handouts: Handout: part 1

Continuing from last week but with even more detailed constructions and Euclidian proofs!

Handouts: Worksheet

Continuing from last week but with even more detailed constructions and Euclidian proofs!

Handouts: Worksheet

This lecture we continue with the same topic. We added a few extra problems

Handouts: Handout

Today we did three challenging problems whose solutions involved Power of a Point in a fundamental way.

Handouts: Handout | Homework
11/22/2020

We will solve the rest of the problems from the previous lesson and

Handouts: Handout | HW

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

Handouts: Handout

Following the introduction to algorithms, we are able to discuss the nuances of the P vs. NP problem.

Handouts: P vs NP
Handouts: finite_geometry

https://docs.google.com/presentation/d/1jZfw6Ng6CNeKB_VKY-KmY9pGI6NsZRV8mVnUZ_htH1s/edit?usp=sharing

This week, we'll finish up Venn Diagrams by going over any of the challenge problems the students may have had difficulty with. Afterward, we'll begin our exploration of a new topic: Egyptian Multiplication. We'll also explore its relationship with binary numbers.

Handouts: Handout: Egyptian_Multiplication | Solutions: Egyptian_Multiplication

This week, we'll finish up Venn Diagrams by going over any of the challenge problems the students may have had difficulty with. Afterward, we'll begin our exploration of a new topic: Egyptian Multiplication. We'll also explore its relationship with binary numbers.

Handouts: Handout: Egyptian_Multiplication | Solutions: Egyptian_Multiplication

To conclude our exploration of Egyptian Fraction Representation, we will complete the Egyptian Fractions part 1 handout and begin Egyptian Fractions part 2. Pages 6-8 EFR part 2 are assigned as homework.

Handouts: EFR part 1 | EFR part 2 | EFR part 1 Key | EFR part 2 Key

To conclude our exploration of Egyptian Fraction Representation, we will complete the Egyptian Fractions part 1 handout and begin Egyptian Fractions part 2. Pages 6-8 EFR part 2 are assigned as homework.

Handouts: EFR part 1 | EFR part 2 | EFR part 1 Key | EFR part 2 Key

We introduce combinatorics, the mathematics of counting.

Handouts: Combinatorics 1

Today we go into a new and really awesome topic: Combinatorics!


Handouts: Combinatorics Worksheet | Solutions

Today we will do practice problems, with no particular theme.

Handouts: Handout

We will introduce Ceva's Theorem, Menelaus' Theorem, the Extended Law of Sines, and some basic length formulas in a triangle.

Handouts: Lecture Notes | Homework
11/29/2020

We talk about functions and some relations between them.

Handouts: Handout | Extra

Happy Thanksgiving! There is no class today, but we do have Math Kangaroo problems to work on!

Handouts: Math Kangaroo

Happy Thanksgiving! There is no class today, but we do have Math Kangaroo problems to work on!

Handouts: Math Kangaroo
12/6/2020
Handouts: Handout

We do a little more analysis of quadratic equations

Handouts: Handout

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.

Handouts: Handout

We prove the Fundamental Theorem of Algebra with a focus on graphing.

Handouts: Fundamental Theorem of Algebra

We continue with the previous worksheet on Finite Geometry.

Handouts: finite_geometry

https://docs.google.com/presentation/d/1yIBsli9ojgR6OKqJ9g21Nwqaewk8cWMPhc4Vt4yRaIs/edit?usp=sharing

This week, we'll continue with our exploration of Egyptian Multiplication and Binary Numbers. If we have additional time, we'll go through the Math Kangaroo problems.

This week, we'll continue with our exploration of Egyptian Multiplication and Binary Numbers. If we have additional time, we'll go through the Math Kangaroo problems.

We will look a method to prove mathematical hypotheses: proof by induction.

Handouts: Induction | Completed Proofs

We will look a method to prove mathematical hypotheses: proof by induction.

Handouts: Induction | Completed Proofs

This week we continue and finish off our introduction to combinatorics.

