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AMC 8

AMC 8 Preparation

A comprehensive, competition-focused curriculum across combinatorics, algebra, number theory, and geometry, built through original explanations, worked examples, and purposeful practice.

Before you begin: Comfort with whole-number arithmetic. Combinatorics, algebra, number theory, and geometry are developed from first principles in the course.

Your learning plan

65 hr 20 min · 98 lessons

30 modules, with explanations, worked examples, and practice.

0 min of planned learning completed

Start learning

32 complete lessons (21 hr 20 min) are ready to study. The outline below shows the full course plan; upcoming lessons are clearly marked.

01

Permutations

3 hr 20 min · 0 / 5 complete

  1. Permutations Definition

    Define a permutation as an ordered arrangement

    AMC 8
    40 minDifficulty 2/10Ready to study

  2. Factorials

    Define n! and compute small factorials

    AMC 8
    40 minDifficulty 2/10Ready to study

  3. Permutations Fundamentals

    Apply the permutation formula P(n,r)

    AMC 8
    40 minDifficulty 3/10Ready to study

  4. Digit Permutations

    Count arrangements of digits into numbers

    AMC 8
    40 minDifficulty 3/10Ready to study

  5. Circular Arrangements

    Count arrangements around a circle

    AMC 8
    40 minDifficulty 4/10Ready to study
02

Combinations

2 hr · 0 / 3 complete

  1. Combinations Fundamentals

    Define a combination as an unordered selection

    AMC 8
    40 minDifficulty 3/10Ready to study

  2. Binomial Identity

    Use the symmetry identity C(n,r) = C(n,n-r)

