Secondary-school-level mathematics

Below are the maths-courses for 11-16 year old pupils, though everyone with the appropriate mathematical background is welcome. There is no particular knowledge requirement to join our initial courses, but we expect pupils to be diligent!

Junior
Intermediate
Senior
School-maths
Extra-maths

Junior school-maths courses revisit the most fundamental things that have probably been covered in your school. And not only we will revisit them, but also answer all the why-s and teach to ask correct questions and properly answer them. Moreover, we will also provide a number of challenging extra-problems for people who find the main content easy.

Schedule (next enrollment)
SM-0, SM-1, SM-2, SM-3: September 16, 20, 23, 27, 30 from either 16:00 till 17:00 GMT or 10:00 till 11:00 GMT.
Junior-school-maths recap week: not planned yet.
Price&payment
One lesson costs 7 US-dollars. The payment for the entire course should be made after your application has been approved and before the second lesson. Thus it is possible to withdraw from the course after the first lesson and not pay anything.
Starting from scratch we will revisit students' knowledge about integers and operations on them. Since the content is quite easy there will be extra problems that are harder.
Before getting down to harder things, we must briefly talk about claims, negations, proofs, logical implications etc... And that is what we will focus on in this course.
First, starting from scratch we will explain variables and basic equations. Second, we will talk about fractions. In both cases we will provide enough material of different levels of difficulty for practice.
This course is dedicated to practising manipulations with numbers, variables and algebraic expressions. Starting from elementary problems we will finish by simplifying complicated expressions and dealing with more involved equations.
This course is designed for you to practice the material covered above. Students will be given 4 problemsheets and there will be 4 discussions of them. No help/hints besides the discussions!

Junior extra-maths courses are aimed at those who have no or little experience in logical/olympiad-style problems. We will talk about things like "proofs", "counterexample" and see a number of simple but cool mathematical ideas. So whether you plan to focus on olympiads, computer science, or you simply want to try out some non-classroom maths, these courses are for you.

This is course is designed for pupils who have at least finished primary school and are interested in extra-curricular maths. Participants will try to come up with different sorts of non-obvious constructions, solve logical puzzles and finally see a cool non-trivial mathematical idea.
In this course participants will also try to come up with interesting constructions and algorithms. Moreover we will give an introduction to formal proofs, in particular "proof by contradiction". Finally, a few cool maths ideas will also be present in this course.
This is a continuation of the EM-1 course but with more focus on olympiad-style problems. Participants will meet (hopefully discover themselves with some guidance) a number of basic maths-olympiad ideas and practise using them to solve problems of various levels.
School-maths
Extra-maths

Intermediate school-maths courses cover most of the basic and fundamental middle-school material. Once you are confident with junior-level material feel free to just do courses below one by one.

This course is dedicated to building up geometrical intuition, solving basic geometrical problems and formalizing our knowledge about Euclidean geometry.
We will start by doing some counting combinatorics and then we will move on to the beginning of probability. We will finish the course by talking about statistics and their interpretation.
We will revisit our knowledge about numbers and make friends with real numbers. We will also talk about square roots, number lines and weird exponents. 
From first principles we will talk about functions, in particular about things like range and domain, graphs and coordinates.
This course is an off-main stream school-maths courses designed for people who need more practise with manipulating with equations and expressions. Being honest, the course is not fun but none-the-less important as you will need to do a lot of such things in algebra. So being guru with this will save you a lot of time later on.
This course is designed to make you more confident with polynomials (especially quadratic functions) by solving all kind of problems involving them.
This time we will talk about similar triangles, con-cyclic quadrilaterals, area and length.
Intermediate Recap week (school-maths)
This course is designed for you to practice things covered above. It will be of a slightly higher standard than the usual school exam as we will give more challenging problems that make sure students' understanding is crystal clear. Students will be given 4 problemsheets and there will be 4 discussions of them. No help/hints besides the discussions!

Intermediate extra-maths courses are aimed at those who have about a year or two of experience in solving non-standard/olympiad-style mathematical problems. Knowing all the theorems/facts/ideas/... from Junior level school&extra maths is a prerequisite.

