More about senior sup-maths courses

Course 7

Prerequisites

This is our first senior level course, thus it's meant to be very demanding and we expect more commitment than for junior and intermediate courses. About 2-2.5 years of problem-solving experience and Olympiad maths is necessary to not get lost:)

Confidence in performing complicated algebraic manipulations, being comfortable with exponentiation and manipulations of functions is required. Understanding of mathematical induction will be assumed in many topics. Geometry at the level of 4-B is essential as well as number theory at the level of 5-B (see relevant course descriptions). Familiarity with graphs at the level of 4-A and other combinatorical ideas from the relevant lessons in 5-A and 6-A are very useful.

In short, successfully completing all intermediate courses, or being confident in all the content there provides sufficient background:)

Confidence in performing complicated algebraic manipulations, being comfortable with exponentiation and manipulations of functions is required. Understanding of mathematical induction will be assumed in many topics. Geometry at the level of 4-B is essential as well as number theory at the level of 5-B (see relevant course descriptions). Familiarity with graphs at the level of 4-A and other combinatorical ideas from the relevant lessons in 5-A and 6-A are very useful.

In short, successfully completing all intermediate courses, or being confident in all the content there provides sufficient background:)

About the content

The course will start with miscellaneous ideas in cell combinatorics, followed by the brief revision of geometry and number theory done in earlier topics. Don't be fooled by the word revision, these lessons will not be any easier than others (in fact, these are some of the harder lessons) and will contain harder problems than in previous courses.

Then we'll see how to cleverly apply the machinery of number theory we have developed so far to solve Diophantine equations, meaning solving equations in integers. We continue with solving a certain type of combinatorial problems called tournament problems, containing many ideas we've seen so far.

The next topic is in geometry -- similar triangles. The concept is fundamental to Olympiad geometry. The next lesson is on pigeonhole principle, the infinite version of it and we highlight the difference between "arbitrarily large" and "infinite".

We will finally introduce polynomials in one real variable and their basic properties, including Bezout's and Vieta's theorems. We split the content into two lessons, as there will be a lot of new material and problems. Polynomials are a central theme in "grown-up" algebra and they're as important in Olympiad mathematics.

We then return to geometry, this time with "cyclic quadrilaterals". We'll consider both angle and length related properties and hopefully by the end of the sheet you'll be able to appreciate the beauty of plane geometry. Our penultimate lesson is on integer grids, this time looking at some geometric (and combinatorial) ideas on it.

The last topic is, as usual, something more fun oriented. We'll discuss infinite series informally, giving famous examples. The highlight of the lesson will be applying analysis to number theory, which mathematicians call analytic number theory.

Then we'll see how to cleverly apply the machinery of number theory we have developed so far to solve Diophantine equations, meaning solving equations in integers. We continue with solving a certain type of combinatorial problems called tournament problems, containing many ideas we've seen so far.

The next topic is in geometry -- similar triangles. The concept is fundamental to Olympiad geometry. The next lesson is on pigeonhole principle, the infinite version of it and we highlight the difference between "arbitrarily large" and "infinite".

We will finally introduce polynomials in one real variable and their basic properties, including Bezout's and Vieta's theorems. We split the content into two lessons, as there will be a lot of new material and problems. Polynomials are a central theme in "grown-up" algebra and they're as important in Olympiad mathematics.

We then return to geometry, this time with "cyclic quadrilaterals". We'll consider both angle and length related properties and hopefully by the end of the sheet you'll be able to appreciate the beauty of plane geometry. Our penultimate lesson is on integer grids, this time looking at some geometric (and combinatorial) ideas on it.

The last topic is, as usual, something more fun oriented. We'll discuss infinite series informally, giving famous examples. The highlight of the lesson will be applying analysis to number theory, which mathematicians call analytic number theory.

Topics

Course 8

Prerequisites

All prerequisites for 7 apply here as well.

Geometry will build up on the previous stuff, so course 7 level geometry is needed.

For combinatorics, algebra and number theory, completion of course 7 is not strictly necessary, but in contrast to prerequisites for course 7, almost every topic in intermediate courses is essential.

Once again, having done and/or knowing the material of every previous course is more than sufficient.

Geometry will build up on the previous stuff, so course 7 level geometry is needed.

For combinatorics, algebra and number theory, completion of course 7 is not strictly necessary, but in contrast to prerequisites for course 7, almost every topic in intermediate courses is essential.

Once again, having done and/or knowing the material of every previous course is more than sufficient.

About the content

We start with more theory on the building block of planar geometry -- triangles. First we discuss the common constructions involving the orthocentre H and the circumcentre O and see how they relate to each other. The next lesson is about the incentre, excentres and angle bisectors. These are one of the most popular themes in Olympiad geometry.

The next topic is extremely important, as it concerns the extremal principle. Indeed, that's one of the fundamental ideas in all of mathematics, and whole theories are built on generalisations of this simple idea. This time we consider its application in number theory (infinite descent) and combinatorics.

