AMC 8 vs. AMC 10: Key Differences, Syllabus Comparison, and Preparation Guidance

As two major competitions in the American Mathematics Competitions (AMC) series, AMC 8 and AMC 10 often cause confusion among Grade 7–8 students and parents.
What are the key differences between AMC 8 and AMC 10?
How should students in Grades 7–8 decide which competition is more suitable?

This article provides a detailed comparison of AMC 8 and AMC 10, including exam structure, content scope, difficulty level, awards, advancement pathways, and selection strategies.

Differences Between AMC 8 and AMC 10

Overview Comparison

AMC 8 vs. AMC 10: Topic Coverage Comparison

AMC 8 Core Topics

• Basic Algebra: integers, rational and irrational numbers, real numbers, number lines, coordinate systems; linear equations, simple quadratic equations, basic inequalities; introductory sequences; fundamental algebraic techniques

• Basic Geometry: elementary constructions; Euclidean plane geometry involving points, lines, triangles, special quadrilaterals, and circles; perimeter and area of regular figures; basic solid geometry

• Solid Geometry: relationships between points, lines, and planes; three-dimensional coordinate systems; regular polyhedra; Euler’s formula

• Basic Number Theory: parity analysis, divisibility rules, greatest common divisor (GCD), least common multiple (LCM), basic congruences

• Basic Combinatorics: Venn diagrams; introductory permutations, combinations, and probability; factorials, binomial coefficients, Pascal’s triangle

AMC 10 Advanced Topics

• Advanced Algebra: polynomials, Remainder Theorem, Vieta’s formulas, relationships between roots and coefficients, special higher-degree equations; advanced inequalities and mean inequalities; introduction to functions including domain and range, quadratic, exponential, logarithmic, and basic trigonometric functions; advanced sequences and algebraic techniques

• Advanced Geometry: advanced constructions; triangle theorems, Law of Sines, Law of Cosines, incircles and circumcircles, Stewart’s Theorem, concurrency and collinearity; cyclic quadrilaterals and tangential quadrilaterals; regular polygons; introductory analytic geometry

• Advanced Number Theory: modular arithmetic, complex congruences, number bases and conversions, basic Diophantine equations

• Advanced Combinatorics: inclusion–exclusion principle, binomial theorem and related results, advanced permutations, combinations, probability, recursion, binary methods, and advanced counting techniques

Newly Added Topics from AMC 8 to AMC 10

Sequences

• Arithmetic and geometric sequences; explicit formulas, recursive formulas, summation methods; applications of complex sequences; non-arithmetic and non-geometric sequences

Linear Functions

• Graphs and properties of linear functions; linear inequalities and applications; coordinate geometry involving lines; word problems solved using systems of linear equations

Permutations, Combinations, and Probability

• Fundamental counting principles (addition and multiplication rules); permutation and combination methods; probability rules and applied problems

Integrated Operations

• Applications of GCD and LCM; sums and products of consecutive integers, odd and even numbers; advanced factorization techniques; exponent rules and equation solving

AMC 8 and AMC 10 Awards

AMC 8 Awards

• Perfect Score Award (25 points)

• Distinguished Honor Roll (Top 1% worldwide)

• Honor Roll (Top 5% worldwide)

• Achievement Roll (Grade 6 and below with a score of 15 or higher)

AMC 10 Awards

• Distinguished Honor Roll (Top 1% worldwide)

• Honor Roll (Top 5% worldwide)

• Achievement Roll (Grade 8 and below with a score of 90 or higher)

• AIME Qualification Certificate (awarded to students who meet the AIME cutoff score)

AMC 8 and AMC 10 Advancement Pathways

• AMC 8 has no direct advancement pathway; however, a Top 1% score provides a strong advantage for middle school advancement and international school applications.

• AMC 10: Approximately the top 2.5% of participants qualify for AIME, which is a required step toward USAMO (USA Mathematical Olympiad). AMC 10 performance is an important academic reference for applications to top universities such as Yale and Stanford.

How Should Grade 7–8 Students Choose Between AMC 8 and AMC 10?

1. Clarify Goals and Academic Foundation

Preparation Goals

• For middle school advancement or international school applications, AMC 8 Top 1% results are often more immediately competitive.

• For overseas high school or university applications, AMC 10 qualification for AIME carries higher academic value.

• Students seeking early exposure and confidence-building may prioritize AMC 8, while those interested in high-level mathematical challenges may attempt AMC 10 directly.

Academic Readiness

• AMC 8 readiness: Solid mastery of elementary competition math (e.g., rate problems, geometric area calculations) and basic middle school algebra (equations and inequalities).

• AMC 10 readiness: In addition to AMC 8 knowledge, students should understand arithmetic sequences, quadratic functions, solid geometry, probability, and demonstrate strong problem-solving speed.

2. Grade-Specific Recommendations

Grade 7 Students

• Weaker foundation: Focus on AMC 8, aiming for Top 5% or Top 1%, then transition to AMC 10 the following year.

• Moderate foundation: Prepare for both AMC 8 (Top 1%) and AMC 10 (Achievement Roll), supplementing AMC 10-specific topics such as combinatorics and number theory.

• Strong foundation: Directly attempt AMC 10 with goals of Top 5% or AIME qualification.

Grade 8 Students

• Average foundation: Prioritize AMC 8 Top 1% while building AMC 10 fundamentals (functions, solid geometry) for future competition.

• Strong foundation: Fully commit to AMC 10, targeting Top 1% or AIME qualification, with AMC 8 used as additional practice if desired.

3. Key Factors to Consider

• Time Commitment: AMC 8 preparation typically requires 3–6 months, while AMC 10 often requires 6–12 months.

• Problem Style Fit: AMC 8 emphasizes logical reasoning and accurate calculation, while AMC 10 requires flexible thinking, especially in the final problems.

• Awards and Long-Term Planning: For short-term recognition, AMC 8 is more accessible; for long-term competition goals such as USAMO, early exposure to AMC 10 is essential.

To help students make informed decisions and prepare effectively, instructors have compiled official AMC 8 and AMC 10 past problems. By practicing with real competition questions, students can better understand the difficulty level and determine whether AMC 8 or AMC 10 is the right fit.