Hong-bing Yu. Problems And Solutions In Mathematical Olympiad: High School 3

WSPC/ECNUP, 2022. — 378 p.The series is edited by the head coaches of China's IMO National Team. Each volume, catering to different grades, is contributed by the senior coaches of the IMO National Team. The Chinese edition has won the award of Top 50 Most Influential Educational Brands in China. The series is created in line with the mathematics cognition and intellectual development levels of the students in the corresponding grades. All hot mathematics topics of the competition are included in the volumes and are organized into chapters where concepts and methods are gradually introduced to equip the students with necessary knowledge until they can finally reach the competition level. In each chapter, well-designed problems including those collected from real competitions are provided so that the students can apply the skills and strategies they have learned to solve these problems. Detailed solutions are provided selectively. As a feature of the series, we also include some solutions generously offered by the members of Chinese national team and national training team.Preface
Permutations and Combinations
Addition Principle and Multiplication Principle
Some Basic Counting Problems
Permutations
Combinations
Illustrative Examples
Binomial Coefficients
Basic Properties of Binomial Coefficients
Combinatorial Identities
Illustrative Examples
Counting: Correspondence and Recursion
Correspondence
Counting: Inclusion-Exclusion Principle
Combinatorial Problems
Exact Division
Greatest Common Divisor
Least Common Multiple
Illustrative Examples
Prime Numbers
Prime Numbers and the Unique Factorization Theorem
Distinguishing Method of Prime Numbers and the Standard Factorization of n!
Illustrative Examples
Congruence (1)
Congruence and Congruence Class
Basic Properties of Euler’s Function
Illustrative Examples
Indeterminate (Diophantine) Equations (1)
Basic Methods of Solving Indeterminate Equations
Some Kinds of Indeterminate Equations
Problems in Number Theory
Operations and Exact Division of Polynomials
Basic Concepts of Univariate Polynomials
Division with Remainder and Greatest Common Factors
Polynomials Modulo a Prime
Illustrative Examples
Zeros of Polynomials
Zeros of Polynomials and the Identity Theorem
Congruence Equation Modulo a Prime Number
Rational Roots of Polynomials with Integer Coefficients
Fundamental Theorem of Algebra
Complex Roots and Divisibility of Polynomials with Rational (Integer) Coefficients
Illustrative Examples
Polynomials with Integer Coefficients
Primitive Polynomials
Eisenstein’s Criterion for Irreducible Polynomials
Complex Roots of Polynomials and their Irreducibility in Z[x]
Illustrative Examples
Interpolation and Difference of Polynomials
Lagrange’s Interpolation Formula
Difference of Polynomials
Integer-Valued Polynomials
Illustrative Examples
Roots of Unity and Their Applications
Illustrative Examples
Generating Function Method
Sets and Families of Subsets
Graph Theory Problems
Congruence (2)
The Order of a Modulo m
Chinese Remainder Theorem
Lucas’ Theorem
Illustrative Examples
Indeterminate Equations (2)
Comprehensive Exercises