AMC8/10/12美国数学竞赛试听课/班课/1v1

美国数学AMC8全程班

适合学生：

本课程适合校内基础扎实并想要参加AMC8竞赛并冲击前5%及以上奖项的学生。

课程教材：

赠送纸制版自研教材

教学服务：

课前诊断评测，课中启发教学

课后学情反馈，作业批改答疑

考前模拟训练，免费考场报名

课程时长：40小时

1. Number Theorem数论（8H）

1.1 factors and prime factorizations质因数分解

Factors因数

Divisors除数

Power幂

Composite number合数

Prime number质数

The Square Root Rule 平方根规则

Relatively prime互质

Prime Factorizations质因数分解

Square平方数

Cube立方数

Number of divisors 因数个数

GCD and LCM 最大公因数和最小公倍数

1.2 Even, Odd and Divisibility 奇偶性，整除技巧

Even and Odd 奇数和偶数

Divisibility 整除

1.3 Base Number and Digits 进制与位值

Base 10 number representations 十进制表达

Remainder 余数

2. Algebra代数（8H）

2.1 Sequences and Series数列与求和

Arithmetic sequence 等差数列

Geometric sequences等比数列

2.2 Ratios, Rates, and Proportions 比率与比例

Ratios 比率

Proportions比例

Continued Ratio 连比

Speed 速度

2.3 Equation and Inequalities 方程与不等式

Fundamental law of fractions 分数的基本法则

Power rules of exponents 指数的幂法则

Properties of radicals 根式的性质

Properties of absolute value 绝对值的性质

Square binomial 二项展开式（二次方）和平方差

One-variable Linear Equations 一元一次方程

Quadratic Equations二次方程的解

Linear Inequalities 一次不等式

2.4 Sets and Venn diagram 集合与韦恩图

Set集合

3. Mid-term Review 阶段复习 （2H）

Mid-term Test期中测试

Mid-term Review阶段复习与总结

4. Geometry几何（9H）

4.1 Triangles and Similarity三角形与相似性

Areas of Triangle三角形面积

Triangle Inequality Theorem三角不等式

Pythagorean Theorem 勾股定理

Similar triangles相似三角形

4.2 Polygons多边形

Polygons多边形

The Pick’s Law皮克定理

4.3 Circles圆

Circumference and Area of Circle圆的周长与面积

Area of sector 扇形的面积

Length of the arc 扇形的弧长

4.4 Volumes体积

Rectangular solid 长方体

Triangular prism 三棱柱

Cylinder 圆柱

Cone 圆锥

Pyramid 棱锥

Octahedron 八面体

5. Probability and Statistic统计与概率（9H）

5.1 Statistics统计

Mean Median Mode and Range平均数，中位数，众数和极差

Factorial 阶乘

5.2 Permutations and Combinations排列组合

Permutation 排列

Combination 组合

Circular Permutations循环排列

5.3 Probability概率

Probability概率

6. Practice and Review复习与练习（4H）

Past Paper Practice 近5年真题练习讲解与分析

备注：

1. 具体课时情况根据学生实际学习效果和进度可能发生微调。

2. 要求学生积极配合老师完成每节课课后的作业，做到预习和复习。

AMC10数学竞赛强化辅导课程

课程须知

强化班：面向有较多数学竞赛学习经验，且在部分比赛中已经获得过一定成绩，目标为AMC10 Distinction（TOP5%）及以上的优秀学员。AMC10前测成绩正确题目介于15题至18题之间（包含）进入强化班学习。

课程目标是帮助学生快速巩固、梳理数学竞赛的核心知识框架，在知识点上做进一步的深化和拓展，增加难题（Q20+）解析的数量和强度，进一步增进学生对于竞赛解题技巧和思维方式的深刻理解和熟练掌握，并为晋级之后的AIME备考打下基础。

AMC10真题及教材

课程大纲

 

1. Number theory（10h）

Elementary

（1）Exponents, Prime factorization, Number of divisors, LCM and GCD

（2）Congruence Theory

（3）Divisibility rules, Venn diagram

（4）Character of digits, Base-n Representation

Advanced

（1）Sum of divisors, Product of divisors, Euclid's Algorithm and *Bezout's Theorem

（2）Euler’s function and theorem, Fermat’s little theorem, *Chinese remainder theorem(CRT)