This course is designed for people with about a year of experience in extra-curricular maths problem solving who want to see what's its like to study more advanced topics at Kvanta. Knowing content from courses EM-1 and EM-2 is a prerequisite. Knowledge of the content of all of the school maths courses SM-0, ..., SM-3 is a must as well .
In this course we have collected mathematical topics and ideas that will be extremely important for programming as well as for maths. Among them: graph theory, mathematical induction, binary representation., etc...
This course is packed with olympiad-style problems and theorems from combinatorics and geometry. Apart from knowing material from previous courses, we also want you to have experience in geometry (e.g course "SM-4").
This course is packed with olympiad-style problems and theorems from number theory and algebra. Even though we will talk about many things like gcd, divisibility, factorization etc. from scratch we will give much harder problems than the ones in EM-2 on similar topics.
This course is off the main stream of extra-maths courses and is designed to show some topics from good-university-level maths. But do not worry, the the main prerequisite is being comfortable with previous courses :)
Intermediate Recap week (extra-maths)
There are a lot of important ideas mentioned in courses EM-3, EM-4, EM-5 and EM-6. So this course is designed to help with that. We will test intermediate school-maths knowledge as well. Students will be given 4 problemsheets and there will be 4 discussions of them. No help/hints besides the discussions!
School-maths
Extra-maths

Senior school-maths courses are mostly aimed to give a high-quality introduction into key high-school/university level topics. Once you are done with these courses, middle-school exams will be ridiculous.

Starting from the very beginning we will talk about vectors, operations on them and their applications.
Starting from first principles we will dive into the world of sin, cos, .... We will solve a great number of problems with them.
These strange numbers are important for mathematics of all sorts and have some beautiful properties that are explored at university. Using the knowledge of SM-10 and SM-11 we will introduce to the complex world and show a few nice facts about it. Interestingly, complex numbers might be helpful for problems from real-world...
The concept of limit of a sequence or a function is fundamental concept for later analysis. We will talk about it and give tons of different examples for pracitse.
We will give you a bit of an insight into a huge and important university level topic: Linear algebra. But at this stage it will all be at the secondary-school-exam level.
Once you feel comfortable with limits (i.e SM-12) we can talk about derivatives and even about integrals. Without going into many boring details, we will work a lot on differentiation.
Mathematicians can do anything and in particular physics. Using the tools from SM-10, ..., SM-13 we will talk about mechanics from scratch but from a mathematical point of view. Believe it or not it should all start making more sense then with the tools.
What is the use of all the tools developed above if there is no important application for them? So we will show you one, which is probability and statistics. And there is real money behind this stuff, trust us...
This course is designed for you to practice things covered above. It is like exam-preparation but harder as the problems will be harder and we also want you to understand things. Things like GSCE are ridiculous compared to this. So this course is designed to help with that. We will test intermediate school-maths knowledge as well. Students will be given 4 problemsheets and there will be 4 discussions of them. No help/hints besides the discussions!

Senior extra-maths courses are for experienced maths-problem-solvers and they last 12 weeks. So apply only if you are prepared to work hard :) The courses are no longer themed; instead they mimic olympiad-training programs given in many countries. Each contains a lot of advanced problems and a number of non-obvious theorems, with the teachers consider course-X as being really hard by any standard. The courses are most suitable for those who would like to take part in olympiads of serious level or explore hard maths. Apart from feeling comfortable with things covered in Junior&Intermediate courses, participants need to make sure they know enough of the relevant school material.

Many of the topics here are an extension of previous courses. Among other topics: introduction to polynomials, similar triangles, cyclic quadrilaterals, Diophantine equations. Problems are of hard-olympiad level. In the end we will discuss infinity and infinite series.
Topics covered: geometry around points I, O, H, radical axes, area-arguments, Chinese remainder theorem, the AM-GM inequality, smoothing, orders and Euler's function. We end with a lecture on cryptography and number theory.
With the help of trigonometry and vectors much of the geometry can be "calculated". So using the knowledge from SM-10 and SM-11 and things we know about some geometrical constructions we will show you how to kill a great number of geometry-problems. This course is off the main stream of extra-maths since this powerful tool is not necessary to become a great geometer. (In fact it might obscure the beauty of geometry...)
Among the topics covered: Bipartite graphs, LTE, factorization in polynomials, symmedians, homotethy, spiral symmetry and movements on the plane as the last lecture.
This course differs in content from the main stream of extra-maths courses and is designed to introduce accessible and interesting topics from university maths. To do this course, students must be on top of the material in previous courses. This is because there are some advanced mathematical topics like Quadratic Residues, Ramsey theory, conics and intuitive topology. And they are quite hard.
Senior recap week (extra-maths)
There are a lot of important ideas mentioned in courses EM-7, EM-8 and EM-9. So this course is designed to help with that. We will test senior school maths knowledge as well. Students will be given 4 problemsheets and there will be 4 discussions of them. No help/hints besides the discussions!
Easy
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Hard
Made on
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