Handouts: Combinatorics Handout

This week we will be continuing what we stared last week and talk more about combinatorics. We will be starting with a brief review of what we spoke about last week, before moving onto completely new problems.

Handouts: Handout

Today we discuss angle chasing, properties of angles inscribed in circles, and a few constructions.

Handouts: Handout

Two proofs of Heron's Formula; Nine-Point Circle, Euler Line, Monge's Theorem

Handouts: Lecture Notes
12/13/2020

No handout, just game!

Handouts: Game

We ran a math dominoes competition.

The students battle it out in a full class Kahoot competition.

Handouts: Problems

https://docs.google.com/presentation/d/1qjqnqtXTYAdYSixph04VEq5T54YDbws0VmPuZgB5raA/edit?usp=sharing

This week, we'll be playing a review game consisting of the topics we've explored this far.

This week, we'll be playing a review game consisting of the topics we've explored this far.

We will continue to explore proof by induction and introduce proof by contradiction. *Last class of fall quarter, Happy Holidays!

We will continue to explore proof by induction and introduce proof by contradiction. *Last class of fall quarter, Happy Holidays!

There is no handout for this class (Instructors can access the private attachments).

We are going to be playing a zoom adapted math circle favorite! I hope everyone has a great time.

We continue with the topic from last time.

Handouts: Handout | Lecture Notes

Simson line configuration (including proof that Simson(P) bisects PH) and incenter touch-point configuration.

Handouts: Homework
Winter 2021 quarter // Filter groups by:
1/10/2021

Today we will be starting on one of the most interesting and important topics in mathematics, at least in my opinion. I hope everyone learns something!

Handouts: Handout

We go from quadratic equations to something more elementary but not easier!

Handouts: Handout

We start the year with problems using weight, logic, and geometry.

Handouts: Handout

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.

Handouts: Competition Problems | Solutions
Handouts: Competition Problems

We study various functions that output lots of prime numbers.

Handouts: Prime_Generating_Functions

Welcome Back! This week, we'll begin our exploration of a new topic: Exponential Functions.

Handouts: Handout: Exponential Functions P1 | Solutions: Exponential Functions P1

Welcome Back! This week, we'll begin our exploration of a new topic: Exponential Functions.

Handouts: Handout: Exponential Functions P1 | Solutions: Exponential Functions P1

Suppose we break a stick into three pieces randomly. What are the chances the resulting pieces will form a triangle?

Handouts: Handout

Suppose we break a stick into three pieces randomly. What are the chances the resulting pieces will form a triangle?

Handouts: Handout

We have an introduction to the complex numbers in this meeting.

Handouts: Complex Numbers Handout (by Kyle Coughlin)

An introduction to something awesome: Complex Numbers.

Handouts: Handout | Solutions

Today we will discuss board problems, involving colorings, tilings and more.

Handouts: Handout

We will go over the winter break geometry homework and review last quarter's material, and do some Olympiad problems from other subjects if time allows.

1/17/2021

We continue our studies in elementary math.

Handouts: Handout

We continue with questions using scales, logic and of course more geometry.

Handouts: Handout

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

Handouts: Handout | Solutions

We will attempt to determine when a given graph is planar, culminating in Kuratowski's Theorem.

Handouts: Planar Graphs
Handouts: Borsuk Ulam Problems

https://docs.google.com/presentation/d/1UAbiQFgUwJIPL8r-7HQq0-6s_z3o87LRb5cIMGCpZS8/edit?usp=sharing

This week, we'll take a deeper look into what happens when we multiply negative numbers and use what we learn to further our exploration of exponential functions.

Handouts: Handout: Multiplying Negative Numbers | Solution: Multiplying Negative Numbers

This week, we'll take a deeper look into what happens when we multiply negative numbers and use what we learn to further our exploration of exponential functions.

Handouts: Handout: Multiplying Negative Numbers | Solution: Multiplying Negative Numbers

We will continue and complete our packet on breaking sticks. In which scenario are we most likely able to form a triangle from our broken sticks?

