📚 What Do AMC 10 and AMC 12 Test? Topics, Format and Difficulty Explained | AMC 10/12 考什么：考点、形式与难度全解析
The AMC 10 and AMC 12 are the two entry-level contests in the American Mathematics Competitions series organised by the Mathematical Association of America (MAA). "What exactly do they test?" is the question students and parents ask most often, and the honest answer is this: they test high-school mathematics up to roughly precalculus level, but they test it in a way that school exams almost never do. This guide breaks the syllabus down topic by topic, explains how the two papers differ, and shows how the 25 questions are actually distributed in difficulty.
AMC 10 与 AMC 12 是美国数学协会（MAA）主办的美国数学竞赛系列中的两项入门赛事。"AMC 10/12 考什么"是学生和家长问得最多的一个问题，而诚实的回答是：它们考的是大约到预备微积分水平为止的高中数学，但考查方式与学校考试几乎完全不同。本文会逐板块拆解考纲，说明 AMC 10 与 AMC 12 的区别，并分析 25 道题在难度上究竟是如何分布的。
1. What Are the AMC 10 and AMC 12? | AMC 10 与 AMC 12 是什么
The AMC 10 and AMC 12 are two separate 25-question multiple-choice papers that share the same date, the same rulebook and much of the same early content. They exist to identify students with strong mathematical reasoning and to act as the first filter on the road to the AIME, the USAMO and ultimately the International Mathematical Olympiad training programme.
AMC 10 与 AMC 12 是两套独立的 25 题选择题试卷，它们同日举行、规则相同，前段题目也有大量重合。它们的作用是筛选出数学推理能力突出的学生，并作为通往 AIME、USAMO 乃至国际数学奥林匹克集训队的第一道关卡。
Eligibility is strict and worth checking before you register. The AMC 10 is open to students in grade 10 or below who are under 17.5 years old on the contest date. The AMC 12 is open to students in grade 12 or below who are under 19.5 years old on the contest date. A strong grade 9 student may therefore sit either paper, while a grade 11 student may only sit the AMC 12.
参赛资格有明确限制，报名前务必确认。AMC 10 面向 10 年级及以下、且在比赛当天未满 17.5 岁的学生；AMC 12 面向 12 年级及以下、且在比赛当天未满 19.5 岁的学生。因此，一名实力强劲的 9 年级学生两套卷都可参加，而 11 年级学生只能考 AMC 12。
Both contests are offered in an A version and a B version, sat on different dates; students may take both and use the higher score.
两项赛事都设有 A 卷和 B 卷，考试日期不同；学生可以两场都参加，取较高的一次成绩。
Calculators of any kind are not permitted. All arithmetic must be done by hand.
任何类型的计算器都不允许使用，所有运算必须手算完成。
Answers are recorded on a machine-readable answer sheet; there is no partial credit for working.
答案填涂在机读答题卡上，过程不给予过程分。
2. Exam Format, Timing and Scoring | 考试形式、时长与计分
Both papers have exactly the same structure: 25 multiple-choice questions, five options per question (A to E), and 75 minutes on the clock. That works out at an average of three minutes per question, but the real time budget is far more uneven than that average suggests, because the last five questions can easily swallow ten minutes each.
两套试卷结构完全一致：25 道选择题，每题 5 个选项（A 至 E），考试时间 75 分钟。平均下来每题约 3 分钟，但真实的时间分配远比这个平均值不均——最后 5 道题每题花掉 10 分钟都很常见。
The scoring system is the detail most newcomers miss. A correct answer earns 6 points, a blank answer earns 1.5 points, and a wrong answer earns 0. The maximum score is therefore 150. Because a blank is worth more than a wrong answer, the paper rewards strategic guessing rather than blind filling: you should always eliminate options before you commit.