    AMC 8
    40 minDifficulty 4/10Ready to study

  3. Tricky Combinations Examples

    Combine multiple selection steps into one count

    AMC 8
    40 minDifficulty 4/10Ready to study
03

Word Rearrangements

1 hr 20 min · 0 / 2 complete

  1. Word Rearrangements Fundamentals

    Count arrangements of a word with all distinct letters

    AMC 8
    40 minDifficulty 3/10Ready to study

  2. Word Rearrangements with Constraints

    Count arrangements where certain letters must stay together

    AMC 8
    40 minDifficulty 4/10Ready to study
04

Probability

4 hr · 0 / 6 complete

  1. Probability Fundamentals

    Define probability as favorable over total outcomes

    AMC 8
    40 minDifficulty 2/10Ready to study

  2. Distinguishability

    Decide whether objects in a probability problem are distinguishable

    AMC 8
    40 minDifficulty 3/10Ready to study

  3. Casework in Probability

    Split a probability problem into disjoint cases

    AMC 8
    40 minDifficulty 4/10Ready to study

  4. Probability of Independent Events

    Define independence between two events

    AMC 8
    40 minDifficulty 3/10Ready to study

  5. Probability of Dependent Events

    Define conditional probability informally

    AMC 8
    40 minDifficulty 4/10Ready to study

  6. Dependent or Independent?

    Decide whether two events in a story are independent or dependent

    AMC 8
    40 minDifficulty 4/10Ready to study
05

Casework

1 hr 20 min · 0 / 2 complete

  1. Casework Fundamentals

    Split a counting problem into exhaustive, non-overlapping cases

    AMC 8
    40 minDifficulty 3/10Upcoming

  2. Harder Casework Examples

    Choose a case split that keeps each case simple

    AMC 8
    40 minDifficulty 5/10Upcoming
06

Complementary Counting

1 hr 20 min · 0 / 2 complete

  1. Complementary Counting Fundamentals

    Count the complement of a condition instead of the condition itself

    AMC 8
    40 minDifficulty 3/10Upcoming

  2. Complementary Counting with Casework

    Combine complementary counting with a case split

    AMC 8
    40 minDifficulty 5/10Upcoming
07

Principle of Inclusion-Exclusion

2 hr 40 min · 0 / 4 complete

  1. PIE Strategies

    Recognize when overlapping sets require inclusion-exclusion

    AMC 8
    40 minDifficulty 4/10Upcoming

  2. PIE for 2 Events

    Apply |A ∪ B| = |A| + |B| - |A ∩ B|

    AMC 8
    40 minDifficulty 4/10Upcoming

  3. PIE for 3 Events

    Extend inclusion-exclusion to three overlapping sets

    AMC 8
    40 minDifficulty 5/10Upcoming

  4. PIE for any Number of Events

    State the general inclusion-exclusion formula for n sets

    AMC 8
    40 minDifficulty 7/10Upcoming
08

Stars and Bars

1 hr 20 min · 0 / 2 complete

  1. Stars and Bars Fundamentals

    Model a distribution problem as stars and bars

    AMC 8
    40 minDifficulty 5/10Upcoming

  2. Stars and Bars with Constraints

    Adjust stars and bars for a minimum-per-group requirement

    AMC 8
    40 minDifficulty 6/10Upcoming
09

Geometric Counting

2 hr 40 min · 0 / 4 complete

  1. Geometric Counting Fundamentals

    Recognize counting problems embedded in geometric figures

    AMC 8
    40 minDifficulty 4/10Upcoming

  2. Number of Squares in a Grid

    Count squares of every size within a square grid

    AMC 8
    40 minDifficulty 5/10Upcoming

  3. Number of Rectangles in a Grid

    Count rectangles by choosing two horizontal and two vertical lines

    AMC 8
    40 minDifficulty 5/10Upcoming

  4. Path Counting

    Count lattice paths between two grid points