It's followed by a lesson on an important tool in modular arithmetic -- Chinese Remainder Theorem. Next, we consider inequalities. Starting with one simple inequality that the square of any real number is non-negative, we derive a set of tools to prove more complicated inequalities. AM-GM is one of the most common and useful inequalities and we can apply the so called "smoothing method" to show many inequalities, including that one.

In combinatorics (in particular in CS), we often encounter ourselves in a situation when to find the answer for the problem with given value, it's enough to know it for previous values, and the answer can be written in terms of previous answers. We now have a recurrence relation. We consider some of the classic examples and see how it can be used to solve a bunch of problems.

Then we continue the lesson "Cyclic quadrilaterals" from the previous course, and see what more can be said about them. This is followed by an introduction to "length ratio arguments", in particular area ratios, Ceva's and Menelaus's theorems.

We then introduce some of the common number-theoretic function, like the number of divisors, sum of divisors and Euler totient function. We investigate some of their properties and then finally show the Fermat-Euler theorem, also investigating orders on the way.

To finish off the course, we again do something not heavily related to Olympiads in the end. This time it's an introduction to cryptography and using number theory in it.

The next topic is extremely important, as it concerns the extremal principle. Indeed, that's one of the fundamental ideas in all of mathematics, and whole theories are built on generalisations of this simple idea. This time we consider its application in number theory (infinite descent) and combinatorics.

It's followed by a lesson on an important tool in modular arithmetic -- Chinese Remainder Theorem. Next, we consider inequalities. Starting with one simple inequality that the square of any real number is non-negative, we derive a set of tools to prove more complicated inequalities. AM-GM is one of the most common and useful inequalities and we can apply the so called "smoothing method" to show many inequalities, including that one.

In combinatorics (in particular in CS), we often encounter ourselves in a situation when to find the answer for the problem with given value, it's enough to know it for previous values, and the answer can be written in terms of previous answers. We now have a recurrence relation. We consider some of the classic examples and see how it can be used to solve a bunch of problems.

Then we continue the lesson "Cyclic quadrilaterals" from the previous course, and see what more can be said about them. This is followed by an introduction to "length ratio arguments", in particular area ratios, Ceva's and Menelaus's theorems.

We then introduce some of the common number-theoretic function, like the number of divisors, sum of divisors and Euler totient function. We investigate some of their properties and then finally show the Fermat-Euler theorem, also investigating orders on the way.

To finish off the course, we again do something not heavily related to Olympiads in the end. This time it's an introduction to cryptography and using number theory in it.

Topics

Course 9

Prerequisites

For number theory and geometry, everything (almost) up to course 8 included is essential.

For algebra, the main prerequisite is polynomials from course 7, in other words everything up to course 7 included.

For combinatorics, we only require all the previous graph-theory related lessons.

(Requirements as for 7 goes without saying)

For algebra, the main prerequisite is polynomials from course 7, in other words everything up to course 7 included.

For combinatorics, we only require all the previous graph-theory related lessons.

(Requirements as for 7 goes without saying)

About the content

LTE lemma is a very useful tool in elementary number theory, we'll apply it to a number of more sophisticated Diophantine equations.

Bipartite graphs are those whose vertices can be coloured with two colours such that no two vertices of the same colour are connected by an edge. We explore some of the properties of such graphs and see where they might arise. Classic results like Hall's marriage theorem, Max-flow min-cut theorem, Ford-Fulkerson algorithm and many others will be discussed.

Some polynomials can be written as a product of other polynomials. We look into this in more detail, discussing famous theorems like Eisenstein's irreducibility criteria and Gauss' lemma.

Homothety is a fancy word for "scaling", and combining that with rotation will lead to many fascinating results which are central in Olympiad geometry.

In number theory, we often need to construct a set of solutions to show that there are infinitely many of them, for example. These and many other kinds of problems need constructions.

Bijection is a formal word for "one-to-one correspondence". We will discuss some of the basic properties of bijective, injective and surjective functions and in particular apply it to problems from olympiad-style mathematics.

The next lesson is on miscellaneous topics on combinatorial geometry, that is extremely popular in olympiads (in particular at IMO). Several key tricks will be discussed at this lesson.

We will then move on to talking about cycles in graphs and in particular talk about Eulerian cycles.

Next, we will talk about geometry. The lesson on geometric inequalities will not contain much theory (our main tool will be just triangle inequality); we will mostly be concerned with solving problems. As for the lesson about symmedians -- despite the name of the lesson, this lesson will be mostly about certain type of quadrilaterals, more precisely: the "harmonic concyclic" ones. It has a number of nice properties that we will talk about in details.

Problems on asymptotic behavior are usually the last problems on olympiads (like IMO P3 2018) even though the main idea is not that hard. The biggest problem is to see "why this idea" and this is indeed usually not obvious.

And we will finish the course by talking about certain type of transformations of the plane, namely the ones that preserve distances between the points (e.g rotation). It turns out that it is not that hard to classify those (even though the definition might sound like there should be infinitely many of those).