计分规则是初学者最容易忽略的细节。答对得 6 分，留空得 1.5 分，答错得 0 分，满分因此是 150 分。由于留空的收益高于答错，这份试卷奖励的是"有策略地猜"而非"盲目填满"：下笔之前一定要先排除选项。
Total score = 6 × (correct) + 1.5 × (blank) + 0 × (wrong), maximum 150
| Item | 项目 | AMC 10 | AMC 10 | AMC 12 | AMC 12 |
| Questions | 题量 | 25 | 25 |
| Time | 时长 | 75 min | 75 min |
| Score per correct | 每题分值 | 6 | 6 |
| Score per blank | 留空分值 | 1.5 | 1.5 |
| Maximum | 满分 | 150 | 150 |
| Eligibility | 资格 | Grade 10 or below, under 17.5 | 10 年级及以下，未满 17.5 岁 | Grade 12 or below, under 19.5 | 12 年级及以下，未满 19.5 岁 |
3. The Five Content Strands | 五大知识板块
Every AMC 10/12 question falls into one of five broad strands. The proportions below are approximate, since the exact mix shifts slightly from year to year, but they are stable enough to guide your revision priorities.
每一道 AMC 10/12 题目都可以归入以下五大板块之一。下面的比例是近似值，具体分布每年会略有浮动，但整体足够稳定，可以用来安排复习优先级。
| Strand | 板块 | Typical share | 大致占比 | Core tools | 核心工具 |
| Algebra & Arithmetic | 代数与算术 | 30-35% | Equations, sequences, functions, ratios | 方程、数列、函数、比例 |
| Geometry | 几何 | 25-30% | Triangles, circles, area, coordinates | 三角形、圆、面积、坐标 |
| Number Theory | 数论 | 15-20% | Divisibility, primes, modular arithmetic | 整除、质数、同余 |
| Combinatorics & Probability | 组合与概率 | 15-20% | Counting, cases, expected value | 计数、分类、期望 |
| Word Problems & Logic | 应用题与逻辑 | overlaps the above | 与上述交叉 | Rate, work, mixture, puzzle setup | 速率、工程、混合、逻辑建模 |
Notice what is not on that list: calculus, matrices, formal proofs, and anything requiring a graphing calculator. The AMC 10/12 is deliberately a contest of clever elementary mathematics, not of advanced machinery.
注意这份清单里没有出现的东西：微积分、矩阵、形式化证明，以及任何需要图形计算器的内容。AMC 10/12 刻意定位为"初等数学的巧思竞赛"，而不是高等工具的比拼。
4. AMC 10 vs AMC 12: What Is Added and What Is Excluded | AMC 10 与 AMC 12 的差异：删减与增补
The two papers overlap heavily, and in a typical year a substantial block of the early questions is common to both. The difference lies in the upper end of the syllabus.
两套试卷重合度很高，通常前段有相当一部分题目是共用的。真正的差别体现在考纲的上限上。
Excluded from AMC 10: trigonometry (sine, cosine, tangent, law of sines and cosines), logarithms and exponents beyond simple integer powers, complex numbers, and advanced polynomial theory.
AMC 10 不考：三角函数（正弦、余弦、正切、正弦定理与余弦定理）、对数以及简单整数幂之外的指数运算、复数、以及较深的多项式理论。
Added in AMC 12: trigonometry including identities and the law of cosines, logarithmic and exponential functions, complex numbers in both rectangular and polar form, De Moivre's theorem, conic sections, and more demanding polynomial and functional equations.
AMC 12 增考：三角函数及恒等式、余弦定理、对数与指数函数、复数的代数形式与极坐标形式、棣莫弗定理、圆锥曲线，以及更难的方程与函数方程问题。
Shared by both: algebra, geometry, number theory, combinatorics and probability, all at contest level.
两者共考：代数、几何、数论、组合与概率，难度均按竞赛标准设置。
A practical consequence: a student who is comfortable with trigonometry should generally sit the AMC 12, because the AIME qualification percentage is more generous there (top 5% versus top 2.5%), even though the raw cutoff score is usually lower.
一个实用的推论是：已经熟练掌握三角函数的学生通常更适合考 AMC 12，因为它的 AIME 晋级比例更宽松（前 5% 对比前 2.5%），尽管其绝对分数线往往更低。
5. Algebra: The Backbone of the Paper | 代数：试卷的主干
Algebra is the single largest strand and the one that quietly appears inside questions that look like geometry or counting problems. Expect linear and quadratic equations, systems of equations, inequalities, absolute values, sequences and series, functions and their transformations, and heavy use of factoring and algebraic manipulation.