Handouts: Breaking Sticks | Answer Key | Homework

We will continue and complete our packet on breaking sticks. In which scenario are we most likely able to form a triangle from our broken sticks?

Handouts: Breaking Sticks | Answer Key | Homework

Along our exploration of the complex numbers and our journey to trigonometry, we take a brief stop at some algebra.

Handouts: Algebra

This week we will continue the worksheet from last week and then move onto a worksheet going into the details of the number systems we went over last week.

Handouts: Handout

Today we cover various methods of solving inequality problems.

Handouts: Handout

Complex numbers and AIME problems

1/24/2021

This week we continue solving problems concerning some of the most interesting areas of school mathematics

Handouts: Handout

We cover more weighings and geometric constructions, as well as look at some problems using invariant solutions.

Handouts: handout

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.

Handouts: Handout

We will explore many interesting results about graph coloring.

Handouts: Graph Coloring

We explore some famous dynamical systems, and study the logistic map and the Mandelbrot set.

Handouts: Dynamics_problems

https://docs.google.com/presentation/d/186hkAvj76yIzn0eznHHnii5rZNhLuLcRMzid1rvrQ0A/edit?usp=sharing

Continue Lesson 9 Cutting Logs

This week, we'll continue with our exploration of exponential functions by picking up where we left off in the packet.

This week, we'll continue with our exploration of exponential functions by picking up where we left off in the packet.

This week we will learn how to convert between fractions and decimals, while also exploring the properties of terminating and non-terminating decimals.

Handouts: Review and Homework

This week we will learn how to convert between fractions and decimals, while also exploring the properties of terminating and non-terminating decimals.

Handouts: Review and Homework

This will be our first time dealing with some trignometry! This will give us the necessary information to talk about Euler's formula next week.

Handouts: Intro to trig

This week we start in the topic of trigonometry

Handouts: Handout

Today we will review similarity of triangles, and we will do problems involving the Menelaus and Ceva theorems.

Handouts: Handout
1/31/2021

We discuss the plots from the previous week in more detail

Handouts: Handout
Handouts: Handout

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

Handouts: Solutions | Handout

We will finish the study of graph coloring and introduce basic ideas in Ramsey Theory.

Handouts: Graph Coloring

We continue exploring dynamical systems.

https://docs.google.com/presentation/d/1ptDmIXjt-pVIM37OgiRHuLuw10l1KD4RoY23BqfGGAc/edit?usp=sharing

This week, we'll continue with our exploration of exponential functions by using its properties to solve a myriad of problems. If we have extra time, we'll work through some Math Kangaroo problems.

Handouts: Handout: Math Kangaroo P2

This week, we'll continue with our exploration of exponential functions by using its properties to solve a myriad of problems. If we have extra time, we'll work through some Math Kangaroo problems.

Handouts: Handout: Math Kangaroo P2

We will continue to explore terminating and non-terminating decimals. WHY do decimals have these properties? How can we write non-terminating decimals as fractions?

Handouts: Class notes | Homework

We will continue to explore terminating and non-terminating decimals. WHY do decimals have these properties? How can we write non-terminating decimals as fractions?

Handouts: Class notes | Homework

We finally arrive at Euler's Formula!

Handouts: Euler's Formula Applications Handout

Today we connect what we learned last week with the complex numbers!

Handouts: Handout
Handouts: Handout
Handouts: Handout
2/7/2021

We continue with mostly invariants and geometric constructions.

Handouts: New handout | Last week's handout

https://docs.google.com/presentation/d/1ehJAFzTun5NkE9QzED1908bas7wy7FpjVaIboCjGyW8/edit?usp=sharing

We cover more explicitly Invariant problems and a couple geometric constructions.

Handouts: Handout
Handouts: Handout from last meeting | New problems

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

Handouts: Handout | Solutions

Continued fractions are an interesting way to represent real numbers. We will develop an algorithm for computing continued fractions and attempt to classify periodic continued fractions.

