Australian Mathematics Competition: Core Topics | 澳大利亚数学竞赛核心考点梳理

📚 Australian Mathematics Competition: Core Topics | 澳大利亚数学竞赛核心考点梳理

The Australian Mathematics Competition (AMC) is one of the most widely participated school mathematics competitions in Australia. It tests quick thinking, precise calculation, and problem-solving strategies across a range of core topics. This article provides a systematic review of the essential knowledge areas that frequently appear in the AMC.

澳大利亚数学竞赛（AMC）是澳大利亚参与人数最多的校内数学竞赛之一。它考查快速思考、精确计算以及跨核心知识的解题策略。本文将系统梳理AMC中经常出现的核心知识板块，帮助你更有针对性地备考。

1. Number Theory | 数论

Number theory is the study of integers and their relationships. In the AMC, early questions often test divisibility, factors, prime numbers, and remainders. A solid grasp of these ideas gives you a quick and confident start.

数论研究整数及其关系。在AMC中，早期题目经常考查整除性、因数、质数和余数。扎实掌握这些概念能让你快速且自信地进入答题状态。

A classic example is counting the divisors of a number. Since 72 = 2³ × 3², the number of positive divisors is (3 + 1)(2 + 1) = 12.

一个经典例子是计算一个数的约数个数。因为 72 = 2³ × 3²，所以正约数个数为 (3 + 1)(2 + 1) = 12。

72 = 2³ × 3², d(72) = (3 + 1)(2 + 1) = 12

You should also know common divisibility rules. For example, a number is divisible by 3 or 9 if the sum of its digits is divisible by 3 or 9. For remainder problems, modular arithmetic provides a compact notation.

你还应掌握常见的整除法则。例如，一个数能被3或9整除，当且仅当它的各位数字之和能被3或9整除。对于余数问题，模运算提供了一种简洁的表达方式。

2. Algebra | 代数

Algebra is a central theme in the AMC. You need to simplify expressions, expand and factorise polynomials, solve linear and quadratic equations, and handle simple inequalities.

代数是AMC的核心主题之一。你需要能够化简表达式、展开和分解多项式、求解线性与二次方程，并处理简单的不等式。

For example, solving an exponential equation often only requires rewriting with the same base. If 2ˣ⁺¹ = 8, then since 8 = 2³, we have x + 1 = 3, so x = 2.

例如，解指数方程通常只需将两边化为同底。若 2ˣ⁺¹ = 8，因为 8 = 2³，所以 x + 1 = 3，从而 x = 2。

2ˣ⁺¹ = 8 ⇒ x + 1 = 3 ⇒ x = 2

Remember that when multiplying or dividing an inequality by a negative number, the inequality symbol reverses. This is a common trap in AMC algebra questions.

请记住：当不等式的两边同时乘以或除以一个负数时，不等号方向必须反转。这是AMC代数题中常见的陷阱。

3. Geometry | 几何

Geometry questions in the AMC cover angles, polygons, triangles, circles, and similarity. A clear understanding of angle sums and the Pythagorean theorem is essential.

AMC中的几何题涉及角、多边形、三角形、圆形和相似。清晰理解角度和以及勾股定理至关重要。

For a polygon with n sides, the sum of interior angles is (n – 2) × 180°. In a regular pentagon, each interior angle is therefore (5 – 2) × 180° ÷ 5 = 108°.

对于 n 边形，内角和为 (n – 2) × 180°。因此，正五边形每个内角为 (5 – 2) × 180° ÷ 5 = 108°。

Interior angle sum = (n – 2) × 180°

For right triangles, the Pythagorean theorem states that a² + b² = c², where c is the hypotenuse. Also remember the triangle inequality: the sum of any two side lengths must be greater than the third.

对于直角三角形，勾股定理表明 a² + b² = c²，其中 c 是斜边。还要记住三角形三边不等式：任意两边长度之和必须大于第三边。

4. Measurement | 度量

Measurement questions connect geometry to real-world quantities. Expect problems involving area, perimeter, volume, surface area, and unit conversion.

度量问题将几何与现实数量联系起来。预计会出现涉及面积、周长、体积、表面积和单位换算的题目。

Common formulas include the area of a circle A = πr², its circumference C = 2πr, and the volume of a cylinder V = πr²h. Always check units before calculating.

常用公式包括圆的面积 A = πr²、周长 C = 2πr，以及圆柱的体积 V = πr²h。计算前务必检查单位是否统一。

A = πr², C = 2πr, V = πr²h

Unit conversion is also important. For example, 1 m² = 10,000 cm², since 1 m = 100 cm and the conversion is squared.

