# 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

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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.

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2. ## Reframing Problems

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

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3. ## Key Strategies

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

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4. ## Color Cube Assembly

Solve an elaborate riddle involving 27 colored cubes.

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5. ## Autobiographical Numbers

Learn how to construct numbers that describe themselves.

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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.

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2. ## Rates and Ratios

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

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

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4. ## Exponents

Master exponents and the rules that simplify them efficiently.

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5. ## Special Functions

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

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6. ## Logarithms

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

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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.

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2. ## AM-GM

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

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3. ## Cauchy-Schwarz

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

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4. 4

## Polynomials

You'll make Vieta proud.

1. ## Roots

Use the properties of polynomials to find their roots.

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2. ## Equations

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

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3. ## Vieta's Formulas

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

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4. ## Transformations

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

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5. 5

## Sequences and Series

Take a look into this telescope...

1. ## Arithmetic Sequences

Practice and strengthen your skills working with arithmetic sequences.

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2. ## Geometric Sequences

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

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3. ## Telescoping Series

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

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## Number Theory Basics

Primes, factors, GCD/LCM, and more.

1. ## Prime Factorization

Stretch the limits of your understanding of prime factors.

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2. ## GCD/LCM

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

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3. ## Counting Factors

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

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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?

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2. ## Fractions

How does modular arithmetic interact with fractions?

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3. ## Units Digit

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

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4. ## Euler's Theorem

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

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## Synthetic Geometry

Triangles, circles, polygons, and more.

1. ## Pythagorean Theorem

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

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2. ## Triangle Areas

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

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3. ## Similar Triangles

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

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4. ## Angle Bisector Theorem

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

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5. ## Power of a Point

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

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7. ## Circles

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

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9. 9

## Analytic Geometry

Coordinates, mass points, and even some complex numbers.

1. ## Coordinate Geometry

Connect your understanding of geometry with algebra.

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2. ## Conics

Explore parabolas, hyperbolas, circles, and ellipses.

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3. ## Mass Points

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

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4. ## Complex Number Geometry

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

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## 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.

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2. ## Law of Cosines

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

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3. ## Law of Sines

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

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4. ## Trigonometric Identities

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

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5. ## Roots of Unity

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

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## 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.

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2. ## Complementary Counting

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

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3. ## Binomial Coefficients

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

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4. ## Principle of Inclusion-Exclusion

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

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5. ## Balls and Urns

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

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## Probability

The probability you'll need this is pretty high.

1. ## Probability

Refresh the essential ideas that underlie probability.

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3. ## Expected Value

Find the average value of a decision.

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4. ## Recursion

Solve big problems by understanding the relationships between small cases.

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5. ## Linearity of Expectation

How do expected values combine?

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6. ## Events with States

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

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## AMC Strategies

These strategies can save the day.

1. ## Casework

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

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2. ## Extreme Cases and Invariants

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

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3. ## Generalization

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

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4. ## Using Symmetry

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

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5. ## Eliminating Choices

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

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6. ## Simplifications

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

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