Techniques for summing various mathematical sequences. 2. Number Theory
Teaching the 'how' and 'why' behind solving problems, not just memorizing formulas.
Volume 1 generally focuses on Algebra and Number Theory, while Volume 2 often covers Geometry and Combinatorics (though this can vary by edition). Below is the typical chapter breakdown for Vol 1:
A defining characteristic of this volume is the caliber of its content. The problems are curated from various prestigious competitions, including the Chinese Mathematical Olympiad, which is known for its extreme difficulty. By working through these notes, students are exposed to the "Chinese style" of problem-solving, which often emphasizes elegant manipulations and deep logical connections. The inclusion of detailed solutions is particularly valuable, as it reveals the thought processes of expert mathematicians, helping students identify their own cognitive gaps.
The Mathematical Olympiad is an international mathematics competition that takes place annually, attracting top students from over 90 countries. The competition is divided into two levels: the Junior Mathematical Olympiad (JMO) for students under 20 years old and the International Mathematical Olympiad (IMO) for students under 20 years old who have completed their secondary education.
: A concise introduction to notations, basic theorems, and core mathematical methods. Worked Examples
Combinatorics in the senior section is a different beast from high school probability. Volume 1 focuses on:
Detailed coverage of classical inequalities like AM-GM, Cauchy-Schwarz, and Holder's inequality.
: Mastering identities and expression transformations.
This module dives into the properties of integers, a crucial skill for Olympiads. Topics include: Divisibility and Modular Arithmetic. Diophantine Equations (Finding integer solutions). Prime Numbers and Prime Factorization. Fermat’s Little Theorem and Euler’s Totient Theorem. 3. Combinatorics
When a theorem is introduced, cover the proof and try to recreate it yourself. Understanding the mechanism behind a tool is crucial for tweaking it during an exam.
Advanced properties of cyclic quadrilaterals, power of a point, and Ptolemy’s Theorem.