Handouts: Continued Fractions

We study how to project 3-dimensional figures onto planes, and find out the intersection of 3d figures with planes.

Handouts: 3D_Sections_problems |

This week, we'll finish up our exploration of exponential functions by finish the remaining challenge problems. Afterward, we'll solve some Math Kangaroo problems.

This week, we'll finish up our exploration of exponential functions by finish the remaining challenge problems. Afterward, we'll solve some Math Kangaroo problems.

We will write and define sequences and numbers through explicit and recursive definitions. We will also explore arithmetic sequences and how we can generalize the sum of n-terms in the sequence.

Handouts: Class Notes

We will write and define sequences and numbers through explicit and recursive definitions. We will also explore arithmetic sequences and how we can generalize the sum of n-terms in the sequence.

Handouts: Class Notes

We investigate and explore some ideas in probability for this class!

Handouts: Probability applied to Geometry

Hello Intermediate 2A students!

Today we will be investigating some ideas in probability in combination with some familiar ideas in geometry. We think today's handout will be great for those that found our previous topics less than exciting compared to usual. We hope that these probability problems will be a good change of pace!

Handouts: Handout | Solutions

Today we are continuing with the topic from last time. Some of the problems in this handout are left-overs from the previous handout.

Handouts: Handout
2/14/2021

We will not meet on 2/14/2021. Happy Valentine's Day/President's Day/long-weekend-in-the-middle-of-the-quarter.

President's Day Weekend. Class on the 21st.

2/21/2021

Problems will appear here with the start of the game

Handouts: Game

We will be playing a game today.

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.

Handouts: Handout | Solutions

We introduce the taxicab metric, a new way of measuring distance in Euclidean space. We will study its properties and figure out how it relates to the Euclidean metric.

Handouts: Taxicab Geometry

We discuss algorithmic complexity, and learn the difference (if indeed there is one) between P and NP problems.

Handouts: PNP_problems

This week, we will begin by working through the remaining math kangaroo problems in the packet. Afterward, we'll begin our exploration of the world of polyhedra.

Handouts: Handout: Polyhedra_P1 | Solution: Polyhedra_P1

This week, we will begin by working through the remaining math kangaroo problems in the packet. Afterward, we'll begin our exploration of the world of polyhedra.

Handouts: Handout: Polyhedra_P1

We will continue our work on arithmetic sequences, then transition to learning about arithmetic series.

Handouts: Class Notes | Homework

We will continue our work on arithmetic sequences, then transition to learning about arithmetic series.

Handouts: Class Notes | Homework

We investigate the Chinese Remainder Theorem in this week's handout.

Handouts: Chinese Remainder Theorem

Hello everyone! This week we will be working on some problems that have to do with number theory! Number theory is a very important subject in math and is considered one of the most difficult!

Handouts: Solutions | Handout
Handouts: Handout

AIME Number Theory problems, which tend to be computational

2/28/2021

We continue with invariants, semi-invariants and geometric constructions.

Handouts: Handout

We are continuing with Invariant question and geometric constructions.

Handouts: Handout

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.

Handouts: Handout | Solutions

We generalize different notions of distances, as motivated by the taxicab distance, into something called a metric space. We then discuss convergence of sequences in metric spaces.

Handouts:
Handouts: ordinals_problems

https://docs.google.com/presentation/d/1ehJAFzTun5NkE9QzED1908bas7wy7FpjVaIboCjGyW8/edit?usp=sharing

This week, we'll continue with our exploration of Polyhedra.

This week, we'll continue with our exploration of Polyhedra.

We will introduce geometric sequences and explore how using geometric series can help us convert non-terminating decimals to fractions.

Handouts: Class Notes | Homework

We will introduce geometric sequences and explore how using geometric series can help us convert non-terminating decimals to fractions.

Handouts: Class Notes | Homework

This week we are continuing our problems on the Chinese Remainder Theorem (CRT) along with a small introduction to ‘if and only if' proofs within the context of remainders and divisibility.

