Interactive Course

Contest Math II

Guided training for mathematical problem solving at the level of the AMC 10 and 12.

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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)!

Topics covered

  • AMC Strategies
  • Analytic Geometry
  • Binomial Coefficients
  • Cauchy-Schwarz Inequality
  • Modular Arithmetic
  • Polynomial Roots
  • Recursion
  • Telescoping Series
  • Trigonometric Identities
  • Vieta's Formulas

Interactive quizzes

61

Concepts and exercises

625+

Course map

Prerequisites and Next Steps

  1. 1

    Introduction

    A taste of what's to come.

    1. Core Topics

      Improve your skills in the core topics of math contests at the level of the AMC.

      1
    2. Reframing Problems

      Explore the art of framing (or re-framing) a situation to make it easier to solve.

      2
    3. Key Strategies

      Learn a few very effective meta-strategies for problem-solving!

      3
    4. Color Cube Assembly

      Solve an elaborate riddle involving 27 colored cubes.

      4
    5. Autobiographical Numbers

      Learn how to construct numbers that describe themselves.

      5
  2. 2

    Algebra Basics

    Exponents, rates, logs, and more.

    1. Systems of Equations

      Look for ways to combine equations in order to solve these systems.

      6
    2. Rates and Ratios

      All rate problems are variations on a single theme, no matter what rate is being measured.

      7
    3. Quadratics

      Quadratics can always be solved algebraically, as long as you know the right techniques.

      8
    4. Exponents

      Master exponents and the rules that simplify them efficiently.

      9
    5. Special Functions

      Floor and ceiling functions are used frequently when the result needs to be an integer.

      10
    6. Logarithms

      Examine the properties of the exponential inverse and use them to unravel these problems.

      11
  3. 3

    Inequalities

    From the basics to AM-GM and Cauchy-Schwarz.

    1. Basic Inequalities

      Inequalities require you to use subtly different techniques than those you'd use with normal equations.

      12
    2. AM-GM

      The arithmetic mean is always greater than or equal to the geometric mean.

      13
    3. Cauchy-Schwarz

      Learn how to use the Cauchy-Schwarz Inequality for a variety of optimization problems.

      14
  4. 4

    Polynomials

    You'll make Vieta proud.

    1. Roots

      Use the properties of polynomials to find their roots.

      15
    2. Equations

      Factoring is usually the key to solving these polynomial equation puzzles.

      16
    3. Vieta's Formulas

      How do the coefficients and the roots of a polynomial relate? Vieta's formulas have an answer.

      17
    4. Transformations

      What happens to the roots when two polynomial functions are composed together?

      18
  5. 5

    Sequences and Series

    Take a look into this telescope...

    1. Arithmetic Sequences

      Practice and strengthen your skills working with arithmetic sequences.

      19
    2. Geometric Sequences

      All geometric sequences follow the same pattern; use it to figure these problems out.

      20
    3. Telescoping Series

      Identify a pattern in the terms to cut out all but the essential information in these series.

      21
  6. 6

    Number Theory Basics

    Primes, factors, GCD/LCM, and more.

    1. Prime Factorization

      Stretch the limits of your understanding of prime factors.

      22
    2. GCD/LCM

      Knowing the common divisors and multiples of a pair of number gives you a lot of information about them.

      23
    3. Counting Factors

      Learn how to efficiently count how many factors a number has.

      24
  7. 7

    Modular Arithmetic

    From units digits to Euler's Theorem.

    1. System of Congruences

      When modular arithmetic reduces the number of integers available, what happens to algebra?

      25
    2. Fractions

      How does modular arithmetic interact with fractions?

      26
    3. Units Digit

      Solve problems that require careful consideration of a number's final digits.

      27
    4. Euler's Theorem

      Understand and learn how to apply Euler's totient function!

      28
  8. 8

    Synthetic Geometry

    Triangles, circles, polygons, and more.

    1. Pythagorean Theorem

      This classical theorem about right triangles shows up in all sorts of situations.

      29
    2. Triangle Areas

      There are many different ways to find the area of a triangle.

      30
    3. Similar Triangles

      How can you use to your advantage the fact that two shapes are the same but for their size?

      31
    4. Angle Bisector Theorem

      If an angle is cut in half, then the triangle it's in is divided in a very particular way...

      32
    5. Power of a Point

      Investigate the relationship between interior points and the circle chords they form.

      33
    6. Cyclic Quadrilaterals

      What is special about quadrilaterals that are inscribed inside of circles?

      34
    7. Circles

      Puzzle out the solutions to these challenging problems full of circles.

      35
  9. 9

    Analytic Geometry

    Coordinates, mass points, and even some complex numbers.

    1. Coordinate Geometry

      Connect your understanding of geometry with algebra.

      36
    2. Conics

      Explore parabolas, hyperbolas, circles, and ellipses.

      37
    3. Mass Points

      Thinking of geometric figures as if they have mass provides some helpful intuition about length ratios.

      38
    4. Complex Number Geometry

      Similar to coordinate geometry, but complex geometry occurs in the complex plane.

      39
  10. 10

    Trigonometry

    The basics, the laws and relationships, and roots of unity.

    1. Trigonometric Functions

      These functions are ratios and understanding them is foundational for what follows.

      40
    2. Law of Cosines

      Use the law of cosines to find the missing information in all kinds of triangles.

      41
    3. Law of Sines

      Beware of ambiguity when solving triangles with the law of sines.

      42
    4. Trigonometric Identities

      Try to solve these puzzles using the extensive relationships between trig functions.

      43
    5. Roots of Unity

      Explore the properties of the many complex roots of the number one.

      44
  11. 11

    Combinatorics

    Counting is a bit harder than 1, 2, 3, ...

    1. Constructive Counting

      Determine how many solutions fit the requirements by counting up one piece at a time.

      45
    2. Complementary Counting

      Sometimes it is much easier to find the opposite of the correct solution.

      46
    3. Binomial Coefficients

      The binomial coefficients aren't just useful for expanding polynomials.

      47
    4. Principle of Inclusion-Exclusion

      Reason about multiple, overlapping groups when you have limited information.

      48
    5. Balls and Urns

      These classic puzzles will put your intuition and counting skills to the test.

      49
  12. 12

    Probability

    The probability you'll need this is pretty high.

    1. Probability

      Refresh the essential ideas that underlie probability.

      50
    2. Conditional Probability

      Having more information doesn't always make problems easier.

      51
    3. Expected Value

      Find the average value of a decision.

      52
    4. Recursion

      Solve big problems by understanding the relationships between small cases.

      53
    5. Linearity of Expectation

      How do expected values combine?

      54
    6. Events with States

      Solve these puzzles using recursive thinking and careful attention to detail.

      55
  13. 13

    AMC Strategies

    These strategies can save the day.

    1. Casework

      Breaking a problem into smaller pieces is sometimes the best approach.

      56
    2. Extreme Cases and Invariants

      Gain intuition about some problems by taking them to their extremes.

      57
    3. Generalization

      When in doubt, solve an easier version of the problem and then generalize.

      58
    4. Using Symmetry

      Find ways to exploit symmetry to efficiently puzzle out these problems.

      59
    5. Eliminating Choices

      Sometimes multiple choice questions can be hacked by ruling out impossible answers.

      60
    6. Simplifications

      Solving the simplest case of a problem can sometimes crack the whole thing wide open!

      61