UCLA Olga Radko Endowed Math Circle

LAMC Meetings Archive • Fall 2007–Summer 2020

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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
Fall 2019 quarter // Filter groups by:
10/6/2019
We start off the new year with some problems from the Moscow Math Olympiad.
Handouts: Handout

We start off the new year with some problems from the Moscow Math Olympiad.

Handouts: Week-1
What kinds of patterns can be used as wallpaper? What are their groups of symmetries, and how can we classify them? How many are there? We will attempt to answer some of these questions and learn how to use Thurston's "orbifold notation" for wallpaper patterns.
Handouts: Wallpaper Symmetries

Introduction to Gaussian integers, as well as prime and irreducible Gaussian integers

Handouts: Gaussian Integers 1
Did you ever want to communicate with your friends in a secret way that only those who knew the secret could understand? A cipher is a tool for writing and reading secret messages. We will study three ciphers this time: the reverse cipher, Polybius cipher, and Casear cipher. You will find a template for making a Caesar cipher disk at the link below.
Handouts: Caesar cipher template
Did you ever want to communicate with your friends in a secret way that only those who knew the secret could understand? A cipher is a tool for writing and reading secret messages. We will study three ciphers this time: the reverse cipher, Polybius cipher, and Casear cipher. You will find a template for making a Caesar cipher disk at the link below.
Handouts: Caesar cipher template
To start out the quarter, we will be developing logic skills by looking at problems with hats and doors. 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 | solutions
To start out the quarter, we will be developing logic skills by looking at problems with hats and doors. 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 | solutions
We will start the BNP curriculum.
We will solve a variety of fun problems.

Today we will be looking at how to optimally divide cakes among any amount of people by using an ancient Egyptian technique of breaking fractions into their unit parts but doing so in a specific way, which we will work with and then prove to be optimal.

Handouts: Egyptian Fractions 1

Today we will be looking at how to optimally divide cakes among any amount of people by using an ancient Egyptian technique of breaking fractions into their unit parts but doing so in a specific way, which we will work with and then prove to be optimal.

Handouts: Egyptian Fractions 1

We will be starting off the new quarter with some algebra activities.

Handouts: Handout | Solutions

We will be starting off the new quarter with some algebra activities.

Handouts: Handout | Solutions
We will discuss a few properties concerning representations of nonnegative integers and rationals in a general base (such as divisibility criteria), and particularly in base 10. Then we will apply this knowledge to several Olympiad-type Number Theory problems.
Handouts: Handout | Solutions
10/9/2019
We will go through a sample AMC exam from 2019.
10/13/2019
We start off probability by introducing some set theory notation and defining sample spaces. Then we apply those to classical probability questions.
Handouts: Handout | Homework
We start off probability by introducing some set theory notation and defining sample spaces. Then we apply those to classical probability questions.
Handouts: Week-2 | Homework-2
We will continue to answer some of the questions posed about wallpaper symmetries. For those that finish the worksheet on wallpaper symmetries, we started discussing the origins and early properties of the p-adic numbers.
Handouts: Wallpaper Symmetries | Absolute Values and p-Adics

Characterization of which positive integers are sums of squares

Handouts: Gaussian Integers 2
An anagram is a word or phrase formed by rearranging letters of another word or phrase. Most often, all the original letters are used once. For example, the words "silent" and "listen" form an anagram. This time, we will have anagrams for a warm-up and then study two ciphers: pigpen cipher and rail fence cipher.
An anagram is a word or phrase formed by rearranging letters of another word or phrase. Most often, all the original letters are used once. For example, the words "silent" and "listen" form an anagram. This time, we will have anagrams for a warm-up and then study two ciphers: pigpen cipher and rail fence cipher.
An alternate method of multiplication based in binary numbers and the basic ideas of the distributive property.
Handouts: Handout | solutions
An alternate method of multiplication based in binary numbers and the basic ideas of the distributive property.
Handouts: Handout | solutions
We will continue the BNP curriculum.
We will solve a variety of fun problems.

This week we continue from last week and try to prove a mathematical algorithm, which is a very important activity in mathematics.

Handouts: Egyptian fractions 2

This week we continue from last week and try to prove a mathematical algorithm, which is a very important activity in mathematics.

Handouts: Egyptian fractions 2

We will discuss board-tiling problems-- is it possible to completely cover a region with a particular set of tiles without overlap? -- and interesting mathematical problems that arise from these puzzles.

Handouts: Handout

We will discuss board-tiling problems-- is it possible to completely cover a region with a particular set of tiles without overlap? -- and interesting mathematical problems that arise from these puzzles.

Handouts: Handout
We will solve a variety of geometry problems involving the computation of a length or an area, or using notions about areas to prove an identity. The problems range in difficulty from introductory to fairly challenging.
Handouts: Handout

We will go over these fundamental concepts and do a range of problems.

Handouts: Handout
10/16/2019
We'll continue going through the solutions for the 2019 AMC exams
10/20/2019
We continue our probability unit with a worksheet on Conditional Probability. This includes the Law of Total Probability and Bayes' Theorem.
Handouts: Handout | Homework

We are continuing solving probability problems. The topics for this class are Conditional Probability, Law of Total Probability, Bayes' Rule.

Handouts: Week-3 | Homework-3

We will study the symmetries of frieze patterns, especially those of unimodular frieze patterns. We work toward an interesting result proved by Conway and Coxeter about polygonal structures in frieze patterns.

Handouts: Triangulated Polygons and Frieze Patterns

We will study the symmetries of frieze patterns, especially those of unimodular frieze patterns. We work toward an interesting result proved by Conway and Coxeter about polygonal structures in frieze patterns.

Handouts: Frieze Patterns 1
We will finish the study of the rail fence cipher and take a quiz on ciphers. Remember, we do not quiz students, we quiz teachers. If you get a low grade, it means we didn't do a good job. Please study for the quiz - don't let us down!
We will finish the study of the rail fence cipher and take a quiz on ciphers. Remember, we do not quiz students, we quiz teachers. If you get a low grade, it means we didn't do a good job. Please study for the quiz - don't let us down!

