The AMC 12 covers a wide range of mathematical topics, from algebra and geometry to number theory and combinatorics. Success on the exam requires not just familiarity with these topics but a deep, working understanding of the key concepts, formulas, and techniques within each area. In this comprehensive guide, we break down the essential mathematical knowledge you need to master for the AMC 12, organized by topic. Whether you are building your foundation or polishing your skills, this reference will serve as your roadmap to competition success.
1. Algebra
Algebra is the backbone of the AMC 12, appearing in roughly 8–10 questions on every exam. Here are the essential algebra topics you must master:
Linear and Quadratic Equations
Quadratic Formula: For ax² + bx + c = 0, x = (-b ± √(b² - 4ac)) / 2a
Discriminant: b² - 4ac determines the nature of roots (positive = 2 real roots, zero = 1 repeated root, negative = 2 complex roots)
Vieta's Formulas: For ax² + bx + c = 0 with roots r and s: r + s = -b/a and r·s = c/a
Completing the Square: A powerful technique for solving quadratics and finding maximum/minimum values
Polynomials
Factor Theorem: If P(a) = 0, then (x - a) is a factor of P(x)
Remainder Theorem: When P(x) is divided by (x - a), the remainder is P(a)
Rational Root Theorem: Rational roots of a polynomial with integer coefficients must be of the form p/q, where p divides the constant term and q divides the leading coefficient
Polynomial Division: Long division and synthetic division techniques
Functions and Their Properties
Function Composition: (f ∘ g)(x) = f(g(x))
Inverse Functions: If f(g(x)) = x and g(f(x)) = x, then f and g are inverses
Absolute Value: |x| = x if x ≥ 0, -x if x < 0
Piecewise Functions: Functions defined by different rules on different intervals
Sequences and Series
Arithmetic Sequences: a_n = a_1 + (n-1)d; Sum = n/2 · (a_1 + a_n)
Geometric Sequences: a_n = a_1 · r^(n-1); Sum = a_1 · (1 - r^n) / (1 - r)
Infinite Geometric Series: Sum = a_1 / (1 - r) when |r| < 1
Telescoping Series: Series where most terms cancel out
Advanced Algebra Topics
Complex Numbers: i² = -1; operations with complex numbers; complex conjugates
Logarithms: Properties of logarithms; change of base formula; solving exponential equations
Exponential Functions: Properties and graphs; solving exponential equations
Inequalities: AM-GM inequality; Cauchy-Schwarz inequality (occasionally)
2. Geometry
Geometry is another major topic, appearing in 6–8 questions per exam. The AMC 12 covers both coordinate geometry and Euclidean geometry, including trigonometry.
Triangles
Special Triangles: 30-60-90 (sides in ratio 1:√3:2) and 45-45-90 (sides in ratio 1:1:√2)
Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)
Law of Cosines: c² = a² + b² - 2ab·cos(C)
Area Formulas: 1/2 · b · h; 1/2 · a · b · sin(C); Heron's formula: √(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2
Triangle Inequality: The sum of any two sides must be greater than the third side
Circles
Arc Length: s = rθ (θ in radians)
Area of a Sector: A = 1/2 · r² · θ
Inscribed Angles: An inscribed angle is half the central angle that subtends the same arc
Power of a Point: For a point P and a circle, if two lines through P intersect the circle at A,B and C,D respectively, then PA·PB = PC·PD
Tangent Lines: A tangent to a circle is perpendicular to the radius at the point of tangency
Coordinate Geometry
Distance Formula: d = √((x₂-x₁)² + (y₂-y₁)²)
Midpoint Formula: M = ((x₁+x₂)/2, (y₁+y₂)/2)
Slope: m = (y₂-y₁)/(x₂-x₁)
Equation of a Line: y - y₁ = m(x - x₁) or y = mx + b
Equation of a Circle: (x-h)² + (y-k)² = r²
Parallel and Perpendicular Lines: Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals
Three-Dimensional Geometry
Volume Formulas: Prism/Cylinder: V = B·h; Pyramid/Cone: V = 1/3·B·h; Sphere: V = 4/3·πr³
Surface Area: Know formulas for common solids (cubes, cylinders, spheres, etc.)
Distance in 3D: d = √((x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²)
3. Trigonometry
Trigonometry is one of the topics that distinguishes the AMC 12 from the AMC 10. You can expect 2–4 trigonometry questions per exam.