    AMC 8
    40 minDifficulty 5/10Upcoming
10

Recursion

2 hr · 0 / 3 complete

  1. Recursion Fundamentals

    Define a sequence by a recursive rule

    AMC 8
    40 minDifficulty 4/10Upcoming

  2. Recursion with Constraints

    Add a restriction to a recursive counting rule

    AMC 8
    40 minDifficulty 5/10Upcoming

  3. Probability Recursions

    Set up a recursion for a probability that depends on prior outcomes

    AMC 8
    40 minDifficulty 6/10Upcoming
11

Ratios and Percentages

1 hr 20 min · 0 / 2 complete

  1. Ratios Fundamentals

    Interpret a ratio as a comparison between quantities

    AMC 8
    40 minDifficulty 2/10Ready to study

  2. Rate and Work

    Define rate as a quantity per unit of another quantity

    AMC 8
    40 minDifficulty 4/10Ready to study
12

Algebraic Manipulations and Equations

2 hr · 0 / 3 complete

  1. System of Equations Basics

    Set up a system of equations from a word problem

    AMC 8
    40 minDifficulty 3/10Ready to study

  2. Advanced Equation Solving Techniques

    Solve equations requiring a clever substitution or grouping

    AMC 8
    40 minDifficulty 5/10Ready to study

  3. Word Problems

    Translate a multi-condition word problem into equations

    AMC 8
    40 minDifficulty 4/10Ready to study
13

Speed, Distance, and Time

40 min · 0 / 1 complete

  1. Speed, Distance, and Time

    Apply distance equals speed times time

    AMC 8
    40 minDifficulty 4/10Ready to study
14

Sequences and Series

2 hr 40 min · 0 / 4 complete

  1. Arithmetic Sequences

    Define an arithmetic sequence by its common difference

    AMC 8
    40 minDifficulty 3/10Ready to study

  2. Special Series

    Recognize sums of consecutive integers, squares, and cubes

    AMC 8
    40 minDifficulty 5/10Ready to study

  3. Geometric Sequences

    Define a geometric sequence by its common ratio

    AMC 8
    40 minDifficulty 4/10Ready to study

  4. Arithmetico-Geometric Sequence

    Recognize a sequence formed by multiplying arithmetic and geometric terms

    AMC 8
    40 minDifficulty 6/10Ready to study
15

Mean, Median, Mode

1 hr 20 min · 0 / 2 complete

  1. Mean, Median, Mode Fundamentals

    Compute the mean, median, and mode of a data set

    AMC 8
    40 minDifficulty 2/10Ready to study

  2. Mean Median Mode Conditions Examples

    Work backward from a stated mean, median, or mode to find missing data

    AMC 8
    40 minDifficulty 5/10Ready to study
16

Telescoping

2 hr 40 min · 0 / 4 complete

  1. Telescoping Basics

    Recognize a sum or product that collapses through cancellation

    AMC 8
    40 minDifficulty 4/10Ready to study

  2. Telescoping Sums

    Apply partial-fraction-style splitting to a telescoping sum

    AMC 8
    40 minDifficulty 5/10Ready to study

  3. Telescoping Products

    Recognize a product that collapses through cancellation

    AMC 8
    40 minDifficulty 5/10Ready to study

  4. Telescoping Equation

    Set up an equation whose side telescopes

    AMC 8
    40 minDifficulty 6/10Ready to study
17

Primes and Divisibility

2 hr 40 min · 0 / 4 complete

  1. Primes

    Define a prime number and list small primes

    AMC 8
    40 minDifficulty 2/10Upcoming

  2. Divisibility Rules

    Apply divisibility rules for small numbers such as 2, 3, 5, 9, and 11

    AMC 8
    40 minDifficulty 3/10Upcoming

  3. Prime Factorization

    Write a number as a product of prime powers

    AMC 8
    40 minDifficulty 3/10Upcoming

  4. Legendre's Formula

    State Legendre's formula for the exponent of a prime in n!