We will continue with different ways to multiplying 2 numbers. This week we will look at Russian Peasant Multiplication, which, surprisingly, has no relation to Russia or Peasants. However, this will be another good way to show the students how to write numbers as sum of powers of 2

Handouts: handout | answers

We will continue with different ways to multiplying 2 numbers. This week we will look at Russian Peasant Multiplication, which, surprisingly, has no relation to Russia or Peasants. However, this will be another good way to show the students how to write numbers as sum of powers of 2

Handouts: handout | answers
We will continue the BNP curriculum.
We will solve a variety of fun problems.

This week we begin on a very very very important topic in mathematics; Modular Arithmetic. Although the importance of this topic is not fully realized until a course in Abstract Algebra, it is still a very important topic because it can instill an understanding that the mathematics or arithmetic that we see commonly is not the only way of operating and that mathematics can be far more general and far reaching.

Handouts: Part 1

This week we begin on a very very very important topic in mathematics; Modular Arithmetic. Although the importance of this topic is not fully realized until a course in Abstract Algebra, it is still a very important topic because it can instill an understanding that the mathematics or arithmetic that we see commonly is not the only way of operating and that mathematics can be far more general and far reaching.

Handouts: Part 1

We will continue to expand on last week's topic of tilings.

Handouts: Handout | Solutions

We will continue to expand on last week's topic of tilings.

Handouts: Handout | Solutions

We solve a couple of introductory problems to Graph Theory and several harder ones. The lesson spans over two weeks.

Handouts: Handout
Using unique factorization into primes and modular arithmetic to solve problems.
10/23/2019
We will go through geometry problems with related to triangles and circles in the context of AMC
10/27/2019

Our third week of probability includes a handout on random variables and expected value, as well as some more interesting conditional probability problems.

Handouts: Handout

Our third week of probability includes a handout on random variables and expected value, as well as some more interesting conditional probability problems.

Handouts: Week-4

We continue our study of the symmetries of unimodular frieze patterns with some challenge problems. We introduce the result proved by Conway and Coxeter connecting polygonal triangulations to frieze patterns.

Handouts: Triangulated Polygons and Frieze Patterns II

We continue our study of the symmetries of unimodular frieze patterns with some challenge problems. We introduce the result proved by Conway and Coxeter connecting polygonal triangulations to frieze patterns.

Handouts: Frieze Patterns 2
We will take the first look at one of the most important concepts in mathematics, that of a function.
We will take the first look at one of the most important concepts in mathematics, that of a function.

This packet introduces the topic of exponents, some of their properties, and size comparison!

Handouts: answers pgs 1-3 | handout

This packet introduces the topic of exponents, some of their properties, and size comparison!

Handouts: answers pgs 1-3 | handout
We will continue the BNP curriculum.
We will solve problems from the Math Kangaroo grades 1-2 contest.

A continuation of next week. Please use this week's sheet as a a set of problems that build stronger arithmetic skills and provide insight into other ways of "counting."

Handouts: Part II

A continuation of next week. Please use this week's sheet as a a set of problems that build stronger arithmetic skills and provide insight into other ways of "counting."

Handouts: Part II

We will be working with some algebra problems this week.

Handouts: Handout | Solutions

We will be working with some algebra problems this week.

Handouts: Handout | Solutions

We continue the lesson from last week with a few Olympiad-type problems in Graph Theory.

Handouts: Handout

Euler's totient function and more number-theoretic functions

Handouts: Handout
10/30/2019

We will go through AMC geometry problems with the emphasis on polygons.

11/3/2019

We will be working through a fun application of probability and graph theory to electrical circuits. No knowledge of physics is necessary, everything needed is contained in the handout.

Handouts: Handout

We will be working through a fun application of probability and graph theory to electrical circuits. No knowledge of physics is necessary, everything needed is contained in the handout.

Handouts: Electrical-Circuits

We work in groups to write rigorous proofs of statements from number theory, graph theory, combinatorics, set theory and other topics.

Handouts: Formal Proofs

Introduction to formal proof writing. Sets, functions, basic combinatorics and number theory proofs

Handouts: Proofs 1

Students will learn the y = f(x) notation.

Students will learn the y = f(x) notation.

Continuing last week's handout!

Handouts: handout | pg 4 answers

Continuing last week's handout!

Handouts: pg 4 answers | handout
We will continue the BNP curriculum.
We will solve problems from the Math Kangaroo grades 1-2 contest.

This week we are working with some logic problems in the guise of "real world" sentences. The hope is that students can see that a basic logical structure is embedded in common speech, and that logical structure is the basis for mathematics and is taken to its most pure form and to its greatest lengths. Please use these worksheets as a warm up for the logical skills needed for induction next week.

Handouts: Handout

This week we are working with some logic problems in the guise of "real world" sentences. The hope is that students can see that a basic logical structure is embedded in common speech, and that logical structure is the basis for mathematics and is taken to its most pure form and to its greatest lengths. Please use these worksheets as a warm up for the logical skills needed for induction next week.

Handouts: Handout

Continuing our work in algebra last week, we will look at remainders and divisibility.

Handouts: Solutions | Handout

Continuing our work in algebra last week, we will look at remainders and divisibility.

Handouts: Solutions | Handout

We discuss various techniques for solving problems involving integer inequalities, specific to Olympiad Number Theory.

Handouts: Handout

NT problems including showing there are infinitely many 4k+1 primes.

Handouts: Handout
11/6/2019

We will cover some of the applications of trigonometry in AMC.

11/10/2019

We dive further into the connection between random walks and electrical circuits, deriving series and parallel circuits. We then discuss electrical circuits on an integer lattice and its connection to Polya's problem.

Handouts: Handout

We dive further into the connection between random walks and electrical circuits, deriving series and parallel circuits. We then discuss electrical circuits on an integer lattice and its connection to Polya's problem.

Handouts: Week-6-Handout

We work in groups to write rigorous proofs of statements from number theory, graph theory, combinatorics, set theory and other topics.

Handouts: Formal Proofs II

Introduction to formal proof writing. Sets, functions, basic combinatorics and number theory proofs

Handouts: Proofs 2

We will continue studying functions.

