📚 AMC 10 Core Concepts: A Comprehensive Review | AMC 10 核心考点精讲
The AMC 10 is a 25-question, 75-minute multiple-choice contest designed to challenge and engage students in grades 10 and below. It covers high school mathematics up to the 10th-grade level, including algebra, geometry, number theory, and combinatorics. Mastering the core topics is essential for a high score. This guide provides a concise yet comprehensive review of the most frequently tested concepts, with examples and key insights to boost your problem-solving toolkit.
AMC 10 是一场包含 25 道题的 75 分钟选择题竞赛，面向 10 年级及以下学生。考查内容涵盖代数、几何、数论和组合数学等高中阶段知识。掌握核心考点是取得高分的关键。本文精讲最常考的核心概念，辅以示例和关键思路，帮助考生增强解题能力。
1. Algebra and Equations | 代数与方程
Linear equations and systems: Solving equations such as ax + b = 0 or using substitution and elimination for two-variable systems is fundamental. Be comfortable rearranging terms and checking solutions.
一次方程与方程组：求解 ax + b = 0 形式的方程以及用代入法、消元法求解二元一次方程组是基础。要熟练掌握移项和验根。
Quadratic equations: For ax² + bx + c = 0 (a ≠ 0), the roots are given by the quadratic formula. The discriminant Δ = b² − 4ac determines the nature of the roots: two real roots if Δ > 0, one real root if Δ = 0, and no real roots if Δ < 0. Factoring and completing the square are also essential techniques.
二次方程：对于 ax² + bx + c = 0 (a ≠ 0)，求根公式为：
x = (−b ± √(b² − 4ac)) / (2a)
判别式 Δ = b² − 4ac 决定根的情况：Δ > 0 时有两个不等实根，Δ = 0 时有重根，Δ < 0 时无实根。因式分解和配方法也必不可少。
Inequalities: When solving linear or quadratic inequalities, remember to reverse the inequality sign when multiplying or dividing by a negative number. For absolute value inequalities like |x − a| < b, rewrite as −b < x − a < b. For |x − a| > b, split into x − a < −b or x − a > b.
不等式：解一次或二次不等式时，乘除负数要记得反转不等号。对于形如 |x − a| < b 的绝对值不等式，可化为 −b < x − a < b；对于 |x − a| > b，则拆分为 x − a < −b 或 x − a > b。
2. Number Theory Foundations | 数论基础
Prime numbers and divisibility: A prime has exactly two distinct positive divisors. The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be uniquely factored into primes. Checking divisibility rules (2, 3, 5, 7, 11) speeds up factorization.
素数与整除：素数恰有两个不同的正约数。算术基本定理表明任意大于 1 的整数可唯一分解为素数乘积。熟记 2, 3, 5, 7, 11 等的整除规则有助于快速分解。
GCD and LCM: Use the Euclidean algorithm to find the greatest common divisor: gcd(a, b) = gcd(b, a mod b). The least common multiple satisfies lcm(a, b) × gcd(a, b) = a × b.
最大公约数与最小公倍数：用欧几里得算法求 gcd：gcd(a, b) = gcd(b, a mod b)。lcm(a, b) × gcd(a, b) = a × b。
Modular arithmetic: a ≡ b (mod m) means m divides a − b. Useful for finding remainders and solving congruences. Key properties: if a ≡ b (mod m) and c ≡ d (mod m), then a + c ≡ b + d and ac ≡ bd (mod m).
同余运算：a ≡ b (mod m) 表示 m 整除 a − b。常用于求余数和解同余方程。基本性质：若 a ≡ b (mod m) 且 c ≡ d (mod m)，则 a + c ≡ b + d 且 ac ≡ bd (mod m)。
3. Geometry of Triangles | 三角形几何
Pythagorean theorem: In right triangle with legs a, b and hypotenuse c, a² + b² = c². Recognize Pythagorean triples like (3,4,5) and (5,12,13). Special right triangles: 45°−45°−90° (leg, leg, leg√2) and 30°−60°−90° (short leg, long leg = short leg√3, hypotenuse = 2 × short leg).
勾股定理：直角三角形两直角边 a, b，斜边 c，满足 a² + b² = c²。熟记勾股数如 (3,4,5) 和 (5,12,13)。特殊直角三角形：45°−45°−90°（腰，腰，腰√2）及 30°−60°−90°（短直角边，长直角边 = 短边√3，斜边 = 2 × 短边）。
Similarity and congruence: Triangles are similar if AA, SAS, or SSS similarity holds. Corresponding side ratios are equal. Angle bisector theorem, Ceva’s and Menelaus’ theorems often appear in advanced geometry problems.
