A Complete Guide to AMC 10: Exam Scope, Difficulty, and Key Topics for 2026 (With Past Papers)

As one of the most recognized high school mathematics competitions worldwide, AMC 10 is not only a “golden ticket” for applications to top universities but also a rigorous test of mathematical thinking and problem-solving skills. How can students prepare efficiently? How can they achieve breakthrough performance? This article reveals the difficulty distribution, core topics, and preparation strategies of AMC 10 to help students maximize their results.

AMC 10 Question Difficulty Distribution

The 25 questions in AMC 10 can be divided into three levels of difficulty. Understanding this distribution helps students allocate time effectively:

1. Foundational Questions (1–10)
These are relatively straightforward and test basic algebra, geometry, and number theory, including integer operations, fractions, decimals, percentages, ratios, simple equation solving, and fundamental properties of plane geometry.

Students with systematic preparation in AMC 10 fundamentals can generally solve these questions efficiently. However, due to strict time limits, quick problem-solving techniques are essential to answer accurately under exam conditions.

2. Intermediate Questions (11–20)
These require stronger reasoning and analytical skills and may challenge many students. Beyond testing basic skills, intermediate questions involve quadratic functions, triangle similarity, and probability concepts.

Solving these problems typically requires integrating multiple concepts and performing multi-step derivations, emphasizing logical consistency throughout the solution process.

3. Advanced Questions (21–25)
These represent the highest difficulty level and are critical for distinguishing top performers. Problems may involve complex sequences, combinatorial challenges, advanced geometric theorems, and multi-step calculations.

Success at this level demands deep mathematical knowledge, exceptional problem-solving intuition, and experience, along with the ability to apply diverse mathematical techniques flexibly.

AMC 10 Exam Syllabus

The AMC 10 syllabus focuses on algebra, geometry, number theory, probability, and combinatorics, excluding calculus. Detailed topics include:

AMC 10 knowledge areas mainly include elementary algebra, basic geometry (Pythagoras theorem, area and volume formulas), number theory, and probability, with algebra and geometry occupying the largest proportion.

AMC 10 Key Topics

Advanced Algebra

• Polynomials, remainder theorem, Vieta’s theorem, root-coefficient relations

• Special higher-order equations

• Advanced inequalities, mean value inequalities

• Functions: domain/range, quadratic, exponential, logarithmic, simple trigonometric functions

• Advanced sequences and series, algebraic techniques

Advanced Geometry

• Triangle constructions, law of sines and cosines

• Inscribed and circumscribed circles, Stewart’s theorem, collinearity and concurrency

• Circle and quadrilateral properties, cyclic quadrilaterals, regular polygons

• Angle, perimeter, and area calculation techniques

• Advanced plane geometry strategies, introductory analytic geometry

Solid Geometry

• Relationships between points, lines, and planes in 3D

• 3D coordinate system, polyhedra, Euler’s formula

• Techniques for special 3D figures

Advanced Number Theory

• Sequences, modular arithmetic, congruences

• Integers, fractions, decimals, base conversions

• Basic Diophantine equations and advanced number theory techniques

Advanced Combinatorics

• Inclusion-exclusion principle, binomial theorem

• Advanced permutations, combinations, and probability

• Recursion, binary method, and advanced combinatorial methods

Focus Areas & Difficulty by Topic

Algebra

• Sequences, equations, quadratic functions, factorization formulas

• Difficulty: simplifying problems and solving polynomial divisibility or quadratic roots

Functions

• Coordinate systems, linear transformations, linear functions, circle equations

• Difficulty: calculating polygon areas using Pick’s theorem or the shoelace formula

Plane Geometry (Triangles, Quadrilaterals, Polygons)

• Triangle congruence, similarity, sine/cosine applications, multi-method area calculation

• Difficulty: irregular shapes, area transformations, and advanced formulas

Circle & Solid Geometry

• Circle properties, 3D volume/surface area, Euler’s formula

• Difficulty: spatial visualization, constructing auxiliary lines

Combinatorics

• Addition/multiplication principles, round-robin arrangements

• Difficulty: avoiding double counting, complex combinatorial scenarios

Probability & Statistics

• Classical probability, geometric probability, conditionals

• Difficulty: analyzing events and conditional probabilities

Number Theory

• Factors and multiples, digit manipulation, primes and composites

• Difficulty: parity analysis, modular arithmetic, custom-defined operations

AMC 10 Knowledge Priorities

Must-Know Topics (Core)

• Algebra: factorization, substitution, quadratic functions and equations, absolute value, sequences, summations, lines and circles

• Geometry: transformations, circle properties

• Number Theory: modular arithmetic, integer equations, positional value and base systems

Optional Topics (Mastery)

• Algebra: polynomials, inequalities, extrema problems

• Geometry: angle calculation, triangle centers

• Combinatorics: advanced counting (recursion, PIE, bijection), combinatorial extrema, geometric probability

Common Mistakes

• Algebra: word problems, absolute value equations, exponential/logarithmic computations

• Geometry: congruent/similar triangles, figure decomposition, spatial visualization

• Counting: distinguish permutations vs combinations, avoid double counting, recursion errors

• Number Theory: prime factorization, last digits, modular arithmetic

• Probability: discrete vs geometric probability