MathClubforKids

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

Olympiad Course for High School

Basic Mathematics

  1. IDENTITIES
    Identities are algebraic equations that hold true for any value of the variable. They are useful for simplifying expressions and solving equations.
  2. INEQUATIONS AND WAVY CURVE
    Inequations describe the relationship between two expressions where one is greater or less than the other. The wavy curve method is used to solve rational inequalities.
  3. BINOMIAL THEOREM
    The binomial theorem provides a way to expand expressions of the form (a+b)n for any positive integer n.

Number Theory

  1. LCM,GCD BASED PROBLEMS
    The least common multiple (LCM) is the smallest number that is a multiple of two or more numbers, while the greatest common divisor (GCD) is the largest number that divides two or more numbers exactly.
  2. CONGRUENCE
    Congruence describes a relationship where two numbers have the same remainder when divided by a given modulus.
  3. FERMAT AND EULER THEOREM
    Fermat’s Little Theorem and Euler’s Theorem are useful for calculating powers of integers modulo prime numbers.
  4. WILSON AND CRT
  5. PERFECT SQUARES, PRIMES
  6. DIOPHANTINE EQUATION
    Diophantine equations require integer solutions.
  7. BASE SYSTEM
    The base system refers to expressing numbers in different bases, such as binary (base 2), octal (base 8), decimal (base 10), and hexadecimal (base 16).
  8. MATHEMATICAL INDUCTION
    Mathematical induction is a method of proving statements that are true for all natural numbers.
  9. PROBLEM SOLVING 1
    Problem solving in number theory often involves a combination of techniques such as factorization, congruence, and properties of prime numbers.
  10. TRIGONOEMTRY- INTRODUCTION
    Trigonometry is the study of the relationships between the angles and sides of triangles, especially right triangles.

Geometry

  1. COMPOUND ,MULTIPLE ANGLES
    Compound and multiple angles involve the use of trigonometric identities to simplify expressions involving sums or multiples of angles.
  2. SINE AND COSINE RULE
  3. PROBLEMS ON ANGLE CHASING
    Angle chasing involves using geometric properties and known angle relationships to find unknown angles in a diagram.
  4. CONGRUENT TRIANGLES
    Two triangles are congruent if their corresponding sides and angles are equal. This can be proven using rules like SSS, SAS, ASA, AAS, and RHS.

    Example 1: Two triangles have sides of lengths 4 cm, 5 cm, and 6 cm. Are they congruent?
    Solution: Yes, if both triangles have corresponding sides of these lengths, they 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.

  5. TRIANGLE INEQUALITIES, MPT,BPT

    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. The Midpoint Theorem (MPT) states that the line joining the midpoints of two sides of a triangle is parallel to the third side and half its length. The Basic Proportionality Theorem (BPT) states that a line parallel to one side of a triangle divides the other two sides proportionally.

    Example 1: Can a triangle have sides of lengths 4 cm, 5 cm, and 10 cm?
    Solution: No, because 4+5=9, which is less than 10. Therefore, the sides do not form a triangle.

    Example 2: In a triangle, a line parallel to one side divides the other two sides into segments of lengths 3 cm and 4 cm. What is the ratio of the divided sides?
    Solution : By the Basic Proportionality Theorem, the sides are divided in the ratio 3:4.

  6. SIMILAR TRIANGLES
    Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional.
  7. PYTHAGORAS, APPOLONIUS, STEWART
    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. Apollonius’ Theorem relates the median of a triangle to its sides, and Stewart’s Theorem relates the sides and cevian of a triangle.
  8. PROBLEMS ON AREAS
    The area of a triangle, rectangle, parallelogram, or any other geometric figure can be calculated using specific formulas based on the dimensions and angles.
  9. CEVAS AND MENELAUS
    Ceva’s Theorem states that for a triangle, three concurrent cevians (lines drawn from the vertices to the opposite sides) divide the sides proportionally. Menelaus’ Theorem relates the lengths of the sides of a triangle cut by a transversal.
  10. CONCEPT OF ROTATION
    Rotation involves rotating a figure by a certain angle around a fixed point.
  11. QUADRILATERALS, CYCLIC QUADRILATERALS
    A quadrilateral is a four-sided polygon. A cyclic quadrilateral is one whose vertices all lie on a circle, and the opposite angles sum to 1800
  12. CIRCLES AND PROPERTIES
    Several theorems relate to the properties of circles, such as the tangent-secant theorem, properties of chords, and angles subtended by arcs.
  13. PTOLEMY AND TANGENTS
    Ptolemy’s Theorem relates the sides and diagonals of a cyclic quadrilateral. The tangent-secant theorem relates a tangent to a secant drawn from the same external point.
  14. CENTROID AND ORTHOCENTER
    The centroid is the point of intersection of the medians of a triangle, while the orthocentre is the point of intersection of the altitudes.
  15. INCENTER AND CIRCUMCENTER

    The incenter is the center of the incircle (a circle inscribed in the triangle), while the circumcentre is the center of the circumcircle (a circle passing through all three vertices).

    Example 1: Where is the circumcenter located in a right triangle?
    Solution: The circumcenter is located at the midpoint of the hypotenuse.

    Example 2: What is the incenter of a triangle?
    Solution : The incenter is the point where the angle bisectors of a triangle intersect.