相似与全等：满足 AA、SAS 或 SSS 相似条件的三角形相似，对应边成比例。在较难的几何题中可运用角平分线定理、塞瓦定理和梅涅劳斯定理。
Area formulas: Area = ½ × base × height, Heron’s formula: √[s(s−a)(s−b)(s−c)] with s semi-perimeter, and the formula using two sides and included angle: ½ ab sin C.
面积公式：面积 = ½ × 底 × 高；海伦公式：√[s(s−a)(s−b)(s−c)]，其中 s 为半周长；两边夹角公式：½ ab sin C。
4. Circles and Polygons | 圆与多边形
Circle basics: Central angle equals intercepted arc measure. Inscribed angle is half the measure of its intercepted arc. Inscribed angles subtending the same chord are equal. A tangent is perpendicular to the radius at the point of tangency.
圆的基础：圆心角等于所对弧的度数。圆周角等于所对弧度数的一半。同弧所对圆周角相等。切线垂直于过切点的半径。
Power of a point: For a point P outside a circle, PA × PB = PC × PD where PAB and PCD are secants. If tangent PT, then PT² = PA × PB.
圆幂定理：圆外一点 P，若 PAB 和 PCD 为割线，则 PA × PB = PC × PD。若 PT 为切线，则 PT² = PA × PB。
Polygons: Sum of interior angles of an n-gon is (n−2) × 180°. Each interior angle of a regular n-gon is (n−2) × 180° / n. Cyclic quadrilaterals have opposite angles summing to 180°.
多边形：n 边形内角和为 (n−2) × 180°。正 n 边形每个内角为 (n−2) × 180° / n。圆内接四边形对角互补（和为 180°）。
5. Counting Principles | 计数原理
Fundamental counting principle: If event A can occur in m ways and event B in n ways, then the number of ways both can occur is m × n. Be careful with independent choices.
基本计数原理：若事件 A 有 m 种方式，事件 B 有 n 种方式，则两者同时发生有 m × n 种方式。注意选择的独立性。
Permutations: The number of ways to arrange n distinct objects in order is n! (n factorial). Arranging r out of n objects: P(n, r) = n! / (n−r)!.
排列：将 n 个不同对象全排列的方法数是 n!。从 n 个中选 r 个排列：P(n, r) = n! / (n−r)!。
Combinations: The number of ways to choose r objects from n without regard to order: C(n, r) = n! / (r!(n−r)!). Often written as “n choose r”. Use combinations for forming teams, selecting items, etc.
组合：从 n 个不同对象中选出 r 个不计顺序的方法数：C(n, r) = n! / (r!(n−r)!)，常记作 “n 选 r”。用于组队、选物品等场景。
Complementary counting: Sometimes it is easier to count the total number of possibilities and subtract the unwanted cases. For example, probability of at least one success = 1 − probability of all failures.
互补计数：有时先计算总情形数再减去不满足的情形更简便。例如，至少成功一次的概率 = 1 − 全部失败的概率。
6. Probability Concepts | 概率概念
Probability = (Number of favorable outcomes) / (Total number of outcomes), assuming equally likely outcomes. For independent events A and B, P(A and B) = P(A) × P(B). For mutually exclusive events, P(A or B) = P(A) + P(B).
概率 = 有利结果数 / 结果总数，基于等可能假设。独立事件 A 和 B：P(A 且 B) = P(A) × P(B)。互斥事件：P(A 或 B) = P(A) + P(B)。
Geometric probability: The probability is the ratio of areas (or lengths) on a geometric figure. For example, a point randomly chosen inside a circle of radius R lands within a concentric smaller circle of radius r with probability r² / R².
几何概型：概率等于几何图形中的面积（或长度）之比。例如，在半径为 R 的圆内随机取一点，落在同心半径为 r 的小圆内的概率为 r² / R²。
Expected value: If random variable X takes values x₁, x₂, …, xₖ with probabilities p₁, p₂, …, pₖ, then E[X] = Σ xᵢ pᵢ. Useful in games and decision problems.
期望值：若随机变量 X 取值 x₁, x₂, …, xₖ，对应概率 p₁, p₂, …, pₖ，则 E[X] = Σ xᵢ pᵢ。常用于博弈和决策问题。
7. Functions and Polynomials | 函数与多项式
Function notation and evaluation: f(x) = expression. Composition (f ∘ g)(x) = f(g(x)). Domain and range are often tested indirectly through problem context.
函数表示与求值：f(x) = 表达式。复合函数 (f ∘ g)(x) = f(g(x))。题目常间接考查定义域和值域。
Polynomial roots and Vieta’s formulas: For a quadratic ax² + bx + c = 0 with roots r₁, r₂: r₁ + r₂ = −b/a, r₁ r₂ = c/a. For a cubic ax³ + bx² + cx + d = 0 with roots r₁, r₂, r₃: r₁+r₂+r₃ = −b/a, r₁r₂ + r₂r₃ + r₃r₁ = c/a, r₁r₂r₃ = −d/a.
