Lecture Notes On Mathematical Olympiad Courses: For Junior Section (In 2 Volumes) - Volume 1

For Junior SectionVolume 1

Jiagu Xu

• Publisher: WSPC
• Year: 2009
• ISBN:
  9789814365192
• Topic: Biological Sciences
• Subtopic: Science General
• Index: Biological Sciences

Olympiad mathematics is not a collection of techniques of solving mathematical problems but a system for advancing mathematical education.

This book is based on the lecture notes of the mathematical Olympiad training courses conducted by the author in Singapore. Its scope and depth not only covers and exceeds the usual syllabus, but introduces a variety concepts and methods in modern mathematics.

In each lecture, the concepts, theories and methods are taken as the core. The examples are served to explain and enrich their intension and to indicate their applications. Besides, appropriate number of test questions is available for reader's practice and testing purpose. Their detailed solutions are also conveniently provided.

The examples are not very complicated so that readers can easily understand. There are many real competition questions included which students can use to verify their abilities. These test questions are from many countries, e.g. China, Russia, USA, Singapore, etc. In particular, the reader can find many questions from China, if he is interested in understanding mathematical Olympiad in China.

This book serves as a useful textbook of mathematical Olympiad courses, or as a reference book for related teachers and researchers.

Contents:

• Operations on Rational Numbers
• Linear Equations of Single Variable
• Multiplication Formulae
• Absolute Value and Its Applications
• Congruence of Triangles
• Similarity of Triangles
• Divisions of Polynomials
• Solutions to Testing Questions
• and other chapters

Readership: Mathematics students, school teachers, college lecturers, university professors; mathematics enthusiasts.

Lecture 1

Operations on Rational Numbers

1. Basic Rules on Addition, Subtraction, Multiplication, Division

Commutative Law:	a + b = b + a ab = ba
Associative Law:	a + b + c = a + (b + c) (ab)c = a(bc)
Distributive Law:	ac + bc = (a + b)c = c(a + b)

2. Rule for Removing Brackets

For any rational numbers x, y,

(i) x + (y) = x + y, x + (−y) = x − y;

(ii) x − (y) = x − y, x − (−y) = x + y.

(iii) x × (−y) = −xy; (−x) × y = −xy; (−x) × (−y) = xy; (−1)ⁿ = −1 for odd n, (−1)ⁿ = 1 for even n.

(iv) If the denominators of the following expressions are all not zeros, then

3. Ingenious Ways for Calculating

• Make a telescopic sum by using the following expressions:

• By use of the following formulae:

Examples

Example 1. Evaluate

Solution

Example 2. There are five operational expressions below:

(ii) (−0.125)⁷ · 8⁸;

(iii) (−11) + (−33) − (−55) − (−66) − (−77) − (−88);

(iv)

(v)

Then the expression with maximal value is

(A) (i), (B) (iii), (C) (iv), (D) (v).

Solution

(ii) (−0.125)⁷ · 8⁸ = −(0.125 × 8)⁷ × 8 = −8;

(iii) (−11) + (−33) − (−55) − (−66) − (−77) − (−88)
= −11 − 33 + 55 + 66 + 77 + 88 = 11 × 22 = 242;

(iv)

(v)

< 1 × 10 = 10;

Thus, the answer is (A)....

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Preface
• Acknowledgments
• Abbreviations and Notations
• Contents
• 1: Operations on Rational Numbers
• 2: Monomials and Polynomials
• 3: Linear Equations of Single Variable
• 4: System of Simultaneous Linear Equations
• 5: Multiplication Formulae
• 6: Some Methods of Factorization
• 7: Absolute Value and Its Applications
• 8: Linear Equations with Absolute Values
• 9: Sides and Angles of a Triangle
• 10: Pythagoras' Theorem and Its Applications
• 11: Congruence of Triangles
• 12: Applications of Midpoint Theorems
• 13: Similarity of Triangles
• 14: Areas of Triangles and Applications of Area
• 15: Divisions of Polynomials
• Solutions to Testing Questions
• Index