📚 AMC10 Math Competition: Core Topics & 6-Month Preparation Plan | AMC10数学竞赛：核心考点与6个月备考规划
The AMC10 is one of the most prestigious high school mathematics competitions in the world, organized by the Mathematical Association of America (MAA). It tests creativity, problem-solving speed, and deep understanding of pre-calculus topics. Whether you aim for AIME qualification or simply want to sharpen your mathematical thinking, a structured plan is essential. This guide breaks down the core topics and provides a detailed 6-month preparation roadmap.
AMC10 是全球最具声望的高中数学竞赛之一，由美国数学协会（MAA）主办。它考验学生的创造力、解题速度和对于前微积分内容的深刻理解。无论你的目标是晋级 AIME，还是仅仅想锤炼数学思维，一份结构化的备考计划都至关重要。本文将拆解核心考点，并提供一份详细的六个月备考路线图。
1. Introduction to AMC10 | AMC10 简介
The AMC 10 is a 25-question, 75-minute multiple-choice contest for students in grades 10 and below, covering mathematics up to the 10th-grade curriculum. It is the first step toward the American Invitational Mathematics Examination (AIME) and the USA Mathematical Olympiad (USAMO). Top scorers earn recognition and a chance to advance to higher-level competitions.
AMC10 是一项包含 25 道题的 75 分钟选择题竞赛，面向 10 年级及以下学生，内容不超过 10 年级数学课程。它是通往美国数学邀请赛（AIME）和美国数学奥林匹克（USAMO）的第一步。高分者能获得荣誉，并有机会晋级更高层级的赛事。
There are two versions, the AMC 10A and AMC 10B, offered in November. Students can take either or both, and the highest score counts toward AIME qualification. The difficulty has increased in recent years, making consistent practice crucial.
竞赛分为 AMC 10A 和 AMC 10B 两个版本，通常在 11 月举行。学生可以选择参加一场或两场，最高分计入 AIME 晋级资格。近年来难度有所上升，因此持续的训练显得尤为关键。
2. Exam Format and Scoring | 考试形式与评分
Each of the 25 problems is worth 6 points for a correct answer, 1.5 points for a blank answer, and 0 points for an incorrect one. This scoring rule discourages random guessing, but leaving an answer blank strategically can boost your score. The maximum score is 150, and the cutoff for AIME typically ranges from about 90 to 105, varying each year.
每道题答对得 6 分，不答得 1.5 分，答错得 0 分。这种评分规则不鼓励盲目猜测，但策略性地空题可以提升总分。满分 150 分，AIME 晋级分数线通常在 90 到 105 分之间，每年有所浮动。
Questions increase in difficulty from 1 to 25. The first 10 are relatively straightforward, the middle 10 require more insight, and the final 5 are very challenging. Time management is critical — many top scorers aim to secure the first 15 problems within 40 minutes to leave time for harder ones.
题目难度从第 1 题到第 25 题逐步加大。前 10 题较为直接，中间 10 题需要更多洞察力，最后 5 题则极具挑战性。时间管理至关重要——许多高分选手会在 40 分钟内确保前 15 题的正确，以留出时间攻克难题。
3. Core Topic 1: Algebra | 核心考点一：代数
Algebra dominates the AMC10, appearing in roughly 40% of problems. Key subtopics include linear and quadratic equations, systems of equations, sequences and series, exponents and logarithms, as well as algebraic manipulation and word problems. Functions, including composition and inverses, also appear regularly.
代数是 AMC10 的重头戏，约占 40% 的题目。重点子主题包括一次与二次方程、方程组、数列与级数、指数与对数，以及代数变形和应用题。函数的复合与反函数也是常客。
You must be fluent in factoring, completing the square, Vieta' = a₁ · rⁿ⁻¹ for geometric ones. Clever substitution often simplifies complex expressions.