Handouts: divisibility_crt_2

This week we are continuing our problems on the Chinese Remainder Theorem (CRT) along with a small introduction to ‘if and only if' proofs within the context of remainders and divisibility

Handouts: Handout
Handouts: Handout
3/7/2021

This week we finish our introduction to geometric constructions and invariants.

Handouts: Handout

More on Invariants and a geometry drawing exercise.

Handouts: Extra | Handout

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

Handouts: Handout

In economics, the Gini index is a measure intended to represent income inequality. We will take a mathematical approach to the study of the Gini index.

Handouts: Gini Index
Handouts: Handout

This week will finish up our packet on polyhedra and practice some Math Kangaroo Problems!

Handouts: Handout: Math Kangaroo P3 | Solution: Math Kangaroo P3

This week will finish up our packet on polyhedra and practice some Math Kangaroo Problems!

Today, we introduce combinatorics: the mathematics of counting.

Handouts: Handout | Key | Homework

Today, we introduce combinatorics: the mathematics of counting.

Handouts: Handout | Key | Homework

We study the GCD and the Euclidean algorithm

Handouts: GCD and Euclid Handout

This week we are continuing some more on divisibility with problems around the greatest common divisor as well as Euclid's division algorithm. The handout is attached below.

Handouts: Handout

We are continuing with the same topic as last time. Problems have been added to the previous handout.

Handouts: Handout
3/14/2021

This week we have a game! Prepare your pen, pencil, paper and spirit!

Handouts:
Handouts: Handout
Handouts: Competition Problems

We have a competition in which we review this term's topics.

Handouts: Problems

https://docs.google.com/presentation/d/1q3ZuUtGr3FlPf8SYaCmkhABypIr9QKtT1c53lRpwCzQ/edit?usp=sharing

We'll continue with our exploration of Polyhedra, solve some math kangaroo problems, and do a short review of exponential functions.

Handouts: Handout: Polyhedra Additional Problem | Solution: Polyhedra Additional Problem

We'll continue with our exploration of Polyhedra, solve some math kangaroo problems, and do a short review of exponential functions.

Handouts: Handout: Polyhedra Additional Problem

We will complete the combinatorics worksheet from last week. Furthermore, we will explore what happens if the order of the items in our list matters.

Handouts: Handout | Key

We will complete the combinatorics worksheet from last week. Furthermore, we will explore what happens if the order of the items in our list matters.

Handouts: Handout | Key

Today we have a math dominoes game for the last meeting of the quarter.

3/21/2021
3/28/2021
Spring 2021 quarter // Filter groups by:
4/4/2021

We are very excited to start a new quarter with all of you starting with this Sunday's topic on random events (random variables).

Handouts: Worksheet

We start a new topic of combinations and Pascal's triangle. Also a problem with triangle similarity.

Handouts: Handout

We start learning basic counting methods in combinatorics.

Handouts: Handout

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.

Handouts: handout

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.

Handouts: handout

We cover several graph theory topics, including planarity, colorability, and Ramsey Theory.

Handouts: Problems

https://docs.google.com/presentation/d/11-UtV4dS7yELTHYrED40r73QbCTKN01WMVQcuneLx6Y/edit?usp=sharing

We finished up the first part of our exploration of Polyhedra and went through the Math Kangaroo problems. This week, we'll explore a new topic: Sets and Venn Diagrams.

Handouts: Handout: Set_Venn_Diagrams_P1 | Solutions: Set_Venn_Diagrams_P1

We finished up the first part of our exploration of Polyhedra and went through the Math Kangaroo problems. This week, we'll explore a new topic: Sets and Venn Diagrams.

Handouts: Handout: Set_Venn_Diagrams_P1 | Solutions: Set_Venn_Diagrams_P1

Can you move the earth with a lever? Today's math circle lesson will require creativity, physics, and a little intuition to balance levers.

Handouts: Key | Handout

Can you move the earth with a lever? Today's math circle lesson will require creativity, physics, and a little intuition to balance levers.

Handouts: Key | Handout

This Sunday we explore random variable and events beyond just geometric probabiltiy (as we've done before).

