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.