We will continue studying functions.

Review what we learned about exponents and continue to build how to compare numbers

Handouts: handout | solutions I | solutions II

Review what we learned about exponents and continue to build how to compare numbers

Handouts: handout | solutions I | solutions II
We will continue the BNP curriculum.
We will solve problems from the Math Kangaroo grades 1-2 contest.

This week we are starting induction! If these problems are challenging (they most likely are) do not panic. These are meant to be attempted to the best of everyone's ability and asking questions is extremely encouraged.

Handouts: Induction

This week we are starting induction! If these problems are challenging (they most likely are) do not panic. These are meant to be attempted to the best of everyone's ability and asking questions is extremely encouraged.

Handouts: Induction

We will continue our work in algebra with GCDs.

Handouts: Handout | Solutions

We will continue our work in algebra with GCDs.

Handouts: Handout | Solutions

We continue the lecture from last time, with a few additional Olympiad-type Number Theory problems.

Handouts: Handout

The useful Lifting the Exponent Lemma, which states that for odd prime p and m,n such that p|(m-n), vp(m^p-n^p) = vp(m-n)+1.

Handouts: Handout | Solutions
11/17/2019

We start a new unit: Complex Numbers! We will see that while complex numbers might seem unnatural, they are actually extremely useful and can simplify problems that have seemingly nothing to do with the square root of -1.

Handouts: Handout

We start a new unit: Complex Numbers! We will see that while complex numbers might seem unnatural, they are actually extremely useful and can simplify problems that have seemingly nothing to do with the square root of -1.

Handouts: Week-7-Handout

We work on Olympiad style problems about Combinatorics, Number Theory, Probability, Geometry, and basic Set and Function Theory.

Handouts: Olympiad Problems
Handouts: Olympiad Style Problems 1 | Solutions

We will review functions. Then students will take quiz 2.

We will review functions. Then students will take quiz 2.

A look at how we can organize multiple sets of objects/people/things using venn diagrams!

Handouts: handout 1 | handout 2 | solutions 1 | solutions 2

A look at how we can organize multiple sets of objects/people/things using venn diagrams!

Handouts: handout 1 | handout 2 | solutions 1 | solutions 2
We will continue the BNP curriculum.
We will solve problems from the Math Kangaroo grades 1-2 contest.

This week we are continuing on induction proofs. It is fully expected that these problems are challenging, but they are extremely important to attempt, not necessarily get correct. The point of these two weeks is to 1. prepare students for induction proofs (the basis of discrete math and many proofs about finite objects) and 2. to challenge them to see the power of mathematical logic to prove an infinite amount of statements in a finite time.

Handouts: Replacements for 3 and 5 | Main worksheet (ignore problems 3 and 5)

This week we are continuing on induction proofs. It is fully expected that these problems are challenging, but they are extremely important to attempt, not necessarily get correct. The point of these two weeks is to 1. prepare students for induction proofs (the basis of discrete math and many proofs about finite objects) and 2. to challenge them to see the power of mathematical logic to prove an infinite amount of statements in a finite time.

Handouts: Replacements for 3 and 5 | Main hand out (ignore problem 3 and 5)

The Euclidean Algorithm is a way to find the greatest common divisor of two numbers. Using what we've learned in algebra in the past weeks, we will investigate how this method works.

Handouts: Handout | Solutions

The Euclidean Algorithm is a way to find the greatest common divisor of two numbers. Using what we've learned in algebra in the past weeks, we will investigate how this method works.

Handouts: Handout | Solutions

We define convexity, convex hulls and triangulations, then solve a few Olympiad-type problems in Geometric Combinatorics.

Handouts: Handout

Techniques for solving equations with the constraint that the variables must take integer values.

Handouts: Handout
11/20/2019

We will go through AMC problems covering several sections of number theory

11/23/2019

We define and discuss basic concepts from trigonometry, including the law of sines and the law of cosines. We then apply these notions to solving several Olympiad-type geometry problems.

Handouts: Handout
11/24/2019

We will explore roots of unity and some geometric aspects of the complex numbers.

Handouts: Notes | Handout

We will explore roots of unity and some geometric aspects of the complex numbers.

Handouts: Notes | Complex-Numbers-II

We do further Olympiad style problems about Combinatorics, Number Theory, Probability, Geometry, and basic Set and Function Theory.

Handouts: Olympiad Problems Hints | Olympiad Problems II | Olympiad Problem Solutions
Handouts: Olympiad Style Problems 2 | Solutions | Hints

This week we will be looking at what it means for shapes to be similar, as well as exploring how we can add on to a shape to create another that is similar to the original.

Handouts: handout 1 | handout 2 | solutions 1 | solutions 2

This week we will be looking at what it means for shapes to be similar, as well as exploring how we can add on to a shape to create another that is similar to the original.

Handouts: handout 1 | handout 2 | solutions 1 | solutions 2
We will continue the BNP curriculum.
We will solve problems from the Math Kangaroo grades 1-2 contest.

This week we are working on finding the perimeter of some interesting shapes. These problems are designed to prepare students to quickly recognize the shapes they are looking at and how they can break them into easily solvable pieces.

Handouts: Worksheet

This week we are working on finding the perimeter of some interesting shapes. These problems are designed to prepare students to quickly recognize the shapes they are looking at and how they can break them into easily solvable pieces.

Handouts: Perimeter

Some more fun with algebra, primes, and GCDs!

Handouts: Handout | Solutions

Some more fun with algebra, primes, and GCDs!

Handouts: Handout | Solutions
Handouts: Handout
12/1/2019

First, students will learn basic facts about polygons. Then they will proceed to build some solids out of cubes and to draw their 2D projections.

First, students will learn basic facts about polygons. Then they will proceed to build some solids out of cubes and to draw their 2D projections.

12/8/2019

For the last meeting of the quarter we will split up into groups and compete!

We're playing a traditional end-of quarter game.

Handouts: Problems with Solutions

A 2 hour long math competition against the other high school group!