  16. PROBLEM SOLVING-2
    Problem-solving in geometry often involves using multiple theorems and techniques to find unknown lengths, areas, or angles.
  17. ANALYTICAL GEOMETRY
    Analytical geometry involves using algebra to study geometric problems by representing points, lines, and curves with equations.

Combinatorics

  1. ARRANGEMENT BASED PROBLEMS
    Arrangement problems involve finding the number of ways to order or arrange objects, often calculated using factorials n!.
  2. nCr
  3. STANDARD CONCEPTS AND PROBLEMS
    Standard combinatronics problems involve basic principles like the pigeonhole principle, inclusion-exclusion, and more.
  4. BEGGAR COIN
    The beggar coin problem involves distributing indistinguishable objects (coins) among distinguishable groups (beggars), often solved using the stars and bars method.
  5. GENERATING FUNCTIONS
    Generating functions are used to solve counting problems by encoding sequences as power series.
  6. DIVISORS
    The number of divisors of a number can be found using its prime factorization.
  7. DISTRIBUTION OF DISTINCT OBJECTS, GROUPING
    This involves distributing or grouping distinct objects into different groups, with or without restrictions.
  8. OBJECTS IN CIRCLES
    Arranging objects in a circle has different rules than arranging them in a row. The number of circular permutations of n objects is (n−1)!.
  9. PHP
    The Pigeonhole Principle (PHP) states that if n items are placed into m containers and n> m, then at least one container must contain more than one item.
  10. RECURRENCE
    A recurrence relation defines each term in a sequence based on previous terms.
  11. COLORING TECHNIQUES

    Colouring techniques are often used in graph theory or combinatronics to solve problems related to partitions, graph colourings, or pattern counting.

    Example 1: In how many ways can the sides of a regular octagon be coloured using 3 different colours if adjacent sides cannot have the same colour?
    Solution: This is a graph colouring problem, and using the chromatic polynomial gives the number of colourings as P=3(3−1)8−1=3×27=384P = 3(3-1)^{8-1} = 3 \times 2^7 = 384P=3(3−1)8−1=3×27=384.

    Example 2: In how many ways can 4 rooms in a house be painted using 5 different colours if no two adjacent rooms can have the same colour?
    Solution: This is a graph colouring problem, and depending on the adjacency of the rooms, the chromatic polynomial is used to count the valid colourings.

  12. PARITY AND INVARIANCE

    Parity refers to whether a number is even or odd, while invariance refers to a quantity that remains unchanged during a process.

    Example 1: If 20 people are seated around a circular table, prove that there must be two people sitting next to each other who are either both men or both women.
    Solution: By parity, if there are an odd number of men and women, there must be at least one pair of the same gender.

    Example 2: Show that in a game where you flip a coin 100 times, there must be a run of consecutive heads or tails of length 6 or more.
    Solution: By invariance, due to the law of large numbers, in a random sequence of 100 flips, a long run of consecutive heads or tails is highly probable.

  13. PROBLEM SOLVING -3

    This section involves complex combinatorics problems that require a combination of techniques like inclusion-exclusion, Pigeonhole Principle, and more.

    Example 1: How many ways can 5 indistinguishable objects be placed into 3 distinct boxes such that no box is empty?
    Solution: Use the inclusion-exclusion principle to account for the restriction that no box is empty. The number of ways is 6.

    Example 2: In how many ways can 7 different books be divided among 3 students such that each student gets at least one book?
    Solution: Using inclusion-exclusion, there are 540540540 ways to divide the books.

Algebra

  1. FACTOR AND REMAINDER THEOREM
  2. RELATION BETWEEN ROOTS AND CORFFICIENTS, NATURE OF ROOTS
  3. RATIONAL AND INTEGRAL ROOTS
  4. GRAPH ANALYSIS AND COMMON ROOTS
    Graph analysis involves studying the behavior of polynomial or algebraic functions, such as their intersections (common roots) and turning points.
  5. MISCLEANEOUS EQUATIONS
    These are equations that may not fit into typical categories, often requiring creative problem-solving techniques.
  6. COMPLEX NUMBERS
  7. SIGMA NOTATION
  8. Arithmetic, Geometric, Harmonic Progressions (AP, GP, HP)
    An arithmetic progression (AP) is a sequence in which the difference between consecutive terms is constant. A geometric progression (GP) has a constant ratio between consecutive terms, and a harmonic progression (HP) is the reciprocal of an arithmetic progression.
  9. OTHER SEQUENCES
    Other sequences include Fibonacci sequences, Lucas sequences, and other specialized sequences with specific recurrence relations.
  10. TELESCOPING
    A telescoping series is a series where most terms cancel out, leaving only a few terms to be added.
  11. MEANS INEQUALITY, MTH POWERS,RMS
    The means inequality states that for any non-negative numbers, the arithmetic mean is greater than or equal to the geometric mean, which is greater than or equal to the harmonic mean. The root mean square (RMS) is the square root of the mean of the squares of the numbers.
  12. CAUCHY SCHWARZ, REARRANGEMENT,TITU'S
  13. FUNCTIONS-INTRO
    A function is a rule that assigns each input exactly one output. Common types of functions include linear, quadratic, polynomial, and trigonometric functions.
  14. MODULUS
  15. GIF, Fractional part
  16. LOGARITHM
  17. FUNCTIONAL EQUATIONS
    A functional equation is an equation in which the unknown is a function.

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

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