Mathematical Database
MATHEMATICAL INDUCTION
1. Introduction
Mathematics distinguishes itself from the other sciences in that it is built upon a set of axioms
and definitions, on which all subsequent theorems rely. All theorems can be derived, or proved,

Mathematical Database
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

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

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

2 0 &
( 1) ( '%$
u e e
"jsy`h#i
tdsp%'j`'ir`'cfw_o hh 'sri'`md`cfw_srvgpnp `r`Cvcz
c i f s e e e w s i s c u i t t q q s t
e l e c g i e t t t s t c q s l c s e e i s s t c
%p%`o`ClqrCvoi'` CydypdQv`iy%'`g#As dj 'hef'QCvct`'jy`Gdb's
wwihrsytu%'`q#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 magnitude only. (e.g. mass, length,
volume) We can describe it

Mathematical Database
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

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

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 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

Mathematical Database
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 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

Mathematical Database
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

Mathematical Database
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

Mathematical Database
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

Mathematical Database
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

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

Mathematical Database
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,

Mathematical Database
INEQUALITIES
UNIT 0
INTRODUCTION
Inequalities are mathematical statements involving one or more inequality signs: >, <, , or
. Readers are assumed to be familiar with the basic operations on inequalities: we may perform
addition or

Mathematical Database
INEQUALITIES
UNIT 3
GEOMETRIC INEQUALITIES
A geometric inequality is an inequality which appears in a geometric context. The simplest
examples are those involving variables which are sides of triangles that we came across in Unit 2.

Mathematical Database
SETS AND FUNCTIONS
1. Introduction of Sets
The concept of sets is the foundation of modern mathematics and definitions of most
fundamental mathematics terms are based on sets. Sets are well-defined collection of objects. Here,
well-d

Mathematical Database
INEQUALITIES
UNIT 2
TECHNIQUES IN PROVING INEQUALITIES
In Unit 1 we learnt several classical inequalities. While the inequalities themselves can be
easily memorised, it is often not easy to apply them to prove other inequalities in p

Mathematical Database
BINOMIAL THEOREM
1. Introduction
When we expand ( x + 1) 2 and ( x + 1)3 , we get
( x + 1) 2 = ( x + 1)( x + 1)
= x2 + x + x + 1
= x2 + 2 x + 1
and
( x + 1)3 = ( x + 1)( x + 1) 2
= ( x + 1)( x 2 + 2 x + 1)
= x3 + 2 x 2 + x + x 2 + 2

Mathematical Database
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

Mathematical Database
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

Mathematical Database
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

Mathematical Database
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

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.

Mathematical Database
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.