Handouts: Problems with Solutions

In this class, students will draw 2D projections of more complicated solids.

In this class, students will draw 2D projections of more complicated solids.

For the last meeting of 2019, we will be playing a Math Review Game!

For the last meeting of 2019, we will be playing a math review game!

We will continue the BNP curriculum.
We will solve problems from the Math Kangaroo grades 1-2 contest.

This week we are playing a Math Circle classic! The rules are attached. Hope everyone has a great time :).

Handouts: How to Play

This week we are playing a Math Circle classic! The rules are attached. Hope everyone has a great time :).

Handouts: How to Play

We will finish out the quarter with some friendly competition!

We will finish out the quarter with some friendly competition!

Review of NT, review of counting and intro to combinatorial identities.

Handouts: Handout
Winter 2020 quarter // Filter groups by:
1/12/2020

We start off the new (calendar) year with something you probably haven't seen before: tropical geometry. Throw out everything you know about addition and multiplication; we define new operations and explore polynomials under the new rules.

Handouts: Handout

We start off the new (calendar) year with something you probably haven't seen before: tropical polynomials. Throw out everything you know about addition and multiplication; we define new operations and explore polynomials under the new rules.

Handouts: Tropical Polynomials | Solutions

We motivate the study of metrics by introducing the taxicab metric (the distance traveled by a taxi in a city with a grid layout). How would you define a circle, line segment between points, or a parabola with a new notion of distance?

Handouts: Taxicab Geometry | Taxicab Geometry Challenge Problems

We introduce a new notion of distance - the taxicab metric. We investigate how geometry behaves with this new distance, and try to find similarities and differences between this and the Euclidean metric.

Handouts: Metrics 1

Students were asked to study the 8th packet, More Solids, during the Winter break. Since the packet is rather challenging for the age, we will through it class.

We started a warm-up discussing auction theory. We will begin our exploration of game theory by starting with the example of subtraction games of varying subtraction sets.

Handouts: handout | solutions

We started a warm-up discussing auction theory. We will begin our exploration of game theory by starting with the example of subtraction games of varying subtraction sets.

Handouts: handout | solutions

This week we will be looking at geometrical numbers and the successive difference found in the sequence of these numbers. Interestingly, these types of numbers have been studied for thousands of years ever sense the Ancient Greeks. These sorts of problems are very useful to build strategies for induction proofs and pattern spotting.

Handouts: Worksheet | Solutions

This week we will be looking at geometrical numbers and the successive difference found in the sequence of these numbers. Interestingly, these types of numbers have been studied for thousands of years ever sense the Ancient Greeks. These sorts of problems are very useful to build strategies for induction proofs and pattern spotting.

Handouts: Worksheet | Solutions

Welcome back to Math Circle! We will kick off the quarter with some applications of graphs and geometry.

Handouts: Handout | Solutions | Homework | Homework Solutions

Welcome back to Math Circle! We will kick off the quarter with some applications of graphs and geometry.

Handouts: Handout | Solutions

We state and prove Ceva's theorem, then solve a few Olympiad-type problems.

Handouts: Handout
Handouts: Handout
1/15/2020

We will go through AMC problems related to statistics and probability theory

1/17/2020
Handouts: Handout
1/19/2020
Handouts: Tropical Orthogonal Representations

Using the taxicab metric as a guide, we define the general notion of metric and give numerous examples.

Handouts: Metrics

We continue our study of metrics, this time considering more abstract examples. We discuss sequences, and we learn that with regards to sequences, the taxicab metric and the Euclidean metric are equivalent.

Handouts: Metrics 2

Students will solve problems from the 2014 Canadian Math Kangaroo test.

Students will solve problems from the 2014 Canadian Math Kangaroo test.

This week we started off class discussing "I cut you choose." We then continued will be continuing our discussion of subtraction games, in particular the Game 21. With 21 sticks, each player can either take away 1,2, or 3 sticks each turn, what is the winning strategy if you don't want to take the last stick? If you want to take the last one? How are these two related?

This week we started off class discussing "I cut you choose." We then continued will be continuing our discussion of subtraction games, in particular the Game 21. With 21 sticks, each player can either take away 1,2, or 3 sticks each turn, what is the winning strategy if you don't want to take the last stick? If you want to take the last one? How are these two related?

Homework

Handouts: Homework Problems

This week we will be working on some very interesting logical problems. These sorts of problems, although not generalization to higher mathematical theory, are very useful in practicing mathematical logic and intuition.

Handouts: Worksheet | Solutions

This week we will be working on some very interesting logical problems. These sorts of problems, although not generalization to higher mathematical theory, are very useful in practicing mathematical logic and intuition.

Handouts: Worksheet | Solutions

More work with graphs and geometry.

Handouts: Handout | Solutions

More work with graphs and geometry.

Handouts: Handout | Solutions

The students took a 2-hour long practice exam for BAMO-type competitions. There were 3 problems (in Geometry, Combinatorics and Number Theory).

Handouts: Handout
Handouts: Handout
1/22/2020

AMC problems related to high school level functions theory

1/26/2020

In the first part of a two (or three) part series on cardinality, we deal with bijections and give a proper definition for the cardinality of a set.

Handouts: Handout | Homework

In the first part of a 3-part series on cardinality, we deal with bijections and give a proper definition for the cardinality of a set.

Handouts: Infinity-1 | Homework

We can extend the integers by including square roots of integers like -1 or 3. Can we predict which prime integers remain prime in the extension?

Handouts: Primes in Extensions of the Integers I

We can extend the integers by including square roots of integers like -1 or 3. Can we predict which prime integers remain prime in the extension?

Handouts: Prime Splitting 1

First, we will go through a few problems from the 9th packet. Then students will take a quiz on polygons and solids. Then we will finish discussing the packet.

First, we will go through a few problems from the 9th packet. Then students will take a quiz on polygons and solids. Then we will finish discussing the packet.

This week, we will be preparing for the Math Kangaroo contest by working out various types of problems!

Handouts: solutions | handouts

This week, we will be preparing for the Math Kangaroo contest by working out various types of problems!

Handouts: handouts | solutions

More work with graph theory and geometry proofs.

