Topics for the AMC
Through carefully crafted problems and guided explanations, you'll improve your skills in the core topics on the AMC.
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)!
A taste of what's to come.
Through carefully crafted problems and guided explanations, you'll improve your skills in the core topics on the AMC.
Explore the art of framing (or re-framing) a situation to make it easier to solve.
Learn a few very effective meta-strategies for problem-solving!
Exponents, rates, logs, and more.
Look for ways to combine equations in order to solve these systems.
All rate problems are variations on a single theme, no matter what rate is being measured.
Quadratics can always be solved algebraically, as long as you know the right techniques.
Master exponents and the rules that simplify them efficiently.
Floor and ceiling functions are used frequently when the result needs to be an integer.
Examine the properties of the exponential inverse and use them to unravel these problems.
From the basics to AM-GM and Cauchy-Schwarz.
Inequalities require you to use subtly different techniques than those you'd use with normal equations.
The arithmetic mean is always greater than or equal to the geometric mean.
Learn how to use the Cauchy-Schwarz Inequality for a variety of optimization problems.
You'll make Vieta proud.
Use the properties of polynomials to find their roots.
Factoring is usually the key to solving these polynomial equation puzzles.
How do the coefficients and the roots of a polynomial relate? Vieta's formulas have an answer.
What happens to the roots when two polynomial functions are composed together?
Take a look into this telescope...
Practice and strengthen your skills working with arithmetic sequences.
All geometric sequences follow the same pattern; use it to figure these problem out.
Identify a pattern in the terms to cut out all but the essential information in these series.
Primes, factors, GCD/LCM, and more.
Stretch the limits of your understanding of prime factors.
Knowing the common divisors and multiples of a pair of number gives you a lot of information about them.
Learn how to efficiently count how many factors a number has.
From units digits to Euler's Theorem.
When modular arithmetic reduces the number of integers available, what happens to algebra?
How does modular arithmetic interact with fractions?
Solve problems that require careful consideration of a number's final digits.
Understand and learn how to apply Euler's totient function!
Triangles, circles, polygons, and more.
This classical theorem about right triangles shows up in all sorts of situations.
There are many different ways to find the area of a triangle.
How can you use to your advantage the fact that two shapes are the same but for their size?
If an angle is cut in half, then the triangle its in is divided in a very particular way...
Investigate the relationship between interior points and the circle chords they form.
What is special about quadrilaterals that are inscribed inside of circles?
Puzzle out the solutions to these challenging problems full of circles.
Coordinates, mass points, and even some complex numbers.
Connect your understanding of geometry with algebra.
Explore parabolas, hyperbolas, circles, and ellipses.
Thinking of geometric figures as if they have mass provides some helpful intuition about length ratios.
Similar to coordinate geometry, but complex geometry occurs in the complex plane.
The basics, the laws and relationships, and roots of unity.
These functions are ratios and understanding them is foundational for what follows.
Use the law of cosines to find the missing information in all kinds of triangles.
Beware of ambiguity when solving triangles with the law of sines.
Try to solve these puzzles using the extensive relationships between trig functions.
Explore the properties of the many complex roots of the number one.
Counting is a bit harder than 1, 2, 3, ...
Determine how many solutions fit the requirements by counting up one piece at a time.
Sometimes it is much easier to find the opposite of the correct solution.
The binomial coefficients aren't just useful for expanding polynomials.
Reason about multiple, overlapping groups when you have limited information.
These classic puzzles will put your intuition and counting skills to the test.
The probability you'll need this is pretty high.
Refresh the essential ideas that underlie probability.
Having more information doesn't always make problems easier.
Find the average value of a decision.
Solve big problems by understanding the relationships between small cases.
How do expected values combine?
Solve these puzzles using recursive thinking and careful attention to detail.
These strategies can save the day.
Breaking a problem into smaller pieces is sometimes the best approach.
Gain intuition about some problems by taking them to their extremes.
When in doubt, solve an easier version of the problem and then generalize.
Find ways to exploit symmetry to efficiently puzzle out these problems.
Sometimes multiple choice questions can be hacked by ruling out impossible answers.
Solving the simplest case of a problem can sometimes crack the whole thing wide open!