Handouts: randomEvents_handout
Handouts: Handout

We will cover linear recurrences and how to solve them

Handouts: Handout
4/5/2021

This week we took a break from our regular scheduled programming for a fun math game!

4/11/2021

We continue studying combinations, introducing Pascal's triangle.

Handouts: Handout

He continue with combinations and how they relate to Pascal's triangle

Handouts: Handout

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


Handouts: Handout

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 cover several graph theory topics, including planarity, colorability, and Ramsey Theory.

https://docs.google.com/presentation/d/10fP6mjiBJ-PTNw5sPwFDYQQbiw-ixFBs9hxcxZieMgY/edit?usp=sharing

This week will start off by learning about the difference between a set and a list as well as some new notation to learn about some special sets such as the empty set and the set of natural numbers.

Handouts: Handout: Set_Venn_Diagrams_P1 | Solutions: Set_Venn_Diagrams_P1

This week will start off by learning about the difference between a set and a list as well as some new notation to learn about some special sets such as the empty set and the set of natural numbers.

Handouts: Handout: Set_Venn_Diagrams_P1 | Solutions: Set_Venn_Diagrams_P1

We will refresh our minds on combinations and permutations from before spring break, and we will also begin to explore probability.

Handouts: questions | class notes

We will refresh our minds on combinations and permutations from before spring break, and we will also begin to explore probability.

Handouts: questions | class notes

We continue some probability theory ideas and introduce the binomial distribution!

Handouts: randomEventsII_handout

Hello Intermediate 2B students!

We are continuing a bit of work from last week and introducing some additional and useful ideas in probability.

Handouts: Handout
Handouts: Handout
Handouts: Handout (3 Weeks)
4/18/2021
Handouts: Game

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.

Handouts: Handout

We will show that certain regions of Euclidean space must contain integer lattice points.

Handouts: Lattices

We explore the field of Incidence Combinatorics, learning about the Szemeredi-Trotter Theorem, sum-product estimates, and Erdos's unit and distinct distances problems.

Handouts: Problems

https://docs.google.com/presentation/d/1P9r9DyJ7LfomvKD1eP5vSxTR8tTKup82SM_UBtwCpg8/edit?usp=sharing

This Sunday, we will continue with our exploration of sets by practicing with the concepts we have learned so far, learning about cardinality, and potentially attempting some challenging problems.

Handouts: Solutions: Set_Venn_Diagrams_P1 | Handout: Set_Venn_Diagrams_P1

This Sunday, we will continue with our exploration of sets by practicing with the concepts we have learned so far, learning about cardinality, and potentially attempting some challenging problems.

Handouts: Handout: Set_Venn_Diagrams_P1 | Solutions: Set_Venn_Diagrams_P1

To test our knowledge of combinatorics that we learned over the past few weeks, we will go over 4 tough problems.

Handouts: Key

To test our knowledge of combinatorics that we learned over the past few weeks, we will go over 4 tough problems.

Handouts: class notes

We introduce some important ideas of functions and proofs, the bread and butter of maths.

Handouts: functions and proofs

This week we introduce some general ideas about functions and proofs.

As a friendly reminder, please change your zoom profile name to your actual ORMC registered name (this will help us identify you at the beginning of class)!

Handouts: Worksheet
Handouts: Handout
4/25/2021

This week we continue our studies in combinatorics. We will finally see and apply the binomial theorem.

Handouts: Handout

We review our last worksheet and continue with Combination questions.

Handouts: Handout

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.

Handouts: Handout | Solutions

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.

Handouts: Smart Codes

We explore the field of Incidence Combinatorics, learning about the Szemeredi-Trotter Theorem, sum-product estimates, and Erdos's unit and distinct distances problems.

https://docs.google.com/presentation/d/1nQ6NQyNWm9-Tk3rhUhX-c-wQUcmYuSQDZfFfvDQI22k/edit?usp=sharing

This week, we'll finish up with the first part of our exploration of sets and connect it to Venn Diagrams.