Handouts: Handout | Solutions

More work with graph theory and geometry proofs.

Handouts: Handout | Solutions

Today we discussed the solutions for Practice Exam 1, and then solved three other problems.

Handouts: Handout
Handouts: Handout | Solutions
1/29/2020

We will go through an actual AMC test, and discuss some of the solutions

2/2/2020

We continue with part 2 of our Infinity sequence. This time we define cardinality through injectivity and bijectivity.

Handouts: Homework | Handout

In the second part of the sequence, we explore more bijections and prove several surprising facts about the cardinalities about the naturals, integers, and the reals.

Handouts: Infinity 2 | Homework

Happy Groundhog Day! With the help of famous results about quadratic reciprocity we will classify primes in most quadratic extensions of the integers.

Handouts: Primes in Extensions of the Integers II

With the help of famous results about quadratic reciprocity we will classify primes in most quadratic extensions of the integers.

Handouts: Primes Splitting 2

We will solve logic problems from the Knights and Liars packet. (Oleg Gleizer)

We will solve logic problems from the Knights and Liars packet. (Oleg Gleizer)

Intro: How many days are in a year? How are leap years counted? Why do we include every 400 years, but not 100, 200, or 300? Hint: a year is technically about 365.25 days - 11 minutes.

In this handout we examine how to systematically perform calculations to find the day of the week (Sunday, Monday, Tuesday, etc.) a particular date is, e.g., your birthday

Handouts: handout | solutions

Intro: How many days are in a year? How are leap years counted? Why do we include every 400 years, but not 100, 200, or 300? Hint: a year is technically about 365.25 days - 11 minutes.

In this handout we examine how to systematically perform calculations to find the day of the week (Sunday, Monday, Tuesday, etc.) a particular date is, e.g., your birthday

Handouts: handout | solutions

This week we will be playing a game with the Math Kangaroo Practice tests. The Math Kangaroo is a math test for young mathematicians and is usually held in the middle of the year.

The game we are playing this week will encourage team work, communication, and dieligent justification of answers.

Handouts: Test 2013 | Test 1998

This week we will be playing a game with the Math Kangaroo Practice tests. The Math Kangaroo is a math test for young mathematicians and is usually held in the middle of the year.

The game we are playing this week will encourage team work, communication, and dieligent justification of answers.

Handouts: Test 2013 | Test 1998

More graphs and geometry.

Handouts: Handout | Solutions | Homework | Homework Solutions

More graphs and geometry.

Handouts: Handout | Solutions

The students took an Olympiad-style practice exam with 3 problems (in Geometry, Algebra, and Combinatorics).

Handouts: Handout
Handouts: Solutions | Handout
2/3/2020

This week we start on the topic of Graph Theory! This topic considers a particular object of mathematics, the Graph, which is defined by its vertices and the edges connecting them. We will be considering the degree of the vertices as well a particular type of Graph, the Bipartite Graph.

Handouts: Graph Theory Sheet | Solutions
2/9/2020
Handouts: Handout

This week we are concluding our conversation about infinity with some more challenge problems on constructing bijections.

We also look into some mathematical paradoxes related to infinity and not only.

Handouts: Infinity-3

We will introduce the generating function, a creative, combinatorial tool that can simply solve many interesting problems. By the end, we will have used generating functions to study the Fibonacci sequence, dice games, and ways to pay the unlucky cashier with coins.

Handouts: Generating Functions | Generating Functions Solutions

We will introduce the generating function, a creative, combinatorial tool that can simply solve many interesting problems. By the end, we will have used generating functions to study the Fibonacci sequence, dice games, and ways to pay the unlucky cashier with coins.

Handouts: Generating Functions

Students will learn the notion of a statement, simple and composite, and how the truth function takes value on statements.

Students will learn the notion of a statement, simple and composite, and how the truth function takes value on statements.

Warm-Up: What is the trick to square numbers ending with 5 and why does it work?

More math kangaroo problems!

Handouts: handouts | solutions

Warm-Up: What is the trick to square numbers ending with 5 and why does it work?

More math kangaroo problems!

Handouts: handout | solutions

This week we start on the topic of Graph Theory! This topic considers a particular object of mathematics, the Graph, which is defined by its vertices and the edges connecting them. We will be considering the degree of the vertices as well a particular type of Graph, the Bipartite Graph.

Handouts: Solutions | Graph Theory Sheet

We will be finishing up our unit on graphs and geometry this week!

Handouts: Homework | Handout | Solutions | Homework Solutions

We will be finishing up our unit on graphs and geometry this week!

Handouts: Handout | Solutions

Today we went through the solutions for the practice exam, and we solved a few extra problems in class.

Next time there will be no class (on President's Day).

Handouts: Handout
Handouts: Handout
2/16/2020

Please use the worksheet from last week.

This week we completed the rest of the Graph Theory packet, going over topics such as the coloring of a graph, which is the least number of colors required to fully "color" a graph. An interesting theory associated with this is that if one considers ANY map that separates the area into counties, states, cities, etc. then one can color that map with four colors such that no two adjacent counties, states, cities, etc. have the same color. This theorem was proved using computers!

Handouts: Graph Theory Sheet | Solutions

Please use the worksheet from last week.

Handouts: Graph Theory Sheet | Solutions
Handouts: Handout
2/23/2020

We start off our two week unit on Point Mass Geometry. This is a topic that lets you get the nice results of planar geometry without the tedious work!

Handouts: Handout | Solutions

We will learn a new method of solving geometry problems with assigning masses to points and using intuition from physics.

Handouts: Handout | Solutions

In the first week of a three week sequence, we introduce basic probability concepts such as conditional probability, Bayes' Rule, and tower property

Handouts: Measure and Probability I
Handouts: Probability 1

We will continue studying the Mathematical Logic mini-course.

We will continue studying the Mathematical Logic mini-course.

Warm-Up: Multiply two-digit and three-digit numbers using the criss-cross method! It really reduces the amount of work required!

Today we will be introducing the topic of Polyhedras and learning the terminology.