（3）Sets, Principle of Inclusion and Exclusion

（4）Infinite decimal

2. Algebra（12h）

Elementary

（1）Arithmetic sequences, Geometric sequences, Periodic sequences

（2）Algebraic manipulations; Polynomials, Division Algorithm, Remainder Theorem

（3）Functions and Graph

（4） Linear equations and Quadratic equations, Vieta’s Theorem

（5）Linear inequalities and system of linear inequalities

Advanced

（1）General recursive sequences

（2）Binomial theorem, Pascal Triangle, Hockey-stick Theorem

（3）Gaussian function

（4） Equations of higher degree, Vieta’s theorem of higher degree

（5）Fundamental inequality, Cauchy's inequality, Absolute value inequality

3. Geometry（10h）

Elementary

（1）Parallel And Similar

（2）Triangles

（3）Polygon (Trapezoid, Parallelogram, Rhombus, Rectangle)

（4）Circles（Chord, Angles, Area）

（5）Solid Geometry

Advanced

（1）Menelaus's theorem, Ceva's theorem, *Stewart Theorem

（2）Heron's formula; Angle Bisector and Median, Centers of Triangles

（3）Four Concyclic Points; Power of a Point Theorem; Ptolemy's theorem

（4） Volume of Frustums

4. Combinatorics（8h）

Elementary

（1）Sum rules and Product rules

（2）Permutations and Combinations

（3）Basic probability Theory and Logic reasoning

Advanced

（1）Geometric Counting Problems

（2）Circular Permutation; Grouping Theorem; Balls into Boxes

（3）Geometric probability, Pigeonhole principle

*（较全程班节奏更快，及更侧重难题练习和知识点的高阶引申）

AMC12数学竞赛强化辅导课程

课程须知

面向有较多数学竞赛学习经验，且在部分比赛中已经获得过一定成绩，目标为AMC12 Distinction（TOP5%）及以上的优秀学员。AMC12前测成绩正确题目介于14题与17题之间（包含）进入强化学习。

课程目标是帮助学生快速巩固、梳理数学竞赛的核心知识框架，在知识点上做进一步的深化和拓展，增加难题（Q20+）解析的数量和强度，进一步增进学生对于竞赛解题技巧和思维方式的深刻理解和熟练掌握，并为晋级之后的AIME备考打下基础。

AMC12真题及教材

课程大纲

1. Number theory

（1）Prime Factorization: Number of divisors, Sum/Product of divsiors; Factorization Method for Solving LCM and GCD; Euclidean Algorithm and *Bezout's Theorem

（2）Congruence Theory and Divisibility:  Modulus and Residue, Properties of Congruence, *Modular Inverse; Divisibility Rules; Principle of Inclusion and Exclusion; *Euler's Theorem/Fermat's little Theorem,  *Chinese Remainder Theorem(CRT), *Wilson's Theorem

（3）Digit Representation and Base conversion, Short Division Algorithm; Infinite repeating decimal

2. Algebra

（1）Arithmetic sequences, Geometric sequences, Periodic sequences; Recursive sequences and *Characteristic Equation Method

（2）Algebraic Manipulation; Pascal Triangle and Binomial Theorem, Hockey-stick Theorem; Polynomials and Division Algorithm, Fundamental Theorem of Algebra, Generalized Remainder Theorem, Rational Root Theorem, Vieta's Theorem for higher degree polynomials

（3）Polynomial Inequalities; Fundamental Inequality, Cauchy's inequality and Extreme Value Problems

（4） Trigonometric Functions and Trigonometric identities; *Product-Sum and Sum-Product Identities

（5）Logarithm and its Calculation

（6）Complex Numbers; Properties of Conjugates and Modulus; Vector representation of Complex Numbers; Polar Form; DeMoivre' Theorem, Roots of unity

3. Geometry

（1）Basics in Geometry; The Law of Sine and the Law of Cosine; Heron's fomula, Area and Area Method

（2）Triangles: Similar Triangles; Angle Bisector and the Angle Bisector Theorem, *Angle bisector length formula; Median and Centroid, Median length formula; Centers of Triangle; Menelaus and Ceva's Theorem, Stewart's Theorem

（3）Circles: Basic geometric properties of circles; Cyclic quadrilaterals; Power of a Point Theorem; *Ptolemy's theorem

（5）Solid Geometry: Box, Cube, Prism; Pyramids; Surface Area and Volume; *Frustums; Cylinder and Sphere; *Theorem of Three Perpendiculars, *Euler's Polyhedron Formula

4. Combinatorics

（1）Basic Counting Principles: Sum rules and Product rules; Geometric Counting Problems

（2）Permutations and Combinations; Circular Permutation; Grouping Theorem; Balls into Boxes; *Advanced Combinatorics Identities, *Recursive Method in Combinatorics

（3）Elementary probability and Simple Stats.

*（较全程班节奏更快，及更侧重难题练习和知识点的高阶引申）