代数是占比最大的板块，也经常"潜伏"在看似几何或计数的题目里。需要掌握一次与二次方程、方程组、不等式、绝对值、数列与级数、函数及其图像变换，以及熟练的因式分解与代数变形。
Three algebraic tools appear so often that they are worth memorising cold.
有三个代数工具出现频率极高，值得背到滚瓜烂熟。
x = (−b ± √b² − 4ac) / (2a)
Sₙ = n(a₁ + aₙ) / 2 and Sₙ = a₁(1 − rⁿ) / (1 − r)
1 + 2 + 3 + … + n = n(n + 1) / 2
Vieta' −b/a and the product is c/a. Many AMC questions that appear to require solving an equation are solved in one line by going straight to the sum or product.
韦达定理同样重要，往往是处理"根的对称式"问题最快的通道。对于二次方程 ax² + bx + c = 0，两根之和为 −b/a，两根之积为 c/a。很多看似需要解方程的 AMC 题目，直接用和与积一行就能解决。
Sequences: recognise arithmetic versus geometric immediately, and watch for telescoping sums.
数列：先迅速判断是等差还是等比，并留意可以"裂项相消"的求和。
Functions: composition, inverse, and the difference between f(x + 1) and f(x) + 1.
函数：复合、反函数，以及 f(x + 1) 与 f(x) + 1 的区别。
Word problems: convert to equations before computing anything.
应用题：先列出方程，再动手计算。
6. Geometry: Triangles, Circles and Coordinates | 几何：三角形、圆与坐标
Geometry is the second-largest strand and the one where a single missing theorem can cost you an entire question. The recurring cast includes similar and congruent triangles, the Pythagorean theorem, special right triangles, angle chasing in polygons, circle theorems (inscribed angles, tangents, chords, power of a point), areas of composite figures, and coordinate geometry with distance and midpoint formulas.
几何是占比第二大的板块，也是"少记一个定理就整题报废"的板块。高频内容包括相似与全等三角形、勾股定理、特殊直角三角形、多边形角度推导、圆的定理（圆周角、切线、弦、圆幂定理）、组合图形面积，以及坐标几何中的距离与中点公式。
d = √((x₂ − x₁)² + (y₂ − y₁)²) A = ½ × b × h A = πr²
A = √(s(s − a)(s − b)(s − c)), where s = (a + b + c) / 2
formula A = ½ab sin C, and the law of cosines c² = a² + b² − 2ab cos C. These convert many "hard geometry" problems into straightforward algebra.
AMC 12 还要在几何中加入三角：sin²θ + cos²θ = 1、面积公式 A = ½ab sin C，以及余弦定理 c² = a² + b² − 2ab cos C。它们能把很多"难题"直接转化为代数运算。
Memorise the Pythagorean triples 3-4-5, 5-12-13, 8-15-17 and 7-24-25; they appear disguised inside larger figures.
牢记勾股数 3-4-5、5-12-13、8-15-17、7-24-25，它们经常伪装在更大的图形中。
When a figure is described in words, sketch it immediately and mark every given length and angle.
当图形只用文字描述时，立刻画出草图，并标注所有已知的边长与角度。
Coordinate geometry is often faster than pure synthetic geometry on the AMC.
在 AMC 中，坐标法往往比纯综合几何法更快。
7. Number Theory: The Quiet Difficulty | 数论：低调的难点
Number theory is the strand most students have never studied formally, and it is where the AMC diverges most sharply from the school syllabus. The good news is that the required toolkit is small and can be learned in a few weeks.
数论是大多数学生从未系统学过的板块，也是 AMC 与学校课程差距最大的地方。好消息是所需工具并不多，几周就能补上。
Divisibility rules for 2, 3, 4, 5, 8, 9, 11, and the meaning of a remainder.
2、3、4、5、8、9、11 的整除规律，以及余数的含义。
Prime factorisation, the number of divisors, and the sum of divisors from the factorisation.
质因数分解、约数个数，以及由分解式求约数之和。
reatest common divisor and least common multiple, linked by gcd(a, b) × lcm(a, b) = a × b.
最大公约数与最小公倍数，二者满足 gcd(a, b)
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