Handouts: handout | solutions

Warm-Up: Multiply two-digit and three-digit numbers using the criss-cross method! It really reduces the amount of work required!

Today we will be introducing the topic of Polyhedras and learning the terminology.

Handouts: handout | solutions

This week we will be solving the instant insanity puzzle. This puzzle is a mathematicians favorite as it is difficult to solve by “brute force,” but permits a very simple and elegant mathematical solution. We will begin by trying brute force tactics and when that fails us we will find a way to use graph theory to solve the puzzle very simply.

here is the amazon link to the puzzle: Winning Moves Games Instant Insanity https://www.amazon.com/dp/B004KCN6EQ/ref=cm_sw_r_cp_api_i_QwSuEb6C13BXG

This week we will be solving the instant insanity puzzle. This puzzle is a mathematicians favorite as it is difficult to solve by “brute force,” but permits a very simple and elegant mathematical solution. We will begin by trying brute force tactics and when that fails us we will find a way to use graph theory to solve the puzzle very simply.

here is the amazon link to the puzzle: Winning Moves Games Instant Insanity https://www.amazon.com/dp/B004KCN6EQ/ref=cm_sw_r_cp_api_i_QwSuEb6C13BXG

We will begin a new unit on game theory.

Handouts: Handout | Homework | Homework Solutions

We will begin a new unit on game theory.

Handouts: Handout

We solve five final practice problems for the Bay Area Mathematical Olympiad.

Handouts: Handout

BAMO 2016

Handouts: Handout
3/1/2020
Handouts: Handout | Solutions
Handouts: Handout

We will continue our study of probability by introducing random variables and distributions.

Handouts: Measure and Probability II
Handouts: Probability 2

We will fist check some problems from the packet 12 homework. Then we will begin studying logic gates.

We will fist check some problems from the packet 12 homework. Then we will begin studying logic gates.

Warm-Up: If I give you two numbers, like 998 and 992, and I ask you to multiply them together using conventional math techniques, you end up writing a lot of numbers to generate the answer. But notice that 998 is just 2 shy of 1000, and 992 is just 8 shy of 1000. If you multiply 2 times 8, you get 16. And if you take 8 away from 998, or you take 2 away from 992, you get 990. And guess what? The correct answer is 990016. Similarly, if I ask you to multiply 990 times 991, you could work it out ... or you could recognize that 990 is 10 below 1000, 991 is 9 below, the product of 9 and 10 is 90, and 990 minus 9 is 981, and 991 minus 10 is also 981. The answer: 981090.

The insight: if you rewrite 998 as (1000-2) and 992 as (1000-8), multiply the two we get 1000*1000 - 2*1000 - 8*1000 + 8*2. Hence we get (1000-2-8)*1000 + 16 = 990,016.

We will be continuing our topic on Polyhedra from last week!

Handouts: handout | solutions

Warm-Up: If I give you two numbers, like 998 and 992, and I ask you to multiply them together using conventional math techniques, you end up writing a lot of numbers to generate the answer. But notice that 998 is just 2 shy of 1000, and 992 is just 8 shy of 1000. If you multiply 2 times 8, you get 16. And if you take 8 away from 998, or you take 2 away from 992, you get 990. And guess what? The correct answer is 990016. Similarly, if I ask you to multiply 990 times 991, you could work it out ... or you could recognize that 990 is 10 below 1000, 991 is 9 below, the product of 9 and 10 is 90, and 990 minus 9 is 981, and 991 minus 10 is also 981. The answer: 981090.

The insight: if you rewrite 998 as (1000-2) and 992 as (1000-8), multiply the two we get 1000*1000 - 2*1000 - 8*1000 + 8*2. Hence we get (1000-2-8)*1000 + 16 = 990,016.

We will be continuing our topic on Polyhedra from last week!

Handouts: handout | solutions

This week we will be working with probability! We will be learning how to calculate the expected value of a given game. This sort of "average" thinking is used all the time by everyone unconsciously and explicitly.

Handouts: Probability sheet

This week we will be working with probability! We will be learning how to calculate the expected value of a given game. This sort of "average" thinking is used all the time by everyone unconsciously and explicitly.

Handouts: Probability sheet

More game theory! We will introduce winning and losing positions.

Handouts: Handout | Solutions | Homework

More game theory! We will introduce winning and losing positions.

Handouts: Handout | Solutions

We introduce and study the basic properties of complex numbers, with the goal of using them to solve geometry problems.

Handouts: Handout
Handouts: Handout
3/4/2020

We will go through some AIME problems from past exams

3/5/2020

We continue our discussion of complex numbers and start applying them to Euclidean geometry problems.

Handouts: Handout
3/8/2020

We continue working on the second handout on point mass geometry. This gets into negative masses and barycentric coordinates.

Handouts: Handout | Solutions
Handouts: Handout

We will continue our study of random variables by introducing the Bernoulli, geometric, and binomial distributions. As a final surprising example, we construct a famous example of a non-measurable set.

Handouts: Measure and Probability I, II, and III

We will finish studying logic gates and will study double negation.

We will finish studying logic gates and will study double negation.

This week we will be continuing our topic of Polyhedras, but this time developing a more rigorous relationship between edges, vertices, and faces!

Handouts: handout

This week we will be continuing our topic of Polyhedras, but this time developing a more rigorous relationship between edges, vertices, and faces!

Handouts: handout

This week we played the famous Math Dominoes Game with an added CHALLENGE ROUND. I hope everyone has a great time!

Continuing our unit on game theory.

Handouts: Handout | Solutions

Continuing our unit on game theory.

Handouts: Handout | Solutions
Handouts:
3/15/2020

For our final (virtual) meeting of the quarter, we will be hosting a Kahoot competition over the Zoom application. Please make sure to download the app at https://zoom.us/download and join the meeting from 4-6 pm at the link sent via email. See you there!

We will prepare for the upcoming Math Kangaroo competition.

We will prepare for the upcoming Math Kangaroo competition.

This week we will be continuing our topic of Polyhedras, but this time developing a more rigorous relationship between edges, vertices, and faces!