Handouts: Handout: Sets_Venn_Diagrams_P2 | Solutions: Sets_Venn_Diagrams_P2

This week, we'll finish up with the first part of our exploration of sets and connect it to Venn Diagrams.

Handouts: Handout: Sets_Venn_Diagrams_P2 | Solutions: Set_Venn_Diagrams_P2

How can we estimate our dartboard score by using probabilities? Today we explore the first 7 problems (pages 1-4) of our probability packet.

Handouts: Handout | class notes

How can we estimate our dartboard score by using probabilities? Today we explore the first 7 problems (pages 1-4) of our probability packet.

Handouts: Handout | class notes

We learn to approximate areas of shapes and graphs; this is basically a small teaser to ideas of integration in calculus.

Handouts: Area Approximation handout
Handouts: Handout

reviewing the 2021 USAJMO

Handouts: Notes p1 | Notes p2 | Notes p3 | main
5/2/2021

This week we will learn the technique called "Stars and Bars" and also solve some funny problems.

Handouts: Handout

We cover the technique of stars and bars

Handouts: Handout

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.

Handouts: Smart Codes
Handouts: Problems

https://docs.google.com/presentation/d/1Q7yFZVRIrwzQDi4PCRmGgZ7n7CXdl_E9FBUKeBc97us/edit?usp=sharing

This week, we'll finish up the final challenge problem in the first part of our exploration of sets and afterward, begin exploring the connection between sets and Venn DIagrams.

Handouts: Handout: Sets_Venn_Diagrams_P2 | Solutions: Set_Venn_Diagrams_P2

This week, we'll finish up the final challenge problem in the first part of our exploration of sets and afterward, begin exploring the connection between sets and Venn DIagrams.

Handouts: Handout: Sets_Venn_Diagrams_P2 | Solutions: Set_Venn_Diagrams_P2

Back to the basics: how can we describe probability for rolling dice and flipping coins? Today we tackle pages 5-10 of our probability worksheet.

Handouts: Handout | Key

Back to the basics: how can we describe probability for rolling dice and flipping coins? Today we tackle pages 5-10 of our probability worksheet.

Handouts: Handout | Key

We go into our breakout rooms to do some problem solving.

This week we introduce some ideas about area approximation, which is very useful in calculus. After this week we will have at least two handouts on one my favorite topics in math: graph theory!

Handouts: Handout
Handouts: Handout
5/9/2021

We continue our studies in combinatorics.

Handouts: Handout

We continue solving problems using the stars and bars method.

Handouts: Handout
Handouts: Handout
Handouts: Competition Problems
Handouts: Problems

https://docs.google.com/presentation/d/1vo2BFKE05-AOGGAuPw6h5FLi8wwllcyPnPY96H1BN34/edit?usp=sharing

This week, we'll continue with our exploration of sets and incorporate the ideas and concepts we've learned so far into Venn Diagrams.

Handouts: Handout: Sets_Venn_Diagrams_P2 | Solutions: Set_Venn_Diagrams_P2

This week, we'll continue with our exploration of sets and incorporate the ideas and concepts we've learned so far into Venn Diagrams.

Handouts: Handout: Sets_Venn_Diagrams_P2 | Solutions: Set_Venn_Diagrams_P2

Today, we will complete the probability packet and explore set notation.

Handouts: Handout | Key

Today, we will complete the probability packet and explore set notation.

Handouts: Handout | Key
Handouts: Handout

This Sunday we are working on some graph theory and geometry problems, with a bit of area approximation remaining from last week.

As a friendly reminder please change your zoom profile name to your actual ORMC registered name (this will help us identify you at the beginning of class)!

Handouts: Worksheet
Handouts: Handout

A few tricks for sums on olympiads

Handouts:
5/16/2021
Handouts: Game

We play another midquarter game

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.

Handouts: Handout

We will define what it means for a sequence to converge in different metric spaces. We will explore many different examples like convergence with the taxicab metric or the discrete metric.

