MathClubforKids

  • 4500 Stonecrest Dr, Ellicott City, MD 21043, USA

Olympiad Course for Class 8

Basic Mathematics

  1. Basic Identities
    Basic algebraic identities are used to simplify expressions and solve equations.
  2. Factorization
    Factorization is the process of expressing a number or an algebraic expression as the product of simpler factors.
  3. Inequalities and Wavy Curve
    Inequalities describe the relationship between two expressions where one may be greater or less than the other. The wavy curve method is used to solve rational inequalities.

Number Theory

  1. GCD & LCM
    The greatest common divisor (GCD) is the largest number that divides two or more numbers exactly. The least common multiple (LCM) is the smallest multiple that is common to two or more numbers.

    Example 1: Find the GCD of 18 and 24.
    Solution: The GCD of 18 and 24 is 6.

    Example 2: Find the LCM of 15 and 20.
    Solution: The LCM of 15 and 20 is 60.

  2. Cyclicity
    Cyclicity refers to the repeating pattern of the last digits of powers of numbers.
  3. Congruence
    Congruence occurs when two numbers have the same remainder when divided by a given number.
  4. Congruence based Problems
    Congruence problems often involve finding solutions to modular arithmetic equations.
  5. Divisibility Tests
    Divisibility rules help determine if one number can divide another without performing the division.

    Example 1: Is 432 divisible by 4?
    Solution: Yes, because the last two digits (32) are divisible by 4.

    Example 2: Is 275 divisible by 11?
    Solution: No, because the alternating sum of the digits 2−7+5=0 is not divisible by 11.

  6. Diophantine Equation

    Diophantine equations are equations where integer solutions are required.

    Example 1: Solve the Diophantine equation 3x+4y=17.
    Solution: One solution is x=3, y=2.

    Example 2: Solve 5x+7y=1.
    Solution: One solution is x=−2, y=1.

  7. Diophantine Equation based Problems

    These are problems requiring integer solutions to linear equations.

    Example 1: Solve 6x+9y=15.
    Solution: One solution is x=0, y=1.

    Example 2: Solve 4x+5y=13.
    Solution: One solution is x=3, y=−1.

  8. Some important theorems and Base System
    Some important theorems in number theory include Fermat’s Little Theorem, Euler’s Theorem, and Wilson’s Theorem.
  9. Problem Solving-1

    Problem solving in number theory often involves applying theorems and methods such as GCD, LCM, and congruence.

    Example 1: Solve 5x+7y=37 for integer values of x and y.
    Solution: One solution is x=4, y=1.

    Example 2: Find the remainder when 135 is divided by 11.
    Solution: 135÷11=12, so the remainder is 3.

Geometry

  1. Angle Sense
    Angle sense involves understanding the relationships between angles in different geometric shapes.
  2. Congruent triangles

    Two triangles are congruent if they have the same shape and size, which means their corresponding sides and angles are equal.

    Example 1: Two triangles have sides of lengths 3 cm, 4 cm, and 5 cm. Are they congruent?
    Solution: Yes, the triangles are congruent by the SSS (Side-Side-Side) rule.

    Example 2: Two triangles have two equal angles and the included side is equal. Are they congruent?
    Solution: Yes, the triangles are congruent by the ASA (Angle-Side-Angle) rule.

  3. Applications Congruent triangles, Triangle Inequalities

    The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

    Example 1: Can a triangle have sides of lengths 6 cm, 8 cm, and 15 cm?
    Solution: No, because 6+8=14, which is not greater than 15. Therefore, these sides do not form a triangle.

    Example 2: Check if sides of lengths 5 cm, 12 cm, and 13 cm can form a triangle.
    Solution: Yes, 5+12>13, 5+13>12, and 12+13>5, so they form a triangle.

  4. MPT,BPT

    The Midpoint Theorem (MPT) states that the line segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length. The Basic Proportionality Theorem (BPT), also known as Thales' Theorem, states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.