多项式根与韦达定理：二次方程 ax² + bx + c = 0 两根 r₁, r₂ 满足 r₁ + r₂ = −b/a，r₁ r₂ = c/a。三次方程 ax³ + bx² + cx + d = 0 三根 r₁, r₂, r₃ 满足 r₁+r₂+r₃ = −b/a，r₁r₂ + r₂r₃ + r₃r₁ = c/a，r₁r₂r₃ = −d/a。
Remainder theorem: When polynomial P(x) is divided by (x − a), the remainder is P(a). Factor theorem: (x − a) is a factor of P(x) iff P(a) = 0.
余式定理：多项式 P(x) 除以 (x − a) 的余式为 P(a)。因式定理：(x − a) 是 P(x) 的因式当且仅当 P(a) = 0。
8. Sequences and Series | 数列与级数
Arithmetic sequences: Each term after the first is obtained by adding a common difference d. The nth term: aₙ = a₁ + (n−1)d. Sum of first n terms: Sₙ = n/2 × (a₁ + aₙ) = n/2 × [2a₁ + (n−1)d].
等差数列：后一项等于前一项加公差 d。第 n 项：aₙ = a₁ + (n−1)d。前 n 项和：Sₙ = n/2 × (a₁ + aₙ) = n/2 × [2a₁ + (n−1)d]。
Geometric sequences: Each term after the first is obtained by multiplying by common ratio r. The nth term: aₙ = a₁ rⁿ⁻¹. Sum of first n terms (r ≠ 1): Sₙ = a₁(1 − rⁿ) / (1 − r). Infinite sum for |r| < 1: S∞ = a₁ / (1 − r).
等比数列：后一项等于前一项乘以公比 r。第 n 项：aₙ = a₁ rⁿ⁻¹。前 n 项和 (r ≠ 1)：Sₙ = a₁(1 − rⁿ) / (1 − r)。当 |r| < 1 时无穷和为 S∞ = a₁ / (1 − r)。
Telescoping sums: Recognize patterns where intermediate terms cancel, e.g., Σ (1/i − 1/(i+1)) = 1 − 1/(n+1). Useful in AMC 10.
裂项相消：识别中间项互相抵消的模式，如 Σ (1/i − 1/(i+1)) = 1 − 1/(n+1)。在 AMC 10 中很实用。
9. Coordinate Geometry | 坐标几何
Distance formula: Between (x₁, y₁) and (x₂, y₂), d = √[(x₂−x₁)² + (y₂−y₁)²]. Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2). Slope: m = (y₂−y₁)/(x₂−x₁).
距离公式：(x₁, y₁) 与 (x₂, y₂) 间距离 d = √[(x₂−x₁)² + (y₂−y₁)²]。中点坐标：( (x₁+x₂)/2, (y₁+y₂)/2 )。斜率：m = (y₂−y₁)/(x₂−x₁)。
Equations of lines: Slope-intercept form y = mx + b; point-slope form y − y₁ = m(x − x₁). Lines are parallel if slopes equal, perpendicular if product of slopes = −1.
直线方程：斜截式 y = mx + b；点斜式 y − y₁ = m(x − x₁)。两直线平行 ⇔ 斜率相等；垂直 ⇔ 斜率之积为 −1。
Circles: Equation of a circle with center (h, k) and radius r: (x − h)² + (y − k)² = r². Recognize completing the square to find center and radius from general equation.
圆方程：以 (h, k) 为圆心、r 为半径的圆：(x − h)² + (y − k)² = r²。遇到一般式时通过配方求圆心和半径。
10. Complex Numbers | 复数
Definition: i = √−1, i² = −1. A complex number is of the form a + bi with real a (real part) and b (imaginary part). Addition/subtraction: (a+bi) ± (c+di) = (a±c) + (b±d)i.
定义：i = √−1，i² = −1。复数形如 a + bi，其中 a 为实部，b 为虚部。加减法：(a+bi) ± (c+di) = (a±c) + (b±d)i。
Multiplication: (a+bi)(c+di) = ac + adi + bci + bdi² = (ac − bd) + (ad + bc)i. Division: multiply numerator and denominator by the conjugate c−di to rationalize.
乘法：(a+bi)(c+di) = ac + adi + bci + bdi² = (ac − bd) + (ad + bc)i。除法：分子分母同乘共轭复数 c−di，使分母实数化。
Conjugate and modulus: Conjugate of z = a+bi is z̅ = a−bi. Modulus |z| = √(a² + b²). Note that z × z̅ = |z|². Complex numbers on AMC 10 mainly involve algebraic manipulation and occasional basic geometry (Argand diagram).
共轭与模：z = a+bi 的共轭为 z̅ = a−bi。模 |z| = √(a² + b²)，且 z·z̅ = |z|²。AMC 10 中的复数题主要涉及代数运算和偶尔出现的复平面基本几何意义。
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