你必须熟练因式分解、配方法、韦达定理和二项式定理。能够识别等差数列的通项 aₙ = a₁ + (n−1)d 和等比数列的通项 aₙ = a₁ · rⁿ⁻¹。巧妙的代换往往能化繁为简。
1/x = 3, find x² + 1/x². (Solution: Square both sides → x² + 2 + 1/x² = 9 → x² + 1/x² = 7)
示例：若 x + 1/x = 3，求 x² + 1/x²。（解答：两边平方 → x² + 2 + 1/x² = 9 → x² + 1/x² = 7）
4. Core Topic 2: Geometry | 核心考点二：几何
Geometry holds a significant share, often blended with algebra or coordinate geometry. Essential topics include triangles (similarity, congruence, Pythagorean theorem, area formulas), circles (arcs, chords, tangents, power of a point), and polygons. The AMC10 does not require formal proofs, but properties like the Angle Bisector Theorem and Stewart's Theorem are heavily tested.
几何也占有重要比重，且常与代数或坐标几何结合。基本内容涵盖三角形（相似、全等、勾股定理、面积公式）、圆（弧、弦、切线、圆幂定理）以及多边形。AMC10 不要求正式证明，但角平分线定理、斯图尔特定理等性质是高频考点。
Coordinate geometry and 3D geometry appear occasionally. In the coordinate plane, distance formula, slope, and equations of lines and circles are your tools. For 3D, visualize cubes, spheres, and pyramids; use Pythagoras in three dimensions.
坐标几何与立体几何偶有出现。在坐标平面中，距离公式、斜率和直线与圆的方程是你的工具。对于立体几何，想象立方体、球体和棱锥，运用三维勾股定理。
Be comfortable with transformations, symmetry, and area dissection. Drawing accurate diagrams is often half the battle. Remember common Pythagorean triples like (3,4,5), (5,12,13), (8,15,17).
要熟悉变换、对称和面积分割。画出准确的示意图往往是成功的一半。记住常用的勾股数，如 (3,4,5)、(5,12,13)、(8,15,17)。
5. Core Topic 3: Number Theory | 核心考点三：数论
Number theory questions test your understanding of integers, divisibility, primes, modular arithmetic, and digit problems. They often look deceptively simple but require elegant insight. Key concepts include greatest common divisor (gcd), least common multiple (lcm), Euclidean algorithm, and the fundamental theorem of arithmetic.
数论题考察学生对整数、整除性、质数、同余算术和数字问题的理解。这类题往往看似简单，实则需精妙思维。核心概念包括最大公约数 (gcd)、最小公倍数 (lcm)、欧几里得算法和算术基本定理。
Modular arithmetic is particularly powerful. For example, 'Find the remainder when 7¹⁰⁰ is divided by 5' is quickly solved using patterns in mod 5. Be familiar with divisibility rules for 2, 3, 4, 5, 8, 9, 11. Euler's totient theorem and Fermat's little theorem sometimes appear in tougher problems.
同余算术尤为强大。例如，“求 7¹⁰⁰ 除以 5 的余数”可通过模 5 的循环规律迅速求解。要熟悉 2、3、4、5、8、9、11 的整除规则。欧拉定理和费马小定理偶尔会在难题中出现。
Diophantine equations (linear and simple quadratic) and base representation also show up. Learn to express numbers in base 2 or base 10 and convert between bases.
不定方程（一次和简单二次）以及进制表示也时有出现。学会用二进制或十进制表示数字，并进行进制转换。
6. Core Topic 4: Combinatorics & Probability | 核心考点四：组合与概率
Combinatorics and probability together form about 15–20% of the AMC10. You will need to count the number of ways to arrange, select, or distribute objects under given conditions. Fundamental counting principles — permutations, combinations, and the 'rule of product' — are the backbone.
组合与概率合起来约占 AMC10 的 15%–20%。你需要计算在给定条件下排列、选取或分配物体的方法数。基本的计数原理——排列、组合与“乘积法则”——是核心支柱。
Binomial coefficients (n choose k) and Pascal's triangle are essential tools. Overcounting and complementary counting are common problem-solving strategies. For instance, it is often easier to count the complement of a set and subtract from the total.
二项式系数（n 选 k）和杨辉三角是必备工具。重复计数与补集计数是常用的解题策略。例如，数一个集合的补集然后从总数中减去往往更容易。
Probability questions may involve equally likely outcomes, geometric probability, or conditional probability. Learn to list sample spaces systematically for small cases. Expected value, though less frequent, can appear in advanced problems.