Basic Trigonometric Functions
Definitions: sin(θ) = opposite/hypotenuse; cos(θ) = adjacent/hypotenuse; tan(θ) = sin(θ)/cos(θ)
Reciprocal Functions: csc(θ) = 1/sin(θ); sec(θ) = 1/cos(θ); cot(θ) = 1/tan(θ)
Unit Circle: Know the values of sin, cos, and tan for common angles (0°, 30°, 45°, 60°, 90° and their radian equivalents)
Trigonometric Identities
Pythagorean Identity: sin²(θ) + cos²(θ) = 1
Other Identities: 1 + tan²(θ) = sec²(θ); 1 + cot²(θ) = csc²(θ)
Double Angle Formulas: sin(2θ) = 2sin(θ)cos(θ); cos(2θ) = cos²(θ) - sin²(θ)
Sum and Difference Formulas: sin(A ± B) = sin(A)cos(B) ± cos(A)sin(B); cos(A ± B) = cos(A)cos(B) ∓ sin(A)sin(B)
Graphs of Trigonometric Functions
Period and Amplitude: For y = A·sin(Bx + C) + D, amplitude = |A|, period = 2π/|B|, phase shift = -C/B, vertical shift = D
Graph Properties: Know the shapes of sin, cos, and tan graphs and their transformations
4. Number Theory
Number theory questions test your ability to work with integers and their properties. You can expect 3–5 number theory questions per exam.
Divisibility and Primes
Divisibility Rules: Know rules for 2, 3, 4, 5, 6, 8, 9, 10, 11
Prime Factorization: Every integer greater than 1 can be uniquely expressed as a product of primes
Prime Testing: To test if n is prime, check divisibility by all primes up to √n
GCD and LCM
Greatest Common Divisor: Use the Euclidean algorithm or prime factorization
Least Common Multiple: LCM(a,b) = (a·b) / GCD(a,b)
Properties: GCD and LCM are useful in problems involving divisibility, remainders, and scheduling
Modular Arithmetic
Basic Properties: If a ≡ b (mod m), then a + c ≡ b + c (mod m) and a·c ≡ b·c (mod m)
Remainders: The remainder when a is divided by m is a mod m
Applications: Useful in problems involving cycles, last digits, and divisibility
Other Number Theory Topics
Number of Divisors: If n = p₁^a₁ · p₂^a₂ · ... · pₖ^aₖ, then the number of divisors is (a₁+1)(a₂+1)...(aₖ+1)
Sum of Divisors: The sum of divisors can be calculated using the prime factorization
Digit Problems: Problems involving the digits of numbers (e.g., sum of digits, reversing digits)
5. Counting and Probability
Counting and probability questions are among the most challenging on the AMC 12, often appearing in the later questions (20–25). You can expect 4–6 questions in this area.
Basic Counting Principles
Addition Principle: If there are m ways to do one thing and n ways to do another (and they are mutually exclusive), there are m + n ways to do either
Multiplication Principle: If there are m ways to do one thing and n ways to do another (in sequence), there are m·n ways to do both
Permutations and Combinations
Permutations: P(n,r) = n! / (n-r)! — the number of ways to arrange r items from n items where order matters
Combinations: C(n,r) = n! / (r!·(n-r)!) — the number of ways to choose r items from n items where order does not matter
Permutations with Repetition: The number of arrangements of n items where some are identical is n! / (n₁!·n₂!·...·nₖ!)
Probability
Basic Probability: P(A) = (number of favorable outcomes) / (total number of outcomes)
Complementary Probability: P(not A) = 1 - P(A) — often the easiest way to solve "at least" problems
Conditional Probability: P(A|B) = P(A and B) / P(B)
Independent Events: If A and B are independent, P(A and B) = P(A)·P(B)
Expected Value: E(X) = Σ(xᵢ · P(xᵢ))
Advanced Counting Techniques
Inclusion-Exclusion Principle: |A ∪ B| = |A| + |B| - |A ∩ B|; for three sets: |A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|
Pigeonhole Principle: If n+1 objects are placed into n boxes, at least one box must contain at least 2 objects
Casework: Breaking a problem into mutually exclusive cases and summing the results
Complementary Counting: Count the complement and subtract from the total
6. Putting It All Together: Problem-Solving Strategies
The hardest AMC 12 problems often require combining knowledge from multiple topics. Here are some general strategies:
Identify the Topic: Before solving, ask yourself: "What topic does this problem fall under? What tools do I have for this type of problem?"
Draw a Diagram: For geometry problems (and many others), a well-labeled diagram can reveal insights that are not obvious from the text alone.
Try Small Cases: For counting and number theory problems, try small examples to identify patterns before attempting the general case.
Use Multiple Approaches: If one approach is not working, try a different one. Many problems can be solved in multiple ways.
Check Your Work: If time permits, verify your answer by plugging it back into the problem or using a different method.
Final Thoughts: Mastery Takes Time
The AMC 12 covers a broad range of mathematical topics, and mastering all of them takes time and consistent effort. Do not try to learn everything at once. Focus on building a strong foundation in each topic, and then gradually work on more challenging problems. Use this guide as a checklist to ensure you have covered all the essential knowledge areas, and refer back to it as you prepare.
Remember, the goal is not just to memorize formulas but to develop a deep, intuitive understanding of the mathematics. When you truly understand a concept, you can apply it creatively to solve problems you have never seen before — and that is the hallmark of a successful AMC 12 competitor.
What topics do you find most challenging? Share your thoughts in the comments below and help build a community of learners supporting each other on the path to AMC 12 success!