Handouts: handout

This week we will be continuing our topic of Polyhedras, but this time developing a more rigorous relationship between edges, vertices, and faces!

Handouts: handout

No in-person class

Handouts:
Spring 2020 quarter // Filter groups by:
4/5/2020

Introduction to formal proof writing. Sets, functions, basic combinatorics and number theory proofs

Handouts: Handout

We'll go over various proof techniques and practice writing rigorous proofs.

Handouts: FormalProofs

We review the material from last quarter.

Handouts: Time Travel

Travel back in time and revisit some of the old topics from Winter 2020

Handouts: Time Travel

We will discuss some problems from packet 14. Then students will take a quiz.

We will discuss some problems from packet 14. Then students will take a quiz.

This helps students see what age-appropriate math problems look like in a different country!

Handouts: handout | solutions

This helps students see what age-appropriate math problems look like in a different country!

Handouts: handout | solutions

This week we solved a mathematical trick using equivalence relations! Equivalence relations are extremely important in mathematics and allow mathematicians to formally categorize "objects" in a particular "Space."

Handouts: Handout | Solutions

This week we solved a mathematical trick using equivalence relations! Equivalence relations are extremely important in mathematics and allow mathematicians to formally categorize "objects" in a particular "Space."

Handouts: Handout | Solutions

Welcome back to (virtual) Math Circle! We're kicking off the quarter with some proofs by induction.

Handouts: Handout | Homework
Handouts: Lecture Notes | Handout
4/12/2020

Introduction to formal proof writing. Sets, functions, basic combinatorics and number theory proofs

Handouts: Handout

We'll continue working on the handout on writing formal and rigorous proofs.

Handouts: FormalProofs-2

We use our knowledge of metrics to study and prove theorems about contraction maps.

Handouts: Fixed Points

We study fixed points, and a few theorems that let us find them.

Handouts: Fixed Points

We will meet inhabitants of a 1-dimesional world, called Lineland.

We will meet inhabitants of a 1-dimesional world, called Lineland.

This week, we will be introducing the distance formula!

Handouts: handout | solutions

This week, we will be introducing the distance formula!

Handouts: handout | solutions

We will continue working on proofs by induction.

Handouts: Handout | Homework

We solve a few more difficult geometry problems using complex numbers.

Handouts: Handout

We prove that for all n there exists a prime number between n and 2*n.

Handouts: Handout | Lecture Notes
4/13/2020

This week we worked on some intuitive mathematics! By that I mean mathematics that can be reliably answered by the physical intuition we have all built up over our lives. This "weighty" way of thinking that we are all seemingly borne with was first formalized mathematically in Ancient Greece and this week we get to explore some of its implications.

Handouts: Worksheet | Solutions
4/17/2020
4/19/2020
Handouts: Handout

We are finishing the handout on writing formal and rigorous proofs.

Sperner's Lemma is an interesting result about the coloring of special graphs. We connect the result with a famous result, the Brouwer fixed-point theorem.

Handouts: Sperner's Lemma

We discuss some interesting graph theory, including the Euler characteristic, and proving Brouwer's Fixed Point Theorem with Sperner's Lemma.

Handouts: Graphs 1

We will continue to study packet 15 starting from the problem 15.10.

We will continue to study packet 15 starting from the problem 15.10.

We will be continuing the idea of rates from last week, but this time applying to trade! What is the best way to optimize the outcome of both parties in trade? When is trade not a win-win situation?

Handouts: handout

We will be continuing the idea of rates from last week, but this time applying to trade! What is the best way to optimize the outcome of both parties in trade? When is trade not a win-win situation?

Handouts: handout

This week we will be taking on some challenging problems from the Russian math Olympiad! These are questions designed to test some of the smartest young mathematicians. I hope everyone is challenged and excited!

Handouts: Handout

This week we will be taking on some challenging problems from the Russian math Olympiad! These are questions designed to test some of the smartest young mathematicians. I hope everyone is challenged and excited!

Proofs by induction in algebra, and some geometry.

Handouts: Handout | Homework | Solutions

We introduce basic notions of probability spaces, outcomes, events and independence, and solve a few related competition-style problems.

Handouts: Handout
Handouts: Handout
4/26/2020

We take a look at the HM-GM-AM-QM inequalities and their applications.

Handouts: Handout

We take a look at the HM-GM-AM-QM inequalities and their applications.

Handouts: Inequalities

We continue working on last week's worksheet, with a few extra problems on Brouwer's Fixed Point Theorem, Euler characteristic, and crossing numbers.

Handouts: More Graphs

We continue working on last week's worksheet, with a few extra problems on Brouwer's Fixed Point Theorem, Euler characteristic, and crossing numbers.

Handouts: More Graphs

We will prepare for the upcoming on-line MK competition.

We will prepare for the upcoming on-line MK competition.

This week, we will continue our discussion on trade, learning about opportunity costs as a way of determining when trade is a win-win situation and when it is not, as well as finding a rate that both traders will be happy with

Handouts: handout

This week, we will continue our discussion on trade, learning about opportunity costs as a way of determining when trade is a win-win situation and when it is not, as well as finding a rate that both traders will be happy with

Handouts: handout

This week we will reviewing the solutions to the Olympiad as well as starting a section on decimals and fractions.

Handouts: Handout
Handouts: Handout

We continue with induction!

Handouts: Handout | Solutions | Homework

We introduce random variables, expectations, variance, independence and functions of random variables, then apply these notions to Olympiad-type problems.

Handouts: Handout
Handouts: Handout
5/3/2020

We continue with more applications of our HM-GM-AM-QM inequalities.

Handouts: Handout | Solutions

We continue the handout on inequalities.

Handouts: Inequalities-2

We introduce braids, knots, and links.

Handouts: Braids and Knots

We study braids and begin to study knots.

Handouts: Knots Week 1

The class will be a mixture of the Flatland handout and MK preparation.

The class will be a mixture of the Flatland handout and MK preparation.

A fun game played on a square grid... a twist on Latin squares!

Handouts: handout | solutions

A fun game played on a square grid... a twist on Latin squares!

