如何高效备考AMC 10/12 | 现代数学启蒙

slug

type

status

1. 明确比赛特点与目标分数

AMC 8：初中难度，重点是基础数论、组合、几何、代数。时间紧（40 分钟 25 题），需要快速计算。

AMC 10/12：难度更高，覆盖高中竞赛数学，涉及更深的数论、组合、解析几何、函数性质等。

目标分数决定策略：

冲击 AIME 资格（AMC10 ≥ 103.5 / AMC12 ≥ 93）→ 要追求高准确率+全覆盖。
稳定中高分 → 要抓易中题+快速放弃陷阱题。

2. 考点体系化

AMC 的考点是有固定分布的，可以按题源分：

模块	占比（AMC10/12）	常见题型
数论	25% 左右	余数、整除、同余、质数分解、位数和、模运算
组合与概率	25% 左右	排列组合、容斥、递推、概率模型、生成函数基础
几何	25% 左右	相似与全等、圆、面积比例、坐标几何
代数与函数	20% 左右	多项式、函数性质、不等式、方程技巧
杂项与技巧题	5-10%	特殊构造、数学归纳、逻辑推理

建议用 Excel 或 Notion 做一个 AMC 知识点表，把每道错题归类到具体知识点，方便查漏补缺。

The American Mathematics Competitions (AMC) covers a broad set of middle–high school mathematics topics, but the exact scope depends on the level:

AMC 8 – for grades 8 and below

AMC 10 – for grades 10 and below

AMC 12 – for grades 12 and below

Here’s a topic breakdown based on official AMC guidelines and past problems.

1. Arithmetic & Number Theory

Integers, factors, multiples, prime numbers

Greatest common divisor (GCD), least common multiple (LCM)

Modular arithmetic (remainders)

Divisibility rules

Base systems (binary, base-n conversions)

Perfect squares, cubes, and other special numbers

Rational numbers, decimals, repeating patterns

Sequences & series (arithmetic, geometric, special sequences)

Properties of remainders and residues

2. Algebra

Algebraic expressions, polynomials, factoring, expansion

Equations and inequalities (linear, quadratic, systems)

Rational expressions & equations

Exponents and radicals

Absolute value equations/inequalities

Functions: evaluation, composition, inverses (basic)

Symmetry and transformations of graphs

Special factorizations (a2−b2,a3+b3,etc.)(a^2 - b^2, a^3 + b^3, etc.)

Simple functional equations

AM–GM inequality, Cauchy–Schwarz (more in AMC 12)

3. Counting & Combinatorics

Basic counting principles (addition & multiplication)

Permutations & combinations

Binomial theorem (expansion & coefficients)

Probability (classical, conditional, complementary counting)

Pascal’s Triangle patterns

Inclusion–exclusion principle

Pigeonhole principle

Arrangements with restrictions

Circular permutations

4. Geometry

Triangles (types, congruence, similarity, Pythagorean theorem)

Circles (chords, tangents, arcs, inscribed angles, power of a point)

Quadrilaterals & polygons (area, angle sums, special types)

Coordinate geometry (distance, midpoint, slope, equation of a line)

Transformations (reflections, rotations, translations, dilations)

3D geometry (prisms, pyramids, spheres, volume, surface area)

Trigonometry basics (ratios, law of sines/cosines for AMC 12)

Geometric loci & constructions

Symmetry and tessellations

5. Logic & Problem-Solving Strategies

Pattern recognition

Working backwards

Guess and check with elimination

Extremal principle (largest/smallest values)

Mathematical induction (AMC 12)

Constructive algorithms

6. Advanced Topics (Mostly AMC 12)

More advanced trigonometry (identities, multiple angles)

Complex numbers in algebra & geometry

Vectors in the plane

Polynomials (roots, Vieta’s formulas, transformations)

Inequalities beyond basics (Jensen’s, Cauchy–Schwarz)

Partial fraction decomposition

Recursion and generating functions (rare but possible)

💡 Difficulty progression:

AMC 8: Focus on basic arithmetic, introductory algebra, geometry, and combinatorics.

AMC 10: Adds more algebra, number theory, and deeper geometry.

AMC 12: Includes higher-level algebra, advanced number theory, and trigonometry.

If you want, I can make you a detailed AMC topic checklist with example problem types for each category so you can track your preparation efficiently. That would make practice much more structured.

3. 基础知识快速巩固（2~3 周）

选一本 AMC 常用教材（如 Art of Problem Solving 系列 AoPS Prealgebra / Algebra / Geometry / Number Theory / Counting & Probability）。