单位换算也很重要。例如，1 m² = 10,000 cm²，因为 1 m = 100 cm，而面积换算需要平方。

5. Combinatorics | 组合数学

Combinatorics is the mathematics of counting. AMC questions frequently use the multiplication principle, the addition principle, permutations, combinations, and the pigeonhole principle.

组合数学是研究计数的数学分支。AMC题目经常使用乘法原理、加法原理、排列、组合以及鸽巢原理。

For example, count the number of three-digit numbers with no repeated digits. The hundreds digit cannot be 0, so there are 9 choices; then 9 choices for the tens digit; then 8 for the units digit. Total = 9 × 9 × 8 = 648.

例如，计算没有重复数字的三位数个数。百位不能为0，因此有9种选择；十位有9种选择；个位有8种选择。总数为 9 × 9 × 8 = 648。

9 × 9 × 8 = 648

The combination formula is C(n, r) = n! / [r!(n – r)!]. It counts selections where order does not matter. If order matters, use permutations of n objects taken r at a time: n! / (n – r)!.

组合公式为 C(n, r) = n! / [r!(n – r)!]，用于计算不考虑顺序的选择数。如果顺序重要，则使用排列数：n! / (n – r)!。

6. Probability | 概率

Probability in the AMC usually involves fair dice, coins, cards, or simple random experiments. The basic formula is P(event) = number of favourable outcomes ÷ total number of outcomes.

AMC中的概率通常涉及公平骰子、硬币、纸牌或简单随机试验。基本公式为 P(事件) = 有利结果数 ÷ 所有可能结果数。

A standard example is rolling two dice and asking for the probability of a sum of 7. There are 6 successful outcomes out of 36 possible pairs, so the probability is 6/36 = 1/6.

一个标准例子是掷两个骰子，求点数和为7的概率。36种等可能结果中有6种成功结果，因此概率为 6/36 = 1/6。

P(sum = 7) = 6/36 = 1/6

Also remember the complement rule: P(not A) = 1 – P(A). For independent events A and B, P(A and B) = P(A) × P(B). This is a common building block in multi-stage probability questions.

同时记住互补事件法则：P(非 A) = 1 – P(A)。对于相互独立的事件 A 和 B，P(A 且 B) = P(A) × P(B)。这是多阶段概率问题中常见的构造基础。

7. Sequences and Series | 数列与级数

Sequences and series appear regularly in the AMC. You should be comfortable with arithmetic progressions, geometric progressions, and their summation formulas.

数列与级数经常出现在AMC中。你需要熟悉等差数列、等比数列及其求和公式。

For an arithmetic sequence, the n-th term is aₙ = a₁ + (n – 1)d, and the sum of the first n terms is Sₙ = n/2 × (a₁ + aₙ).

对于等差数列，第 n 项为 aₙ = a₁ + (n – 1)d，前 n 项和为 Sₙ = n/2 × (a₁ + aₙ)。

For a geometric sequence, aₙ = a₁ rⁿ⁻¹, and the sum is Sₙ = a₁(1 – rⁿ) / (1 – r) when r ≠ 1.

对于等比数列，aₙ = a₁ rⁿ⁻¹，当 r ≠ 1 时前 n 项和为 Sₙ = a₁(1 – rⁿ) / (1 – r)。

Sₙ = n/2 × (a₁ + aₙ), Sₙ = a₁(1 – rⁿ) / (1 – r)

A special case is the sum of the first n positive integers: 1 + 2 + 3 + … + n = n(n + 1)/2. This result is very useful in AMC counting and pattern questions.

一个特殊情况是前 n 个正整数的和：1 + 2 + 3 + … + n = n(n + 1)/2。这个结果在AMC计数和找规律题中非常有用。

8. Functions and Equations | 函数与方程

Function concepts appear in higher-level AMC papers. You need to know the meaning of domain and range, how to graph linear and quadratic functions, and how to solve equations involving them.

函数概念出现在AMC较高年级的试卷中。你需要了解定义域和值域的含义，会画线性函数和二次函数的图像，并会解相关方程。

The quadratic formula is essential: for ax² + bx + c = 0, the solutions are x = [-b ± √(b² – 4ac)] / (2a).

二次公式是必须掌握的：对于 ax² + bx + c = 0，解为 x = [-b ± √(b² – 4ac)] / (2a)。

x = [-b ± √(b² – 4ac)] / (2a)

Graph transformations are also tested: adding a constant shifts the graph vertically, adding a constant inside the argument shifts it horizontally, and a negative sign reflects the graph. Inverses require swapping x and y and then solving for y.

图像变换也是考点：

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Australian Mathematics Competition: Core Topics | 澳大利亚数学竞赛核心考点梳理