
In this lesson, we shall learn about the concept of area and how to use it to solve problems
from AMC 8, MATHCOUNTS, and similar contests.
1. Area of a rectangle
2. Area of a circle
3. Areas of triangles having shared bases or shared heights.
4. Area of an equilateral triangle.
5. Area of a sector.
The focus of this lesson will be to study the concepts of tangents and secants of a circle. We shall give their definitions and also discuss some properties.
1. Tangents
2. Secants
3. A property of tangent lines.
In this lesson, we shall discuss the premise of coordinate geometry and also see some properties of the basic geometric objects, such as points, lines, and circles.
1. The basic idea of coordinate geometry
2. Lines and circles
3. The section formula
4. The distance formula
5. Area of a triangle in terms of its vertices.
We shall begin this lesson by introducing the concept of a transformation of a plane. We shall then define symmetries in terms of these transformations.
1. Transformations of the plane
2. Basic isometries
3. Types of symmetries.
In this lesson, we shall discuss the concepts of ratios and percentages, which are fundamental to all of Mathematics.
1. Ratios.
2. Percentages.
Strictly speaking, the measures of central tendency and dispersion belong to the field of Statistics. They are used to get quantitative information about sets of data. In this lesson, we shall see the very basics of this theory.
1. Mean
2. Median
3. Mode
4. Range
Polynomial equations of small degrees are the first objects that one encounters in Algebra. In this lesson, we shall discuss the question of their solvability and also see explicit expressions for the solutions.
1. Linear equations
2. Quadratic equations
3. The discriminant of a quadratic equation.
The idea of telescoping products and sums is best explained with examples. Indeed, there is no general theory for these concepts so it is hard to define them rigourously. We shall see a few examples of each.
1. Telescoping sums
2. Telescoping products.
In this lesson, we shall describe the basic counting principles that are used in enumerative combinatorics.
1. Addition principle
2. Multiplication principle
3. Bijection principle.
Permutations and combinations are, in a sense, extensions of the basic counting principles. Here we introduce the notation, discuss some basic identities and solve some problems.
1. Permutations
2. Combinations
3. Permutations with repetitions
The principle of inclusion and exclusion is the last of the basic counting principles. Due to the intricacies of the statement, a separate lesson has been dedicated to it.
1. The statement of the PIE.
2. Illustrative example.
The pigeonhole principle is deceptively simple; however, its applications are not. In this lesson, we shall give some formulations of the principle and use them to solve problems.
1. Two different formulations of the principles
2. Handshake theorem.
Though Probability theory is an area of Mathematics in its own right, the very basics can be
stated in combinatorial terms. In this lesson, we shall see the combinatorial definition of
probability and some properties.
1. Sample space
2. Definition of probability
3. Conditional probability
We use this lesson to revisit some basic concepts of number theory, like digits of numbers, division, and primes.
1. Decimal digits
2. Rules for checking divisibility by certain integers
3. Prime numbers
4. The greatest common divisor
5. The division algorithm
6. The Fundamental Theorem of Arithmetic