    Example 1: In a triangle, the midpoint of one side is joined to the midpoint of another side. What is the relationship between this line and the third side?
    Solution: The line is parallel to the third side, and its length is half the length of the third side (MPT).

    Example 2: In a triangle, a line is drawn parallel to one side, dividing the other two sides. What is the relationship between the divided sides?
    Solution: The line divides the two sides proportionally (BPT).

  5. Similar Triangles

    Two triangles are said to be similar if their corresponding angles are equal and their corresponding sides are proportional.

    Example 1: Are two triangles with angles 30o, 600, and 900 similar?
    Solution: Yes, by the AA (Angle-Angle) similarity criterion.

    Example 2: Two triangles have sides of lengths 5 cm, 7 cm, and 9 cm, and 10 cm, 14 cm, and 18 cm, respectively. Are they similar?
    Solution: Yes, the triangles are similar by the SSS (Side-Side-Side) similarity rule, as the sides are proportional.

  6. Angle Bisector Theorem ,Pythagoras
    The Angle Bisector Theorem states that the angle bisector of a triangle divides the opposite side into segments that are proportional to the other two sides. The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
  7. AREAS
    The area of a geometric figure, such as a triangle, rectangle, or parallelogram, can be found using specific formulas based on their dimensions.
  8. Points of Triangles

    Important points in a triangle include the centroid, circumcentre, incenter, and orthocentre.

    Example 1: What is the centroid of a triangle?
    Solution: The centroid is the point where the medians of the triangle intersect. It divides each median into two segments, one of which is twice the length of the other.

    Example 2: Where is the circumcentre of a right triangle located?
    Solution: The circumcentre is at the midpoint of the hypotenuse.

  9. Circles and Related Theorems
    Many theorems relate to the properties of circles, such as the tangent-secant theorem and the angle subtended by a chord theorem.
  10. Tangents and Power
    The Power of a Point Theorem states that for a point outside a circle, the product of the lengths of the segments of a secant line equals the square of the length of the tangent drawn from that point.

Algebra

  1. Factor and Remainder Theorem
  2. Quadratic Equations
  3. Nature Of Roots and Concept of iota
  4. Analysis of Graph of Quadratic
  5. System Of Equations
  6. Problems on Quadratic Equations
  7. Arithmentic Progressions
    An arithmetic progression (AP) is a sequence of numbers in which the difference between consecutive terms is constant.
  8. Geometric Progressions
    A geometric progression (GP) is a sequence where each term is found by multiplying the previous term by a fixed number called the common ratio.
  9. Telescoping
    A telescoping series is a series in which most terms cancel out, leaving only a few terms.

Combinatorics

  1. Arrangements
    Arrangements involve determining how many ways a set of items can be ordered or arranged. This often involves the use of factorials (n!).
  2. Selections
    Selections involve choosing a certain number of items from a larger group, typically without regard to order. This is often calculated using combinations  .
  3. Gap and Block Method
    The gap and block method is used in combinatorics to arrange items with gaps or blocks, particularly in problems involving restrictions.
  4. Beggar Coin
    The Beggar Coin method involves distributing indistinguishable items (like coins) to distinguishable groups (like beggars) with or without restrictions.
  5. Divisors
    The number of divisors of a number can be found by factoring the number into its prime factors and applying the formula for the number of divisors.

Trigonometry

  1. Trigonomentry Ratios
    Trigonometric ratios are functions of an angle commonly used in right triangles. The basic trigonometric ratios are sine (sin), cosine (cos), and tangent (tan).
  2. Identities and Standard Angles
  3. Trigonometry-Big Picture
    Trigonometry is essential for studying the relationships between angles and sides in triangles, particularly in navigation, physics, and engineering.
  4. Areas and He ight and Distance
    Trigonometry is often used to calculate the areas of triangles and to solve height and distance problems.

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$100 / month

  • Course Duration : 12 weeks/ 3 months
Subscription for Math Olympiad Preparation

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₹2000 / month

Admission fee : ₹2000 (for first time only)

  • Course Duration : 12 weeks/ 3 month

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