Handouts: handout | solutions
Handouts: Handout
Handouts: Worksheet

This week we will introduce our unit on bipartite graphs.

Handouts: Handout | Solutions | Homework

We introduce the basic notions of convergence for sequences and series, and solve a few related problems.

Handouts: Handout
Handouts: Handout
5/10/2020
Handouts: Handout

We introduce Reidemeister moves and the idea of a knot invariant. We work up to an interesting knot invariant that has the structure of a quandle.

Handouts: Knot Invariants and Quandles

We study a knot invariant, with the structure of a quandle.

Handouts: Knots Week 2

Students will review the Lineland and Flatland material and take a quiz.

Students will review the Lineland and Flatland material and take a quiz.

This week we will introduce the idea of Prisoner's Dilemma!

Handouts: teaching notes

This week we will introduce the idea of Prisoner's Dilemma!

Handouts: teaching notes
Handouts: Handout
Handouts: Worksheet

More work with bipartite graphs and geometry.

Handouts: Handout | Solutions | Homework

We continue talking about sequences and series, and solve a few harder related problems.

Handouts: Handout

Definitions, examples, subgroups and isomorphisms, and some philosophy

Handouts: Handout
5/17/2020

We introduce our most powerful knot invariants yet: the normalized Kauffman bracket and the Jones polynomial.

Handouts: Kauffman Bracket and Jones Polynomial

We study a more difficult knot invariant - the Jones polynomial.

Handouts: Knots Week 3

Students will learn to solve problems working backward.

Students will learn to solve problems working backward.

This week, we will talk about a cool application of combinations and permutations!

Handouts: homework | teaching notes

This week, we will talk about a cool application of combinations and permutations!

Handouts: teaching notes | homework
Handouts: Handout

Today we pivot from fractions and decimals and focus on percents, a related and equally important topic.

Handouts: Handout

This week, we focus on geometry.

Handouts: Homework | Handout | Solutions

We introduce continuous and differentiable functions, and study a few of their basic properties.

Handouts: Handout

Lagrange's theorem, quotients, maps between groups

5/24/2020

Memorial Day Holiday

5/31/2020

We will continue studying backward reasoning.

We are finishing the Inversion handout.

Handouts: Problems
Handouts: Game Theory

We will continue studying backward reasoning.

We will continue studying backward reasoning.

We will be going over the hw from last time, and working through a new application!

Handouts: handout | solution sketches

We will be going over the hw from last time, and working through a new application!

Handouts: handout | solution sketches
Handouts: Handout

Time to put your Math Circle knowledge to the test! We will be revisiting some of the topics we've covered this past year.

Handouts: Handout | Solutions

We finalize our discussion of differentiability with more rigor, and apply it to related Olympiad-type problems.

Handouts: Handout

Group actions and Burnside's Lemma

6/7/2020

We finish off the school year with a Kahoot individual competition emphasizing problems and techniques that we have done this year.

Handouts: Problems
Handouts: Problems

We will help a mouse and two ants to travel through and around a cube. The journeys reveal a wealth of ideas and serve as an intro to studying nets of cubes.

We will help a mouse and two ants to travel through and around a cube. The journeys reveal a wealth of ideas and serve as an intro to studying nets of cubes.

We will be reviewing the concepts we have learned for the quarter

Handouts: questions

We will be reviewing the concepts we have learned for the quarter

Handouts: questions

Finishing out the quarter with some friendly competition.

We introduce generating functions and apply them to combinatorial sequences such as binomial coefficients, the Fibonacci numbers and partitions.

Handouts: Handout
Summer 2020 quarter // Filter groups by:
6/27/2020

Here are the two handouts for meeting 1 on 6/28.

Handouts: |
6/28/2020

We will begin to study Roman numerals.

We will begin to study Roman numerals.

We will study the Polybius and Caesar ciphers.

We will study the Polybius and Caesar ciphers.

Handouts: Handout

LAMC class for intermediate 2b. We work on Math kangaroo problems!

Handouts: 1998 AMC 8 math problems!

We'll start off this summer with a lecture on basic test-taking techniques. Students will take the 2018 AMC 8 as a diagnostic test. You can access the problems and solutions from that exam on the AoPS website.

7/12/2020

We will use craft sticks to solve more challenging Roman numerals problems.

We will use craft sticks to solve more challenging Roman numerals problems.

We will check the first homework and move on to study the second packet.

We will check the first homework and move on to study the second packet.

Handouts: Handout

We'll look at the principle of mathematical induction and how we can use it to prove various algebraic and geometric statements.

Handouts: Induction 1

We introduce mathematical induction in this class. Handouts are scheduled to be sent out the morning of.

Handouts: Induction 1

After week one's diagnostic test, we start the official AMC 8 prep portion of the course with an introductory lesson on combinatorics. See the attached documents below for resources used in class that can be helpful for future study.

Handouts: Lecture Notes | AMC 8 Combinatorics Problems + Links to Solutions
7/19/2020

We will study more complex Roman numerals.

We will study more complex Roman numerals.

We will review anagrams and then study the pigpen and rail fence cipher.

We will review anagrams and then study the pigpen and rail fence cipher.

Handouts: Handout

We'll continue looking at induction and solve problems involving more complicated sequences and the Fibonacci numbers.

Handouts: Induction 2
Handouts: Induction 2 (Strong induction) | Induction 1 (Recap, reading)

This week, we'll be using the combinatorial skills from week two to solve more advanced problems in statistics and probability. All handouts and resources should be posted below.

Handouts: In-Class Slides | AMC 8 Stats & Probability Problems + Links to Solutions
7/26/2020

We will review the rail fence cipher and the ciphers we have studies earlier.

We will review the rail fence cipher and the ciphers we have studies earlier.

Handouts: Handout

We'll look at the properties of looking at integers mod m.

Handouts: Modular Arithmetic 1

We introduce graph theory.

Handouts: Part 1 Introduction to Graph Theory

This week, we'll once again be building on our previous knowledge by learning about number theory, the mathematical study of integers. All associated resources used in class should be posted below.

Handouts: Lecture Notes | AMC 8 Number Theory Problems + Links to Solutions