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 to a continued fraction and how this
method is related 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 data stream.
Data translation: to change the form of repr
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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 statement, or proposition, is a sentence that is readabl
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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 are possible and many questions can be asked. One of t
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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, 39, 45
We remarked that 39 is the odd one out, for it
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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 equations in contemporary mathematics is perhaps those
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. Those results are simple, but
readers should pay attenti
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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 b = ca. We also say b is divisible by a, b is a multip
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 area ratio of two triangles with a
common side. With th
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 physical concepts
related to vector applications. We will no
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 are classical and standard,
their usefulness are illustra
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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 properties of numbers. For most of the time we
shall be
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 magnitude only. (e.g. mass, length,
volume) We can describe it
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u e e
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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 nature. For a long time these two disciplines have been
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 and its
application to finding relative extrema; and fi
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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. Unless you have looked into it
thoroughly, it is hard to
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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. Actually there are many other
similar results. For example,
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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 afternoon? In answering
these questions, we need to determi
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, personality, nationality, etc.),
there are many ways to
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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 differentiation in a rigorous yet elementary way. If you h
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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 phrase
lim f ( x ) = L .
xa
In this article we will pursue
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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 function when we know that the
conditions it satisfies.
1.
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
= 2n a1a2
Dividing both side by 2n , we have 1 a1a2
2.
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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 counting problems, we are often interested in
counting the
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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 one is asked to fill in a missing number among a chain 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 he travelled 16 km, and in the second hour he
travelled
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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. To accommodate the case of no
real roots, i.e., to prov