Handouts: Convergence in Metric Spaces I
Handouts: Problems

https://docs.google.com/presentation/d/18iNIzBPqQ0R65gpMm-ZIxQwQ5pn91AiPpFQScuYWCwA/edit#slide=id.gd170f84eb4_0_10

This week we'll go through the challenge problems in the second sets handout and then take a short quiz on everything we've learned so far. If we have extra time, we'll work on some Math Kangaroo problems and Futoshiki puzzles.

Handouts: Handout: Math Kangaroo P4 w/ Futoshiki | Solutions: Math Kangaroo P4 w/ Futoshiki

This week we'll go through the challenge problems in the second sets handout and then take a short quiz on everything we've learned so far. If we have extra time, we'll work on some Math Kangaroo problems and Futoshiki puzzles.

Handouts: Handout: Math Kangaroo P4 w/ Futoshiki | Solutions: Math Kangaroo P4 w/ Futoshiki

How can probability be applied to more than dartboards, dice, and coins? Genetics! Today we will explore how we can use probability in biological applications to figure out blood types, hair colors, nose shapes, and more!

Handouts: Handout

How can probability be applied to more than dartboards, dice, and coins? Genetics! Today we will explore how we can use probability in biological applications to figure out blood types, hair colors, nose shapes, and more!

Handouts: Handout

We visit bipartite graphs!

Handouts: Bipartite graphs

This week we are starting to look at the Taxi-cab metric: an awesome part of analysis

Handouts: Handout
Handouts: Handout
5/23/2021

We introduce a new method to encode partitions of positive integers -- Young diagrams, and use them to prove some beautiful bijections on partitions.

Handouts: Handout

We will continue with combinatorics and in particular, the method of using a Young tableau

Handouts: Handout
Handouts: Handout

We define the notion of a continuous function using limits in arbitrary metric spaces. We apply the Intermediate Value Theorem to a range of functions on the real line.

Handouts: Convergence in Metric Spaces II
Handouts: With extra problems

https://docs.google.com/presentation/d/1QJVrmny8BNaV9ZIywmfFKkUITCwH7IOYbmCughXH0O4/edit?usp=sharing

This week, we'll review the quiz problems and work on some additional Venn diagram problems.

Handouts: Handout: Interperting Venn Diagrams | Solutions: Interpreting Venn Diagrams

This week, we'll review the quiz problems and work on some additional Venn diagram problems.

Handouts: Handout: Interpreting Venn Diagrams | Solutions: Interpreting Venn Diagrams

We will finish our genetics worksheet from last week and then start with a new topic: herd immunity! What needs to happen in a population to reach herd immunity? How can we approach it mathematically?

Handouts: Handout | Key | Herd Immunity Notes

We will finish our genetics worksheet from last week and then start with a new topic: herd immunity! What needs to happen in a population to reach herd immunity? How can we approach it mathematically?

Handouts: Handout | Key | Herd Immunity Notes

Today we study some ideas in propositional logic.

Handouts: Logic1

This week we are studying invariants!

Handouts: Worksheet | Worksheet II
Handouts: Handout

Also discussion of finite abelian groups

Handouts:
5/30/2021

There is no class today due to Memorial Day weekend.

There is no class today due to Memorial Day weekend!

6/6/2021

Game!

Handouts:

We play an end of quarter game.

Handouts: Competition
Handouts: Competition Problems
Handouts: Competition_problems.pdf

This Sunday, we'll experience some Russian Math.

This Sunday, we'll explore some Russian Math

For our last class, we will review all of the best problems and topics from the school year.

For our last class, we will review all of the best problems and topics from the school year.

We have math dominoes problems by Nikita Gladkov.

Handouts: Math Dominos Game with solutions
Handouts: Handout

Dirichlet's theorem on arithmetic progressions, an comprehensive outline thereof

This involves many standard manipulations of sums, complex numbers, and bounding arguments

Handouts:
6/13/2021
Handouts: Total | Measurement Errors | Smart Codes | Sequences