AMC 10/12 American Mathematics Competitions: Key Topics and Test-taking Strategies | AMC 10/12 美国数学竞赛：核心考点与应试技巧

📚 AMC 10/12 American Mathematics Competitions: Key Topics and Test-taking Strategies | AMC 10/12 美国数学竞赛：核心考点与应试技巧

The AMC 10/12 is the first step in the prestigious American Mathematics Competitions series, opening doors to AIME and USAMO. This guide covers the essential topics, question types, and proven strategies to maximize your score.

AMC 10/12 是美国数学竞赛系列的第一关，为晋级 AIME 和美国数学奥林匹克打开大门。本指南涵盖核心考点、题型分布，以及经过验证的应试策略，助你冲刺高分。

1. Exam Structure and Scoring | 考试结构与评分规则

Both AMC 10 and AMC 12 are 75-minute, 25-question multiple-choice tests. Each correct answer awards 6 points, each blank 1.5 points, and wrong answers receive 0. This scoring system rewards carefulness and strategic omissions.

AMC 10 和 AMC 12 均为 75 分钟、25 道选择题的考试。答对得 6 分，不答得 1.5 分，答错得 0 分。这鼓励谨慎作答并有策略地留空。

AMC 10 is open to students in grade 10 and below, covering algebra, geometry, number theory, and combinatorics up to a tenth-grade level. AMC 12 is open to students in grade 12 and below, adding trigonometry, complex numbers, logarithms, and advanced precalculus topics.

AMC 10 面向十年级及以下学生，涵盖十年级水平的代数、几何、数论和组合。AMC 12 面向十二年级及以下，额外增加三角、复数、对数和高级预微积分内容。

The difficulty generally increases from question 1 to 25. The first 10 questions are relatively easy, the middle 10 require deeper insight, and the last 5 are very challenging.

难度大致从第 1 题到第 25 题递增。前 10 题相对简单，中间 10 题需要更深入思考，最后 5 题极具挑战性。

2. Algebra: Equations, Functions, and Polynomials | 代数：方程、函数与多项式

Quadratic equations appear frequently. For ax² + bx + c = 0, the solutions are given by the quadratic formula:

二次方程频繁出现，对于 ax² + bx + c = 0，求根公式为：

x = (-b ± √(b² – 4ac)) / (2a)

x = (-b ± √(b² – 4ac)) / (2a)

Vieta’s formulas link the roots and coefficients: sum of roots = -b/a, product = c/a for a quadratic. For higher-degree polynomials, the sum of roots taken one at a time, two at a time, etc., follows sign-alternating patterns.

韦达定理联系根与系数：二次方程根之和 = -b/a，根之积 = c/a。对于更高次多项式，根的一次和、两两积之和等遵循交错符号规律。

Functions involve domain, range, composition, and inverses. Be ready to solve f(g(x)) = something, or find bounds for x given inequalities with absolute values and radicals.

函数涉及定义域、值域、复合和反函数。准备好求解 f(g(x)) 表达式，或处理包含绝对值和根式的不等式求 x 范围。

Polynomial identities like difference of squares and sum/difference of cubes are vital. Recognize a³ – b³ = (a – b)(a² + ab + b²).

多项式恒等式如平方差、立方和／差至关重要。熟记 a³ – b³ = (a – b)(a² + ab + b²)。

3. Geometry: Triangles, Circles, and Coordinates | 几何：三角形、圆与坐标

Similar triangles, the Pythagorean theorem, and properties of right triangles (30°-60°-90° and 45°-45°-90°) are tested heavily. Know that in a circle, the inscribed angle is half the central angle subtending the same arc.

相似三角形、勾股定理以及特殊直角三角形的性质（30°-60°-90° 和 45°-45°-90°）是高频考点。牢记在同圆中，圆周角等于所夹弧对应圆心角的一半。

Area formulas include Heron’s formula: for a triangle with sides a, b, c and semiperimeter s = (a+b+c)/2, area = √(s(s-a)(s-b)(s-c)). Also, area = ½ ab sin C.

面积公式包括海伦公式：三角形三边长 a, b, c，半周长 s = (a+b+c)/2，则面积 = √(s(s-a)(s-b)(s-c))。同时也常用面积 = ½ ab sin C。

Power of a point: If two chords intersect at P inside a circle, PA·PB = PC·PD. For tangents, tangent² = external segment × whole secant.

圆幂定理：若两弦在圆内交于点 P，则 PA·PB = PC·PD。对于切线，切线长的平方等于圆外线段与该割线全长之积。

Coordinate geometry: distance formula, midpoint formula, slope, equations of lines and circles. Many geometry problems can be solved by placing the figure on a coordinate plane.

坐标几何：距离公式、中点公式、斜率、直线与圆的方程。许多几何题可通过将图形置于坐标系中简化。

4. Number Theory Foundations | 数论基础

Divisibility rules for 2,3,4,5,6,8,9,11 are useful. Prime factorization helps find the number of divisors: if N = ∏ pᵢᵉᵢ, then the number of positive divisors is ∏(eᵢ+1).

2,3,4,5,6,8,9,11 的整除规则非常实用。质因数分解可求出正因数个数：若 N = ∏ pᵢᵉᵢ，则正因数个数为 ∏(eᵢ+1)。

Modular arithmetic simplifies remainders. For example, to find the last digit of 7²⁰²⁵, compute 7¹ mod 10 = 7, 7²

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