概率题可能涉及等可能结果、几何概型或条件概率。对于小规模情况，学会系统地列出样本空间。期望值虽然频率较低，但也会出现在进阶题目中。
7. The 6-Month Study Plan Overview | 六个月备考规划总览
A structured 6-month plan gives you enough time to cover every topic, build problem-solving stamina, and take multiple mock exams. We divide the timeline into three phases: Foundation Building (Months 1–2), Intensive Practice (Months 3–4), and Final Sprint with Mock Exams (Months 5–6). Adjust the pace based on your initial diagnostic test.
一份结构化的六个月计划让你有充足的时间覆盖每个知识点、增强解题耐力，并进行多次模拟考试。我们将时间线分为三个阶段：基础夯实（第 1–2 个月）、强化训练（第 3–4 个月）以及冲刺与模考（第 5–6 个月）。你可以根据初始诊断测试调整进度。
Consistency is more important than intensity. Aim for at least 5–6 hours per week, divided into daily sessions. Mix topic study with mixed-problem sets to avoid forgetting earlier material. Keep an error log to track mistakes and patterns.
持之以恒比强度更重要。目标为每周至少 5–6 小时，分摊到每天进行。将专题学习与混合题集相结合，避免遗忘先前内容。建立错题本追踪错误和规律。
8. Phase 1: Foundation Building (Months 1–2) | 第一阶段：夯实基础（第 1–2 个月）
Start with a diagnostic AMC10 test (from a recent year) to identify weaknesses. Dedicate the first month to reviewing core algebra and geometry because they form the bulk of the exam. Use a structured resource like the AoPS Introduction series or the AMC 10 Preparation book by MAA.
先做一套近年的 AMC10 诊断测试，找出薄弱环节。第一个月集中复习核心代数和几何，因为它们占据考卷的大头。使用系统性教材，如 AoPS 入门系列或 MAA 出版的 AMC 10 备考书籍。
Break each week into mini-units: Week 1 – linear and quadratic equations; Week 2 – polynomials and Vieta; Week 3 – triangle geometry; Week 4 – circles and coordinate geometry. Solve at least 15–20 targeted problems per topic. Focus on accuracy, not speed.
将每周拆分为小单元：第一周——一次与二次方程；第二周——多项式与韦达定理；第三周——三角形几何；第四周——圆与坐标几何。每个专题至少做 15–20 道针对练习。此时重点是准确率，而非速度。
In month 2, add number theory and basic combinatorics. Learn modular arithmetic from scratch if needed. Reinforce algebra and geometry with mixed review sets every weekend. By the end of Phase 1, you should confidently solve problems 1–15 on a typical AMC10 paper.
第二个月加入数论和基础组合。如果需要，从头学习同余算术。每周末通过混合复习题集巩固代数和几何。到第一阶段结束时，你应该能自信地解答典型 AMC10 试卷的第 1–15 题。
9. Phase 2: Intensive Practice (Months 3–4) | 第二阶段：强化训练（第 3–4 个月）
Now shift to medium and challenging problems (roughly AMC10 #16–22). Work through past AMC10 papers from 2010–2020 systematically. Do two complete problem sets per week under timed conditions (75 minutes) to build speed.
现在转向中等和挑战题（大致 AMC10 第 16–22 题）。系统性地刷 2010–2020 年的 AMC10 真题。每周在计时条件下完成两整套题目（75 分钟），以提升速度。
Deepen your understanding of combinatorics (permutations with repetition, stars-and-bars, inclusion-exclusion) and advanced number theory (Chinese Remainder Theorem, Euler's φ function). Learn common algebraic tricks like Simon's Favorite Factoring Trick and telescoping series.
深化组合（有重复的排列、星星与柱子法、容斥原理）和进阶数论（中国剩余定理、欧拉 φ 函数）的理解。学习常见的代数技巧，如西蒙分解技巧和裂项相消法。
After each timed practice, spend at least the same amount of time analyzing your mistakes. Categorize errors: conceptual gap, calculation mistake, or time pressure. Review the relevant concept immediately. Maintain your error log diligently.
每次计时练习后，至少花同样多的时间分析错题。将错误归类：概念漏洞、计算失误，还是时间压力。立即回顾相关知识点。认真记录错题本。
10. Phase 3: Final Sprint and Mock Exams (Months 5–6) | 第三阶段：冲刺与模考（第 5–6 个月）
In month 5, take one full-length AMC10 mock exam every week under realistic conditions. Use the official MAA AMC10 papers from 2020–2024. Simulate the 8:00 AM test start time and take only a 10-minute break between halves, even though the actual test is continuous. This builds mental endurance.
