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Brilliant

Contest Math II

via Brilliant

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Overview

This course is here to guide you through the "magic", revealing the thought processes that lead to clever solutions to beautiful problems.

You’ll become a better mathematical problem-solver across several exciting topics, including algebra, geometry, number theory, and discrete math. You’ll be able to connect the dots between various strategies, so that you can tackle advanced math competition problems (even the ones that don't look like problems you've seen before)!

Syllabus

  • Introduction: A taste of what's to come.
    • Core Topics: Improve your skills in the core topics of math contests at the level of the AMC.
    • Reframing Problems: Explore the art of framing (or re-framing) a situation to make it easier to solve.
    • Key Strategies: Learn a few very effective meta-strategies for problem-solving!
    • Color Cube Assembly: Solve an elaborate riddle involving 27 colored cubes.
    • Autobiographical Numbers: Learn how to construct numbers that describe themselves.
  • Algebra Basics: Exponents, rates, logs, and more.
    • Systems of Equations: Look for ways to combine equations in order to solve these systems.
    • Rates and Ratios: All rate problems are variations on a single theme, no matter what rate is being measured.
    • Quadratics: Quadratics can always be solved algebraically, as long as you know the right techniques.
    • Exponents: Master exponents and the rules that simplify them efficiently.
    • Special Functions: Floor and ceiling functions are used frequently when the result needs to be an integer.
    • Logarithms: Examine the properties of the exponential inverse and use them to unravel these problems.
  • Inequalities: From the basics to AM-GM and Cauchy-Schwarz.
    • Basic Inequalities: Inequalities require you to use subtly different techniques than those you'd use with normal equations.
    • AM-GM: The arithmetic mean is always greater than or equal to the geometric mean.
    • Cauchy-Schwarz: Learn how to use the Cauchy-Schwarz Inequality for a variety of optimization problems.
  • Polynomials: You'll make Vieta proud.
    • Roots: Use the properties of polynomials to find their roots.
    • Equations: Factoring is usually the key to solving these polynomial equation puzzles.
    • Vieta's Formulas: How do the coefficients and the roots of a polynomial relate? Vieta's formulas have an answer.
    • Transformations: What happens to the roots when two polynomial functions are composed together?
  • Sequences and Series: Take a look into this telescope...
    • Arithmetic Sequences: Practice and strengthen your skills working with arithmetic sequences.
    • Geometric Sequences: All geometric sequences follow the same pattern; use it to figure these problems out.
    • Telescoping Series: Identify a pattern in the terms to cut out all but the essential information in these series.
  • Number Theory Basics: Primes, factors, GCD/LCM, and more.
    • Prime Factorization: Stretch the limits of your understanding of prime factors.
    • GCD/LCM: Knowing the common divisors and multiples of a pair of number gives you a lot of information about them.
    • Counting Factors: Learn how to efficiently count how many factors a number has.
  • Modular Arithmetic: From units digits to Euler's Theorem.
    • System of Congruences: When modular arithmetic reduces the number of integers available, what happens to algebra?
    • Fractions: How does modular arithmetic interact with fractions?
    • Units Digit: Solve problems that require careful consideration of a number's final digits.
    • Euler's Theorem: Understand and learn how to apply Euler's totient function!
  • Synthetic Geometry: Triangles, circles, polygons, and more.
    • Pythagorean Theorem: This classical theorem about right triangles shows up in all sorts of situations.
    • Triangle Areas: There are many different ways to find the area of a triangle.
    • Similar Triangles: How can you use to your advantage the fact that two shapes are the same but for their size?
    • Angle Bisector Theorem: If an angle is cut in half, then the triangle it's in is divided in a very particular way...
    • Power of a Point: Investigate the relationship between interior points and the circle chords they form.
    • Cyclic Quadrilaterals: What is special about quadrilaterals that are inscribed inside of circles?
    • Circles: Puzzle out the solutions to these challenging problems full of circles.
  • Analytic Geometry: Coordinates, mass points, and even some complex numbers.
    • Coordinate Geometry: Connect your understanding of geometry with algebra.
    • Conics: Explore parabolas, hyperbolas, circles, and ellipses.
    • Mass Points: Thinking of geometric figures as if they have mass provides some helpful intuition about length ratios.
    • Complex Number Geometry: Similar to coordinate geometry, but complex geometry occurs in the complex plane.
  • Trigonometry: The basics, the laws and relationships, and roots of unity.
    • Trigonometric Functions: These functions are ratios and understanding them is foundational for what follows.
    • Law of Cosines: Use the law of cosines to find the missing information in all kinds of triangles.
    • Law of Sines: Beware of ambiguity when solving triangles with the law of sines.
    • Trigonometric Identities: Try to solve these puzzles using the extensive relationships between trig functions.
    • Roots of Unity: Explore the properties of the many complex roots of the number one.
  • Combinatorics: Counting is a bit harder than 1, 2, 3, ...
    • Constructive Counting: Determine how many solutions fit the requirements by counting up one piece at a time.
    • Complementary Counting: Sometimes it is much easier to find the opposite of the correct solution.
    • Binomial Coefficients: The binomial coefficients aren't just useful for expanding polynomials.
    • Principle of Inclusion-Exclusion: Reason about multiple, overlapping groups when you have limited information.
    • Balls and Urns: These classic puzzles will put your intuition and counting skills to the test.
  • Probability: The probability you'll need this is pretty high.
    • Probability: Refresh the essential ideas that underlie probability.
    • Conditional Probability: Having more information doesn't always make problems easier.
    • Expected Value: Find the average value of a decision.
    • Recursion: Solve big problems by understanding the relationships between small cases.
    • Linearity of Expectation: How do expected values combine?
    • Events with States: Solve these puzzles using recursive thinking and careful attention to detail.
  • AMC Strategies: These strategies can save the day.
    • Casework: Breaking a problem into smaller pieces is sometimes the best approach.
    • Extreme Cases and Invariants: Gain intuition about some problems by taking them to their extremes.
    • Generalization: When in doubt, solve an easier version of the problem and then generalize.
    • Using Symmetry: Find ways to exploit symmetry to efficiently puzzle out these problems.
    • Eliminating Choices: Sometimes multiple choice questions can be hacked by ruling out impossible answers.
    • Simplifications: Solving the simplest case of a problem can sometimes crack the whole thing wide open!

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