每天做少量高质量例题（每类 2-3 题），重点掌握：

常用结论（如模运算公式、三角函数特殊值、组合数恒等式）
常见构造方法（如补集法、对称法、极端原理）

用 错题本 + 知识卡片（Anki） 固化记忆。

AoPS专题训练

https://artofproblemsolving.com/wiki/ （输入Combinatorics等关键词）,可以找到分类真题！！！

示例:

https://artofproblemsolving.com/wiki/index.php/Combinatorics

https://artofproblemsolving.com/wiki/index.php/Category:Introductory_Combinatorics_Problems

https://artofproblemsolving.com/wiki/index.php/Complex_number

https://artofproblemsolving.com/wiki/index.php/Function

https://artofproblemsolving.com/wiki/index.php/Probability

4. 提升解题速度与技巧（3~4 周）

AMC 限时解题，速度很重要：

限时训练：一次做 5 题，定时 10~12 分钟，强迫快速思考。

估算与排除法：能口算或快速估算的不要代数化；遇到难题先排除不可能选项。

跳题策略：

先做前 15 题（通常较简单），保证全对。
后面先找自己擅长的题型（如几何），最后再碰最难的。

常用技巧：坐标化几何、代换简化、构造对称、反证、拆点法、方程化数论等。

5. 真题模拟与分析（4~6 周）

从 近 10 年 AMC 真题 开始做（可用 AoPS 题库 / amc-pastpapers）。

模拟时必须：

严格限时（AMC8：40 分钟；AMC10/12：75 分钟）。

模拟考试环境（不用草稿以外资料）。

做完立刻分类分析错题（是知识不会？方法不会？粗心？时间不够？）。

错题二次训练：

当天再做一次相似题。
一周后回顾同类题，确保不会再错。

6. 心态与考试策略

AMC 评分机制：错题倒扣分（1.5 分 → 0 分），所以不会的题要果断放弃，避免瞎猜。

考前一周做 2~3 次完整模拟，适应节奏。

考试中：

前 5 分钟快速扫题，先做秒杀题。
不陷入长时间计算，超过 4 分钟直接跳。
最后 5 分钟做全局检查，防止低级错误。

1. Understand the Contest Format & Your Target Score

AMC 8: Middle-school level, focuses on basic number theory, combinatorics, geometry, and algebra. Short time limit (40 minutes, 25 questions), requires fast calculation.

AMC 10/12: Higher difficulty, covers high-school competition math, including deeper number theory, combinatorics, analytic geometry, and functions.

Target score affects your strategy:

AIME qualification goal (AMC10 ≥ 103.5 / AMC12 ≥ 93) → aim for high accuracy + full coverage.
Stable mid-high score → secure easy and medium problems first, skip traps.

2. Systematize the Syllabus

AMC problems follow a stable distribution of topics:

Topic	Share (AMC10/12)	Common Question Types
Number Theory	~25%	Remainders, divisibility, congruences, prime factorization, digit sums, modular arithmetic
Combinatorics & Probability	~25%	Permutations, combinations, inclusion-exclusion, recursion, probability models, basic generating functions
Geometry	~25%	Similarity/congruence, circles, area ratios, coordinate geometry
Algebra & Functions	~20%	Polynomials, function properties, inequalities, equation techniques
Miscellaneous & Creative	5–10%	Special constructions, induction, logic puzzles

Tip: Use Excel or Notion to build a “topic checklist” and classify every mistake into its exact topic — great for spotting gaps.

3. Rapidly Consolidate the Basics (2–3 weeks)

Choose a standard AMC prep series (e.g., Art of Problem Solving Prealgebra / Algebra / Geometry / Number Theory / Counting & Probability).

Do a few high-quality examples per topic daily (2–3 per category), focusing on:

Key formulas & theorems (e.g., modular formulas, special triangle ratios, binomial identities)
Standard methods (e.g., complement counting, symmetry, extreme principle)

Use a mistake log + spaced repetition flashcards (Anki) to lock in concepts.

4. Improve Problem-Solving Speed & Tricks (3–4 weeks)

AMC is timed — speed matters:

Timed drills: Do 5 problems in 10–12 minutes to force fast thinking.

Estimation & elimination: If you can solve mentally or estimate, don’t fully algebra-ize; use elimination for multiple choice.

Skipping strategy:

First clear the first 15 problems (usually easier).
Then solve topics you’re strongest in (e.g., geometry), save the hardest for last.

Common tricks: Coordinate bash for geometry, variable substitution, symmetry construction, proof by contradiction, transforming number theory into equations.

5. Past Papers & Deep Review (4–6 weeks)

Start with the past 10 years of AMC papers (AoPS or official MAA archive).

When simulating:

Strict timing (AMC8: 40 mins; AMC10/12: 75 mins)

Real exam setting (no outside references)

Classify errors immediately — knowledge gap? wrong method? careless? time issue?

Error recycling:

Redo similar problems the same day.
Revisit in 1 week to ensure retention.

6. Mindset & Test-Day Strategy

AMC scoring: Wrong answers lose points (from 1.5 → 0), so skip if truly unsure — don’t guess randomly.

In the last week, do 2–3 full simulations to lock in pacing.

In the contest:

First 5 min: scan for “fast kills”.
Never spend >4 minutes stuck — skip and return.
Last 5 min: check for silly mistakes.

Recommended Resources

Books:

AoPS Series (Counting & Probability, Number Theory, Geometry, Algebra)
USA Mathematical Talent Search Problems and Solutions

Problem banks:

AoPS online past problem database
MAA AMC official PDFs

Training platforms:

Art of Problem Solving (AoPS) Online
Brilliant.org (concepts + quizzes)
Math Kangaroo CN / other curated problem sets

If you want, I can also prepare a 12-week AMC crash plan with weekly topic focus + daily problem quotas + past-paper scheduling, so you can just follow it step-by-step for maximum efficiency.