Contest Math II

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

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)!

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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.

2. Reframing Problems

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

3. Key Strategies

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

4. Color Cube Assembly

Solve an elaborate riddle involving 27 colored cubes.

2. 2

Algebra Basics

Exponents, rates, logs, and more.

1. Included with

Systems of Equations

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

2. Included with

Rates and Ratios

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

3. Included with

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

4. Included with

Exponents

Master exponents and the rules that simplify them efficiently.

3. 3

Inequalities

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

1. Included with

Basic Inequalities

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

2. Included with

AM-GM

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

3. Included with

Cauchy-Schwarz

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

4. 4

Polynomials

You'll make Vieta proud.

1. Included with

Roots

Use the properties of polynomials to find their roots.

2. Included with

Equations

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

3. Included with

Vieta's Formulas

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

4. Included with

Transformations

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

5. 5

Sequences and Series

Take a look into this telescope...

1. Included with

Arithmetic Sequences

Practice and strengthen your skills working with arithmetic sequences.

2. Included with

Geometric Sequences

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

3. Included with

Telescoping Series

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

6. 6

Number Theory Basics

Primes, factors, GCD/LCM, and more.

1. Included with

Prime Factorization

Stretch the limits of your understanding of prime factors.

2. Included with

GCD/LCM

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

3. Included with

Counting Factors

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

7. 7

Modular Arithmetic

From units digits to Euler's Theorem.

1. Included with

System of Congruences

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

2. Included with

Fractions

How does modular arithmetic interact with fractions?

3. Included with

Units Digit

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

4. Included with

Euler's Theorem

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

8. 8

Synthetic Geometry

Triangles, circles, polygons, and more.

1. Included with

Pythagorean Theorem

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

2. Included with

Triangle Areas

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

3. Included with

Similar Triangles

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

4. Included with

Angle Bisector Theorem

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

9. 9

Analytic Geometry

Coordinates, mass points, and even some complex numbers.

1. Included with

Coordinate Geometry

Connect your understanding of geometry with algebra.

2. Included with

Conics

Explore parabolas, hyperbolas, circles, and ellipses.

3. Included with

Mass Points

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

4. Included with

Complex Number Geometry

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

10. 10

Trigonometry

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

1. Included with

Trigonometric Functions

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

2. Included with

Law of Cosines

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

3. Included with

Law of Sines

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

4. Included with

Trigonometric Identities

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

11. 11

Combinatorics

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

1. Included with

Constructive Counting

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

2. Included with

Complementary Counting

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

3. Included with

Binomial Coefficients

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

4. Included with

Principle of Inclusion-Exclusion

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

12. 12

Probability

The probability you'll need this is pretty high.

1. Included with

Probability

Refresh the essential ideas that underlie probability.

2. Included with

Conditional Probability

Having more information doesn't always make problems easier.

3. Included with

Expected Value

Find the average value of a decision.

4. Included with

Recursion

Solve big problems by understanding the relationships between small cases.

13. 13

AMC Strategies

These strategies can save the day.

1. Included with

Casework

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

2. Included with

Extreme Cases and Invariants

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

3. Included with