During the period 2016-2022, I taught students not only olympiad mathematics but some interesting topics in Topology, Abstract Algebra, Differential Geometry, Probability in Combinatorics etc. In each case, I tried to simplify a language and gave some intuition of main ideas without loss of strictness as much as possible.

2021-2022

I taught 9-11 grades students (a common group). Discussed themes were closely related to the following:

1. Benford’s Law. Slides
2. Recurrence Relations
3. Conic Sections
4. Projective Plane $\mathbb{R}P^2$ and Projective Transformations. Notes
5. Pole and Polar, Cross-Ratio, Pascal’s Theorem, Brianсhon’s Theorem. Notes
6. Algebra of Quaternions $\mathbb{H}$. Task
7. Geometry of Quaternions, homomorphsim $\mathrm{Sp}(1)\to SO(3)$. Task
8. Probabilities, Buffon’s Needle Problem. Task
9. Probabilistic Method, Card Shuffling Problem
10. Knots and Their Inavriants. Task
11. Pick’s theorem, Continued Fractions and Dirichlet’s Approximatopn Theorem. Notes
12. Hurwitz’s Theorem. Notes
13. Cyclic Groups, Root of Unity Groups and Euler’s Formula $e^{x+iy} = e^x\left ( \cos y + i\sin y \right )$
14. Cyclotomic Polynomials $\Phi_n$ and Weak Variant of Dirichlet’s Progressions Theorem for Primes of The Form $p\cong 1\ (\mathrm{mod}\ n)$. Notes
15. Vieta Jumping

2020-2021

I taught 9-11 grades students (a common group). Discussed themes were closely related to the following:

1. Brunn-Minkowski Inequality. Slides
2. Averaging in Geometry. Slides
3. Geometry of Numbers. Slides
4. Сolors in Combinatorics
5. Arrow’s Impossibility Theorem. Slides
6. Simmetric Polynomials. Slides
7. Three Convex Hull Theorems. Slides
8. Center of Mass. Slides
9. Divisibility and Combinatorics
10. Algebra of Complex Numbers $\mathbb{C}$. Task
11. Geometry of Complex Numbers, Chasles’ theorem, Napoleon’s theorem. Notes
12. Areas. Task
13. Catalan Numbers. Task
14. Fundamental Theorem of Algebra and Vector Fields on a Plane
15. Aztec Diamond

2019-2020

I taught 9-11 grades students (a common group). Discussed themes were closely related to the following:

1. Number Theory: Pythagorean triples, Pell’s equation, Markov’s equation, Continued Fractions
2. Abstract Algebra: Gaussian numbers, rings and fields, ideals, principle domaines, Euclidean rings, Fermat’s Christmas Theorem
3. Differential Geometry: Cavalieri’s principle, computing volumes of solids via Cavalieri’s principle, Crofton formula, On the Sphere and Cylinder approach by Archimedes
4. Solid geometry: Pyramids, Invertion in 3D
5. Combinatorics: [Pólya’s Theorem](https://magisterlud.github.io/files/little-mechmath/(Pólya’s-%20theorem.pdf), Chromatic Polynomial

2018-2019

I taught 9-11 grades students (a common group). Discussed themes were closely related to the following:

1. Plane Geometry: inversion and its applications
2. Solid Geometry: polyhedra, circumscribed and inscribed spheres
3. Spherical Geometry: lines, spherical triangles, law of cosines, law of sines
4. Hyperbolic Geometry: Poincaré disc model, Poincaré’s half-plane model, Klein’s disc model, hyperboloid model, isometries, cross-ratio, hyperbolic disstance, areas, hyperbolic law of cosines, hyperbolic law of sines, Thurstone’s theorem, Mostow rigidity theorem
5. Algebra: Cubic Equations

2017-2018

I taught 8th grade students. Discussed themes were closely related to the following:

1. Graph Theory: mathings in graps, bipartite graph, Hall’s marriage theorem, complete graphs, Turàn’s theorem, Euler’s characteristic in 2-dimensional sphere case, five color theorem
2. Number Theory: Euler’s function, Euler’s theorem, Chinese remainder theorem
3. Geometry: areas of figures, similar triangles, circles and their proeprties, homothety, rotations, Euler’s circle, Euler’s line, Ceva’s and Menelaus’s Theorems, Center of Mass
4. Abstract Algebra: groups, first examples of groups, subgroups, Lagrange’s theorem
5. Algebra: polynomes, little Bézout’s theorem, Lagrange polynomial, Vieta’s formulas for polynomials
6. Calculus: Continued Functions, Functions Properties

2016-2017

I taught 7th grade students. Discussed themes were closely related to the following:

1. Graph Theory: property of trees, Euler’s theorems on cycles in graphs, planar graphs, Euler’s characteristic in 2-dimensional sphere case
2. Game Theory: numeral systems, Nim game, simmetry strategies
3. Number Theory: modular arithmetic, Euclid’s algorithm, Fermat’s little theorem, linear integral numbers equations
4. Geometry: convex polygons, Helley’s theorem, geometric inequalities
5. Combinatorics: sets and operations over them, first simple observations about finite sets, binomial coefficients, binomial theorem