Students who have just completed the AMC 8 may already be turning their attention to the AMC 10. To achieve strong results in the AMC 10, the first and most important step is gaining a clear understanding of the exam content. In this article, we take a comprehensive look at the AMC 10 Mathematics Competition, helping students plan their preparation more efficiently and effectively.
1. Overview of the AMC 10 Competition
The AMC 10 is designed for students in Grade 10 and below. The exam consists of 25 multiple-choice questions, with a scoring system of +6 points for a correct answer, −6 points for an incorrect answer, and 0 points for unanswered questions. The maximum score is 150 points, and the total exam duration is 75 minutes.
The questions are generally divided into three difficulty levels:
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Questions 1–10: Focus on fundamental concepts and core knowledge, emphasizing conceptual understanding.
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Questions 11–20: Test integrated problem-solving and logical reasoning, requiring flexible use of multiple methods.
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Questions 21–25: High-difficulty problems that differentiate top performers and often require creative or non-standard solutions.
Although the AMC 10 is intended for students up to Grade 10, many students in Grades 8–9 achieve outstanding results. This demonstrates that strategic preparation is often more important than grade level.
2. AMC 10 Exam Content
Approximately 90% of the AMC 10 syllabus overlaps with the content of China’s middle school mathematics curriculum. However, for students in Grade 10 and below, the AMC 10 extends slightly beyond standard coursework and includes a limited amount of Grade 10–11 material. Therefore, students aiming for strong performance are advised to consolidate school mathematics first and then engage in moderate advanced preparation.
Knowledge Areas and Topics
Basic Number Theory:
Prime numbers, prime factorization, divisor-counting formulas, greatest common divisor (GCD), least common multiple (LCM), Euclidean algorithm
Congruences and Divisibility:
Congruences, divisibility rules, Diophantine equations
Advanced Theorems and Number Bases:
Euler’s theorem, Fermat’s Little Theorem, Wilson’s theorem, Chinese Remainder Theorem, digit sums, number bases, repeating decimals
Basic Geometry:
Triangles, area, perimeter
Advanced Geometry:
Similar triangles, point-line relationships within triangles
Circles:
Fundamental circle properties and advanced theorems
Solid Geometry:
Lines, planes, angles, coordinate-based solid geometry, polyhedra
Coordinate Geometry:
Lines and circles
Geometric Transformations:
Translations, reflections, rotations
Addition and Multiplication Principles:
Fundamental counting principles
Permutations and Combinations:
Permutations, circular permutations, combinations, grouping, Vandermonde’s identity, inclusion–exclusion principle
Probability:
Classical probability, geometric probability, Markov chains, recursion
Sequences:
Arithmetic sequences, geometric sequences, and other sequence types
Polynomials:
Fundamental Theorem of Algebra, generalized Vieta’s formulas, rational root test, synthetic division, long division, Descartes’ Rule of Signs, remainder theorem, factor theorem
Functions and Graphs:
Common functions and their graphs, Gaussian function, ceiling function
Inequalities:
Linear inequalities, higher-degree polynomial inequalities, quadratic inequalities, Cauchy–Schwarz inequality, mean inequalities
3. Detailed Topic Analysis by Module
(1) Algebra (Approximately 35–40%)
Algebra is the core component of the AMC 10. It tests not only computational skills but also the ability to understand and construct algebraic relationships.
Key Topics:
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Polynomial operations and factorization (especially higher-degree polynomials)
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Systems of equations, including nonlinear systems
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Function properties and graph analysis (quadratic, exponential, logarithmic functions)
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Sequences and series (arithmetic and geometric)
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Solving and proving inequalities (AM-GM, Cauchy–Schwarz, etc.)
Preparation Advice:
Avoid rote memorization of formulas. Focus on understanding underlying principles and applying them flexibly in real problems. Learn to translate word problems into algebraic models.
(2) Geometry (Approximately 30–35%)
Key Topics:
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Triangle properties (similarity, congruence, special points, triangle inequality)
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Circle theorems (inscribed angles, tangents, power of a point)
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Polygons and area computation (dissection and rearrangement methods)
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Introductory solid geometry (volume, surface area, spatial relationships)
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Applications of trigonometric ratios and functions
Preparation Advice:
Combine visual reasoning with analytical thinking. Draw accurate diagrams and use visualization to support abstract reasoning.
(3) Number Theory and Combinatorics (25–30%)
These topics often determine score separation and are among the most challenging parts of the exam.
Key Topics:
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Divisibility, primes and composites, modular arithmetic
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Counting techniques (permutations, combinations, inclusion–exclusion, recursion)
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Probability models and calculations
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Logical reasoning and strategy-based problems
Preparation Advice:
There is no single fixed approach for these topics. Begin with a solid understanding of fundamental concepts and build intuition through extensive practice. Many problems admit elegant solutions—avoid unnecessary complexity.
4. Which Grades Are Suitable for the AMC 10?
In China, AMC 10 participants typically range from Grades 7 to 10. Many international schools treat AMC 10 results as an informal benchmark for admissions and may even include AMC-style questions in entrance exams. As a result, many students aim to complete the AMC 10 by Grades 7–9.
Students who begin preparation in Grade 7 can bridge into AMC 10 content through pre-AMC 10 coursework, with the goal of achieving a high score and qualifying for the AIME within one year. By Grade 8, these students often possess knowledge equivalent to middle school graduation and parts of high school mathematics, allowing for a smoother and more confident academic progression.