NUMBER THEORY
UNIT 5
CONTINUED FRACTIONS
1. Introduction
What is a continued fraction? Two examples are
1
1+
1
1
and 4 +
3+
1
1+
2
.
1
2+
1
1
In this unit, we shall see how a number could be changed t
Coding in communication system
In the engineering sense, coding can be classified into four areas:
Encryption: to encrypt information for security purpose.
Data compression: to reduce space for the da
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LOGIC
One of the most important criteria in studying mathematics is having a logical mind. In the
following texts, some fundamental ideas will be presented here.
1. Statement
A s
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NUMBER THEORY
UNIT 4
SOME SPECIAL NUMBERS
1. Introduction
In the previous units we have come across various properties of the integers. The theory is so
rich that many variations
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NUMBER THEORY
UNIT 2
CONGRUENCES
1. Basic Properties
Recall that in Unit 0, we came across the problem of finding the odd one out among the
following group of numbers:
13, 17, 25
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NUMBER THEORY
UNIT 3
DIOPHANTINE EQUATIONS
1. Introduction
A Diophantine equation is an equation for which integral solutions are to be found. The most
famous type of Diophantine
CHAPTER 2
POINTS, LINES, AND TRIANGLES
In this chapter we will highlight a small core of basic results related to triangles that proves
useful in a large number of problems in Mathematics Olympiads. T
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NUMBER THEORY
UNIT 1
DIVISIBILITY
1. Divisibility
Definition 1.1.
Let a and b be integers, a 0 . We say that a divides b, denoted by a | b, if there exists an integer c
such that
CHAPTER 1
CEVAS THEOREM AND MENELAUSS THEOREM
The purpose of this chapter is to develop a few results that may be used in later chapters. We
will begin with a simple but useful theorem concerning the
VECTORS IN PHYSICS
Vectors represent directions and magnitudes. Indeed, many concepts in Physics can be defined
or represented by vectors. This set of notes will serve as an introduction of some physi
CHAPTER 3
PROPERTIES OF CIRCLES
We will present in this chapter a few of the most interesting properties of circles and related
problems in Mathematics Olympiads. The materials in the first section ar
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NUMBER THEORY
UNIT 0
INTRODUCTION
The history of number theory dates back to almost as far as the history of mathematics. As its
name implies, number theory is the study of the p
VECTOR
- as an essential tool for 3-dimensional coordinate geometry 1. Introduction
We begin with some basic definitions about vectors.
1. Scalar:
A scalar is a quantity which is specified by magnitud
2 0 &
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CHAPTER 10
PHYSICAL TREATMENTS OF SOME MATHEMATICAL PROBLEMS
Mathematics is the study of the relationships between numbers, quantities and space, whereas
physics is a science that deals with laws of n
DIFFERENTIATION II
In this article we shall investigate some mathematical applications of differentiation. We shall
be concerned with a rate of change problem; we shall discuss the Mean Value Theorem
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PIGEONHOLE PRINCIPLE
Pigeonhole principle is a fundamental but powerful tool in combinatorics. Unlike many other
strong theorems, the principle itself is exceptionally simple. Un
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RAMSEYS THEORY
You probably have heard of this interesting fact: among any six people in the world, there exist
three who know each other or three who dont know each other. Actua
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PROBABILITY
In everyday life, we would frequently encounter questions like What is the chance that our
team will win in todays football match? or Is it likely to rain in the afte
NUMBER SYSTEMS
1. The Real Numbers
Readers are assumed to be familiar with the real numbers: 2, 47, 3, 1 ,
3
2,
3
5 , , e, Just
like there are many ways to classify people (according to gender, age, p
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INTRODUCTORY THEORY OF DIFFERENTIATION
This article is written for beginners. No previous knowledge of differentiation is assumed, and
we shall treat the theory of limits and dif
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CONVERGENCE AND LIMITS
1. Introduction
In Introductory Theory of Differentiation we introduced the theory of limits of functions. In
particular we defined the meaning of the phra
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FUNCTIONAL EQUATIONS
Functional equations are equations for unknown functions instead of unknown numbers. In this
chapter, we will try to explore how we can find the unknown func
Inequalities (Unit 1 Unit 3)
Solutions to Exercises
Inequalities (Unit 1)
1.
By the AM-GM inequality, we have
1 + ai
1 ai , i.e. 1 + ai 2 ai for all i. Hence
2
2n = (1 + a1 )(1 + a2 )
(
2 a1
)( 2
a2
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GENERATING FUNCTIONS
1. Introduction
The concept of generating functions is a powerful tool for solving counting problems.
Intuitively put, its general idea is as follows. In cou
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SEQUENCES AND
RECURRENCE RELATIONS
1. Introduction
Our very first experience with sequences probably came along with the following type of
numerical reasoning puzzles, in which o
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INEQUALITIES
UNIT 1
CLASSICAL INEQUALITIES
1. Inequality of the Means
To motivate our discussion, lets look at several situations.
(A) A man drove for 2 hours. In the first hour
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COMPLEX NUMBERS
When we are solving quadratic equations with real roots, the roots of the equations exhibit
three cases: two distinct real roots, a double root or no real roots.