第五个月，每周在逼真条件下做一次完整的 AMC10 模拟考。使用 2020–2024 年 MAA 官方 AMC10 试卷。模拟早上 8 点的考试开始时间，虽然正考是连续作答，但可以在中途给自己 10 分钟休息，以锻炼心理耐力。
Refine your guessing strategy. Since a blank gives 1.5 points, only guess if you can eliminate at least two answer choices. Practice 'bail-out' discipline — if you spend more than 3 minutes on a problem without progress, mark it and move on.
完善猜测策略。既然空题得 1.5 分，只在能够排除至少两个选项时才猜。练习“及时止损”的纪律——如果某题超过 3 分钟仍无进展，标记并继续前进。
In the final two weeks, review your error log and re-solve the hardest problems you encountered. Reduce the volume of new problems; instead, focus on strengthening weak spots. Memorize essential formulas: Vieta's, Law of Sines/Cosines, area of a triangle (1/2 ab sin C), angle bisector theorem, and key number theory results.
最后两周，回顾错题本并重做你遇到过的最难题目。减少新题的数量，转而强化薄弱点。熟记必备公式：韦达定理、正弦定理/余弦定理、三角形面积 (½ ab sin C)、角平分线定理以及关键数论结论。
11. Recommended Resources | 推荐资源
High-quality resources make a huge difference. The Art of Problem Solving (AoPS) books, particularly the 'Introduction to Algebra,' 'Introduction to Geometry,' and 'Introduction to Number Theory/Combinatorics,' are the gold standard. Their online community and Alcumus adaptive system offer endless free practice.
优质的资源能带来巨大帮助。《The Art of Problem Solving》系列，尤其是《Introduction to Algebra》《Introduction to Geometry》和《Introduction to Number Theory/Combinatorics》，是公认的黄金标准。其在线社区和 Alcumus 自适应系统提供海量免费练习。
For official problems, the MAA's AMC 10 Contest Problem Books (available on the MAA website) compile past years with solutions. The 'AMC 10/12 Preparation' YouTube channels, such as those by Richard Rusczyk and Professor Loh, provide free video solutions.
对于官方题目，MAA 网站上提供的 AMC 10 竞赛试题集汇编了历年真题及解答。Richard Rusczyk 和罗教授等 YouTube 频道的'AMC 10/12 Preparation'系列提供免费视频解析。
Supplement with competition problem databases like The Official AMC 10/12 Archive and the AoPS Wiki. Flashcards for quick formulas and a 'problem-of-the-day' routine keep concepts fresh.
辅以竞赛题目数据库，如官方 AMC 10/12 档案和 AoPS 维基。制作公式闪卡，并保持“每日一题”的习惯，让概念常记常新。
12. Tips for Test Day and Beyond | 考试当天及后续建议
Get a full night's sleep before the exam. Eat a protein-rich breakfast. Arrive early with sharpened pencils, an eraser, and a fully approved calculator (check MAA guidelines). Read the first few problems carefully — easy points are often lost due to misreading.
考前好好睡一觉。吃一顿富含蛋白质的早餐。提前到场，带好削好的铅笔、橡皮和符合 MAA 规定的计算器。仔细阅读前几道题——简单题的分数常常因为误读而丢失。
During the test, fill in answers neatly on the bubble sheet as you go, rather than waiting until the end. If time runs short, focus on confirming your earlier answers rather than tackling a new hard problem. Remember, a blank answer gives 1.5 points, so do not guess wildly.
考试过程中，边做边在答题卡上清晰涂写，不要等到最后。如果时间不够，专注于确认前面的答案，而非攻克一道全新的难题。记住，空题得 1.5 分，千万不要胡乱猜测。
After the AMC10, if you qualify for AIME, celebrate — but then start preparing for a much longer (3-hour) and harder 15-question exam. If you do not qualify, treat the experience as a stepping stone and aim higher next year. The mathematical maturity you build will benefit all your future studies.
AMC10 之后，如果你晋级 AIME，庆祝一下——但随即开始准备一项时长更长（3 小时）、难度更高的 15 题考试。如果未能晋级，将这次经历视为垫脚石，来年再战。你在此过程中建立的数学素养将惠及未来的所有学业。
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