📚 IGCSE Cambridge Additional Mathematics: International Competition Preparation Guide | IGCSE 剑桥进阶数学：国际竞赛备战攻略
This guide connects the Cambridge IGCSE Additional Mathematics (0606) syllabus with the problem-solving skills needed for international mathematics competitions such as the UKMT Intermediate and Senior Challenges, AMC 10/12 and regional olympiad qualifiers.
本攻略将 Cambridge IGCSE 进阶数学（0606）大纲与国际数学竞赛所需的解题能力结合，适用于 UKMT 中级与高级挑战赛、AMC 10/12 及地区奥林匹克预选赛等。
1. Core Syllabus and Competition Overlap | 核心大纲与竞赛考点重合
The Additional Mathematics syllabus includes functions, quadratic theory, equations and inequalities, indices and surds, polynomials, simultaneous equations, logarithmic and exponential functions, straight-line graphs, circular measure, trigonometry, permutations and combinations, binomial expansion, vectors, differentiation and integration.
进阶数学大纲涵盖函数、二次函数、方程与不等式、指数与根式、多项式、联立方程、对数与指数函数、直线图像、弧度制、三角学、排列组合、二项式展开、向量、微分与积分。
Competitions rarely test new topics but require you to apply these core ideas more flexibly, often with a non-calculator arithmetic twist.
竞赛很少考查全新主题，而是要求你更灵活地运用这些核心知识，并且常常带有心算或非计算器的算术巧思。
2. Building Algebraic Fluency | 打造代数熟练度
Speed and accuracy in expanding, factorising, completing the square and using the discriminant are essential because competition questions often hide a simple quadratic inside a more complex expression.
展开、因式分解、配方法以及判别式的运用速度和准确性至关重要，因为竞赛题常把简单二次式藏在更复杂的表达式中。
den quadratic: let y = x², then y² − 5y + 4 = 0 gives y = 1 or y = 4, so x = ±1 or x = ±2.
例如 x⁴ − 5x² + 4 = 0 就是一个隐藏二次方程：令 y = x²，则 y² − 5y + 4 = 0，解得 y = 1 或 y = 4，因此 x = ±1 或 x = ±2。
Practice simplifying surds such as √12 = 2√3 and rationalising denominators, because these manipulations often appear in exact-value competition answers.
练习化简根式，如 √12 = 2√3，以及分母有理化，因为竞赛答案常要求精确值。
3. Functions and Graph Transformations | 函数与图像变换
You must be able to find composite and inverse functions, and state domain and range restrictions. In competitions, domain errors are a common trap.
你必须能求复合函数与反函数，并写出定义域和值域限制。竞赛中定义域错误是常见陷阱。
= 2x + 1 and g(x) = x², the composite fg(x) = f(g(x)) = 2x² + 1, while gf(x) = (2x + 1)².
设 f(x) = 2x + 1，g(x) = x²，则复合函数 fg(x) = f(g(x)) = 2x² + 1，而 gf(x) = (2x + 1)²。
+ a) (horizontal shift), y = af(x) (vertical stretch) and y = f(ax) (horizontal stretch).
图像变换遵循以下规律：y = f(x) + a（纵向平移）、y = f(x + a)（横向平移）、y = af(x)（纵向伸缩）和 y = f(ax)（横向伸缩）。
4. Equations, Inequalities and Modulus | 方程、不等式与绝对值
Solving inequalities involving quadratics requires a sign diagram or interval test. For x² − 4x + 3 > 0, the critical values are 1 and 3, giving x < 1 or x > 3.
解二次不等式需要画符号图或区间测试。对 x² − 4x + 3 > 0，临界值为 1 和 3，因此解为 x < 1 或 x > 3。
Modulus equations and inequalities should be converted into cases. For example, |x − 3| ≤ 5 becomes −5 ≤ x − 3 ≤ 5, hence −2 ≤ x ≤ 8.
绝对值方程和不等式应转换为分段情形。例如 |x − 3| ≤ 5 变成 −5 ≤ x − 3 ≤ 5，因此 −2 ≤ x ≤ 8。
Watch for excluded values and squared-equation extraneous roots, especially when both sides of an equation are squared.
注意排除不可取的值以及平方后产生的增根，尤其是方程两边同时平方时。
5. Trigonometry Beyond the Basics | 进阶三角学
hematics. Use arc length s = rθ and sector area A = ½r²θ whenever the angle is in radians.
弧度制是进阶数学的核心。只要角度以弧度为单位，就可用弧长公式 s = rθ 和扇形面积公式 A = ½r²θ。
Key identities include sin
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