    AMC 8
    40 minDifficulty 6/10Upcoming
18

Factors

2 hr · 0 / 3 complete

  1. Number of Factors

    Compute the number of positive divisors from a prime factorization

    AMC 8
    40 minDifficulty 4/10Upcoming

  2. Sum of Factors

    Compute the sum of positive divisors from a prime factorization

    AMC 8
    40 minDifficulty 5/10Upcoming

  3. Product of Factors

    Compute the product of all positive divisors of a number

    AMC 8
    40 minDifficulty 5/10Upcoming
19

GCD and LCM

2 hr 40 min · 0 / 4 complete

  1. GCD and LCM Fundamentals

    Define greatest common divisor and least common multiple

    AMC 8
    40 minDifficulty 3/10Upcoming

  2. GCD and LCM Product

    State the identity GCD(a,b) times LCM(a,b) equals a times b

    AMC 8
    40 minDifficulty 4/10Upcoming

  3. More GCD/LCM Properties

    Apply GCD and LCM properties to sums and multiples

    AMC 8
    40 minDifficulty 5/10Upcoming

  4. Euclidean Algorithm

    Apply the Euclidean algorithm to compute a GCD

    AMC 8
    40 minDifficulty 5/10Upcoming
20

Modular Arithmetic

3 hr 20 min · 0 / 5 complete

  1. Modular Arithmetic Fundamentals

    Define congruence modulo m

    AMC 8
    40 minDifficulty 3/10Upcoming

  2. Product Rule

    Multiply congruences modulo m

    AMC 8
    40 minDifficulty 4/10Upcoming

  3. Exponent Rule

    Find the cycle of remainders for powers of a base

    AMC 8
    40 minDifficulty 5/10Upcoming

  4. Multiple Modular Congruences

    Combine two or more congruence conditions on one integer

    AMC 8
    40 minDifficulty 5/10Upcoming

  5. Digit Cycles

    Find the repeating cycle of a units digit under repeated multiplication

    AMC 8
    40 minDifficulty 4/10Upcoming
21

Diophantine Equations

2 hr 40 min · 0 / 4 complete

  1. Quadratic Factorizations

    Rewrite an integer equation using a quadratic factorization identity

    AMC 8
    40 minDifficulty 5/10Upcoming

  2. Simon's Favorite Factoring Trick

    Add a constant to both sides to complete a factorable product

    AMC 8
    40 minDifficulty 6/10Upcoming

  3. Interesting Examples

    Combine factoring tricks with bounding arguments on a harder integer equation

    AMC 8
    40 minDifficulty 6/10Upcoming

  4. Cubic Factorizations

    Apply sum- and difference-of-cubes identities to an integer equation

    AMC 8
    40 minDifficulty 7/10Upcoming
22

Miscellaneous Number Theory

2 hr 40 min · 0 / 4 complete

  1. Palindromes

    Define a palindrome and describe its digit symmetry

    AMC 8
    40 minDifficulty 4/10Upcoming

  2. Money Problems

    Model coin and bill combinations as integer equations

    AMC 8
    40 minDifficulty 4/10Upcoming

  3. Integer Operations

    Apply floor and ceiling reasoning to integer problems

    AMC 8
    40 minDifficulty 5/10Upcoming

  4. Chicken McNugget Theorem

    State the Chicken McNugget (Frobenius) theorem for two relatively prime numbers

    AMC 8
    40 minDifficulty 6/10Upcoming
23

Angle Chasing

2 hr 40 min · 0 / 4 complete

  1. Angle Chasing Basics

    Apply vertical, supplementary, and parallel-line angle relationships

    AMC 8
    40 minDifficulty 3/10Upcoming

  2. Inscribed Angles

    Apply the inscribed angle theorem

    AMC 8
    40 minDifficulty 5/10Upcoming

  3. Polygons

    Compute the sum of interior angles of a polygon

    AMC 8
    40 minDifficulty 4/10Upcoming

  4. Advanced Circle Angle Chasing Theorems

    Apply the tangent-chord angle theorem

    AMC 8
    40 minDifficulty 7/10Upcoming
24

Triangles

3 hr 20 min · 0 / 5 complete

  1. Area of a Triangle

    Apply the base-height area formula

    AMC 8
    40 minDifficulty 4/10Upcoming

  2. Special Triangles

    Apply the side ratios of 45-45-90 and 30-60-90 triangles

    AMC 8
    40 minDifficulty 5/10Upcoming

  3. Pythagorean Theorem

    Apply the Pythagorean theorem to find a missing side

    AMC 8
    40 minDifficulty 3/10Upcoming

  4. Triangle Properties

    Apply the triangle inequality to test side-length feasibility

    AMC 8
    40 minDifficulty 4/10Upcoming

  5. Angle Bisector Theorem

    State the angle bisector theorem

    AMC 8
    40 minDifficulty 6/10Upcoming
25

Quadrilaterals

1 hr 20 min · 0 / 2 complete

  1. Quadrilateral Properties

    Apply the defining properties of squares, rectangles, and rhombi

    AMC 8
    40 minDifficulty 3/10Upcoming

  2. Breaking Quadrilaterals into Triangles

    Decompose a quadrilateral into triangles to find its area

    AMC 8
    40 minDifficulty 5/10Upcoming
26

Circles

2 hr · 0 / 3 complete

  1. Circle Properties

    Apply the relationships between radius, diameter, and chord

    AMC 8
    40 minDifficulty 3/10Upcoming

  2. Circular Area

    Apply the area formula for a full circle

    AMC 8
    40 minDifficulty 4/10Upcoming

  3. Length inside Circles

    Apply the arc length formula

    AMC 8
    40 minDifficulty 5/10Upcoming
27

Similar Triangles

1 hr 20 min · 0 / 2 complete

  1. Congruent Triangles

    Apply SSS, SAS, and ASA congruence criteria

    AMC 8
    40 minDifficulty 3/10Upcoming

  2. Similar Triangles

    Apply AA, SAS, and SSS similarity criteria

    AMC 8
    40 minDifficulty 4/10Upcoming
28

Area and Length of Complex Shapes

2 hr · 0 / 3 complete

  1. Regular Hexagons and Octagons

    Decompose a regular hexagon into equilateral triangles for area

    AMC 8
    40 minDifficulty 5/10Upcoming

  2. Area of Complex Shapes

    Decompose an irregular figure into familiar shapes

    AMC 8
    40 minDifficulty 5/10Upcoming

  3. Length of Complex Shapes

    Find the perimeter of an irregular figure by tracking each boundary piece

    AMC 8
    40 minDifficulty 5/10Upcoming
29

3D Geometry

2 hr · 0 / 3 complete

  1. Cubes, Prisms, and Pyramids

    Apply volume and surface area formulas for cubes and prisms

    AMC 8
    40 minDifficulty 4/10Upcoming

  2. Cylinders, Cones, and Spheres

    Apply volume and surface area formulas for a cylinder

    AMC 8
    40 minDifficulty 5/10Upcoming

  3. Similar Triangles in 3D

    Extend similarity ratios to three-dimensional figures

    AMC 8
    40 minDifficulty 6/10Upcoming
30

Additional Techniques and Strategies

2 hr · 0 / 3 complete

  1. Meta-solving Techniques

    Use answer choices to eliminate impossible options

    AMC 8
    40 minDifficulty 3/10Upcoming

  2. Silly Mistakes

    Identify the most common arithmetic and sign-error slip-ups

    AMC 8
    40 minDifficulty 2/10Upcoming

  3. Other Strategies to Maximize Your Score

    Allocate time across a fixed-length contest

    AMC 8
    40 minDifficulty 2/10Upcoming

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