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

For Junior SectionVolume 1

Jiagu Xu

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eBook - ePub

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

For Junior SectionVolume 1

Jiagu Xu

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About This Book

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.

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Information

Publisher
WSPC
Year
2009
ISBN
9789814365192

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 + (βˆ’y) = x βˆ’ y;
(ii) x βˆ’ (y) = x βˆ’ y, x βˆ’ (βˆ’y) = x + y.
(iii) x Γ— (βˆ’y) = βˆ’xy; (βˆ’x) Γ— y = βˆ’xy; (βˆ’x) Γ— (βˆ’y) = xy; (βˆ’1)n = βˆ’1 for odd n, (βˆ’1)n = 1 for even n.
(iv) If the denominators of the following expressions are all not zeros, then
images
3. Ingenious Ways for Calculating
β€’ Make a telescopic sum by using the following expressions:
images
β€’ By use of the following formulae:
images
Examples
Example 1. Evaluate
images
Solution
Images
images
Example 2. There are five operational expressions below:
images
(ii) (βˆ’0.125)7 Β· 88;
(iii) (βˆ’11) + (βˆ’33) βˆ’ (βˆ’55) βˆ’ (βˆ’66) βˆ’ (βˆ’77) βˆ’ (βˆ’88);
(iv)
images
(v)
images
Then the expression with maximal value is
(A) (i), (B) (iii), (C) (iv), (D) (v).
Solution
images
(ii) (βˆ’0.125)7 Β· 88 = βˆ’(0.125 Γ— 8)7 Γ— 8 = βˆ’8;
(iii) (βˆ’11) + (βˆ’33) βˆ’ (βˆ’55) βˆ’ (βˆ’66) βˆ’ (βˆ’77) βˆ’ (βˆ’88)
= βˆ’11 βˆ’ 33 + 55 + 66 + 77 + 88 = 11 Γ— 22 = 242;
(iv)
images
(v)
images
< 1 Γ— 10 = 10;
Thus, the answer is (A)....

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