Bipartite graphs are those whose vertices can be coloured with two colours such that no two vertices of the same colour are connected by an edge. We explore some of the properties of such graphs and see where they might arise. Classic results like Hall's marriage theorem, Max-flow min-cut theorem, Ford-Fulkerson algorithm and many others will be discussed.

Some polynomials can be written as a product of other polynomials. We look into this in more detail, discussing famous theorems like Eisenstein's irreducibility criteria and Gauss' lemma.

Homothety is a fancy word for "scaling", and combining that with rotation will lead to many fascinating results which are central in Olympiad geometry.

In number theory, we often need to construct a set of solutions to show that there are infinitely many of them, for example. These and many other kinds of problems need constructions.

Bijection is a formal word for "one-to-one correspondence". We will discuss some of the basic properties of bijective, injective and surjective functions and in particular apply it to problems from olympiad-style mathematics.

The next lesson is on miscellaneous topics on combinatorial geometry, that is extremely popular in olympiads (in particular at IMO). Several key tricks will be discussed at this lesson.

We will then move on to talking about cycles in graphs and in particular talk about Eulerian cycles.

Next, we will talk about geometry. The lesson on geometric inequalities will not contain much theory (our main tool will be just triangle inequality); we will mostly be concerned with solving problems. As for the lesson about symmedians -- despite the name of the lesson, this lesson will be mostly about certain type of quadrilaterals, more precisely: the "harmonic concyclic" ones. It has a number of nice properties that we will talk about in details.

Problems on asymptotic behavior are usually the last problems on olympiads (like IMO P3 2018) even though the main idea is not that hard. The biggest problem is to see "why this idea" and this is indeed usually not obvious.

And we will finish the course by talking about certain type of transformations of the plane, namely the ones that preserve distances between the points (e.g rotation). It turns out that it is not that hard to classify those (even though the definition might sound like there should be infinitely many of those).

Topics

Course X

Prerequisites

The content of this course is quite advanced but most of the topics are self-contained. However we still ask you to know the material covered before. In particular, we ask you to revise basic probability, bijection-related things and number-theory.

About the content

The content of this course is focused on showing what serious mathematics looks like. Therefore only several courses are of olympiad-style.

The first lesson will be dedicated to permutations of numbers 1, 2, ...,*n* and facts around them. Permutations are vitally important for something known as "group theory" (usually covered at university). However we will not talk about this in this lesson, but rather focus on solving problems.

Next we will move on to taking first steps into the world of analysis. It is possible that students will already know the key definitions convergence and continuity but we will give a number of hard problems.

Functional equations and Helly's theorem are both olympiad-style topics and are self-contained. The first one is algebraic and later on you will deal with it much more, whereas the second one is just a fact from combinatorial geometry that has a number of interesting applications.

Countability is related to bijections and infinite sets, so you can see it as a combinatorial topic. It will prove to be useful later on in university (if you decide to do mathematics of course).

Then we will talk about how to use probability to prove some claims. It should look surprising at first, because "how come talking about something uncertain can explain something that is certain?". We will show how. And we will apply this idea a lot in graph-theory problems, in particular Ramsey theory -- one of the graph theory topics.

The lesson on quadratic residues is a lesson on number-theory. It will be hard and in particular we will prove a very advanced theorem known as the "law of quadratic reciprocity". Gauss was very proud of it and came up with several proofs of this law. We will probably have time for only one of them..

The last few lessons are very distant from olympiad mathematics but are cool. In particular "using physics to solve mathematical problems" lesson is exactly what it says it should be: we will indeed use some of the physics laws to prove some of the mathematical facts like Pythagoras theorem.

The first lesson will be dedicated to permutations of numbers 1, 2, ...,

Next we will move on to taking first steps into the world of analysis. It is possible that students will already know the key definitions convergence and continuity but we will give a number of hard problems.

Functional equations and Helly's theorem are both olympiad-style topics and are self-contained. The first one is algebraic and later on you will deal with it much more, whereas the second one is just a fact from combinatorial geometry that has a number of interesting applications.

Countability is related to bijections and infinite sets, so you can see it as a combinatorial topic. It will prove to be useful later on in university (if you decide to do mathematics of course).

Then we will talk about how to use probability to prove some claims. It should look surprising at first, because "how come talking about something uncertain can explain something that is certain?". We will show how. And we will apply this idea a lot in graph-theory problems, in particular Ramsey theory -- one of the graph theory topics.

The lesson on quadratic residues is a lesson on number-theory. It will be hard and in particular we will prove a very advanced theorem known as the "law of quadratic reciprocity". Gauss was very proud of it and came up with several proofs of this law. We will probably have time for only one of them..

The last few lessons are very distant from olympiad mathematics but are cool. In particular "using physics to solve mathematical problems" lesson is exactly what it says it should be: we will indeed use some of the physics laws to prove some of the mathematical facts like Pythagoras theorem.

Topics