Problems And Solutions In Mathematical Olympiad (high School 3)

✍ Scribed by Hong-bing Yu

• Publisher: WSPC/ECNUP
• Year: 2022
• Tongue: English
• Leaves: 378
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Contents
Editorial Board
Preface
1. Permutations and Combinations
Addition Principle and Multiplication Principle
Some Basic Counting Problems
1. Permutations
2. Combinations
Illustrative Examples
Exercises
Group A
Group B
2. Binomial Coefficients
Basic Properties of Binomial Coefficients
Combinatorial Identities
Illustrative Examples
Exercises
Group A
Group B
3. Counting: Correspondence and Recursion
Correspondence
Exercises
Group A
Group B
4. Counting: Inclusion-Exclusion Principle
Exercises
Group A
Group B
5. Combinatorial Problems
Exercises
Group A
Group B
6. Exact Division
Exact Division
Greatest Common Divisor
Least Common Multiple
Illustrative Examples
Exercises
Group A
Group B
7. Prime Numbers
Prime Numbers and the Unique Factorization Theorem
Distinguishing Method of Prime Numbers and the Standard Factorization of n!
Illustrative Examples
Exercises
Group A
Group B
8. Congruence (1)
Congruence and Congruence Class
Basic Properties of Euler’s Function
Illustrative Examples
Exercises
Group A
Group B
9. Indeterminate (Diophantine) Equations (1)
Basic Methods of Solving Indeterminate Equations
Some Kinds of Indeterminate Equations
Exercises
Group A
Group B
10. Problems in Number Theory
Exercises
Group A
Group B
11. Operations and Exact Division of Polynomials
Basic Concepts of Univariate Polynomials
Division with Remainder and Greatest Common Factors
Polynomials Modulo a Prime
Illustrative Examples
Exercises
Group A
Group B
12. 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
Exercises
Group A
Group B
13. Polynomials with Integer Coefficients
Primitive Polynomials
Eisenstein’s Criterion for Irreducible Polynomials
Complex Roots of Polynomials and their Irreducibility in Z[x]
Illustrative Examples
Exercises
Group A
Group B
14. Interpolation and Difference of Polynomials
Lagrange’s Interpolation Formula
Difference of Polynomials
Integer-Valued Polynomials
Illustrative Examples
Exercises
Group A
Group B
15. Roots of Unity and Their Applications
Illustrative Examples
Exercises
Group A
Group B
16. Generating Function Method
Exercises
Group A
Group B
17. Sets and Families of Subsets
Exercises
Group A
Group B
18. Graph Theory Problems
Exercises
Group A
Group B
19. Congruence (2)
The Order of a Modulo m
Chinese Remainder Theorem
Lucas’ Theorem
Illustrative Examples
Exercises
Group A
Group B
20. Indeterminate Equations (2)
Exercises
Group A
Group B
Comprehensive Exercises
Solutions
1. Permutations and Combinations
2. Binomial Coefficients
3. Counting: Correspondence and Recursion
4. Counting: Inclusion-Exclusion Principle
5. Combinatorial Problems
6. Exact Division
7. Prime Numbers
8. Congruence (1)
9. Indeterminate (Diophantine) Equations (1)
10. Problems in Number Theory
11. Operations and Exact Division of Polynomials
12. Zeros of Polynomials
13. Polynomials with Integer Coefficients
14. Interpolation and Difference of Polynomials
15. Roots of Unity and their Applications
16. Generating Function Method
17. Sets and Families of Subsets
18. Graph Theory Problems
19. Congruence (2)
20. Indeterminate Equations (2)
Comprehensive Exercises