每年仅在北美地区正式登记参加比赛的学生就超过30万人次。AMC数学竞赛参赛者年龄从小学覆盖到高中，AMC考试包括AMC8、AMC10、AMC12、AIME、USJMO、USAMO。今天一起来了解下AMC系列竞赛中AMC2、AMC10、AMC12的考试大纲。
AMC8 考试大纲
基础代数：整数，有理数，无理数，实数，数轴和直角坐标系；多元一次方程，简单二次方程，简单不等式；简单数列；基本代数技巧
基础几何：基础几何作图；平面欧氏几何，点、线、三角形、特殊四边形、圆；规则图形的周长和面积；基本平面几何技巧；规则立体几何图形
基础数论：奇偶分析，整除的性质，最小公倍数和最大公约数，同余问题
基础组合：韦恩图；排列、组合和概率入门；阶乘和二项式系数，杨辉三角形
AMC10 考试大纲
进阶代数：多项式，余数定理，韦达定理，根与系数的关系，特殊高次方程；进阶不等式、均值不等式；函数入门，定义域和值域、二次函数、指数函数、对比函数、简单三角函数；数列进阶；代数技巧进阶
进阶几何：进阶几何作图；三角形进阶、正弦定理、余弦定理、内切圆和外切圆，斯图瓦尔特定理，共点和共线；圆和四边形，四点共圆，圆的外切四边形；正多边形，角度，周长和面积；进阶平面几何技巧；解析几何入门
立体几何：点、线、面的关系，三维坐标系；立体几何作图；正多面体，欧拉公式；特殊的立体几何图形，立体几何技巧
进阶数论：数，数组和序列；模运算，复杂同余问题；整数、分数和小数，进制转换；基本丢番图方程，进阶数论技巧
进阶组合：容斥原理；二项式定理及相关结论；进阶排列、组合和概率；期望入门，递推、二分法，进阶组合方法
AMC12考试大纲
进阶代数：复杂不等式、调和不等式、轮换不等式、柯西不等式；复杂函数问题，反函数和符合函数，三角函数和差化积、积化和差，万能公式；复数，复平面，欧拉公式，蒂莫夫公式；数学归纳法、复杂数列和极限
进阶几何：圆相关几何进阶；数形结合，二维、三维图形的函数表达，进阶解析几何；不规则二维、三维图形的处理；二维向量、三维向量
进阶数论：二次余数，高次余数、费马圣诞节定理、费马小定理；各类丢番图方程的解法
进阶组合：随机过程和期望；复杂组合问题技巧、基本综合问题
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, 
, and 
lie in that order along a straight path where the distance from 
meters. Ina runs twice as fast as Eve, and Paul runs twice as fast as Ina. The three runners start running at the same time with Ina starting at 
, 
, and 
, and for 
define 
recursively to be the remainder when 
is divided by 
. Find 
.
such that the base-
integer 
is a perfect square and the base-
is a perfect cube.
, 
and 
. The self-intersecting octagon 
enclosed six non-overlapping triangular regions. Let 
be the area enclosed by 
, where 
and 
.
, 
, and 
are complex numbers such that 
, 
, and 
, where 
. Then there are real numbers 
. Find 
.![$[-20, 18]$](https://latex.artofproblemsolving.com/5/e/d/5ed3fd299f7f005431bef3d44f28cf5f7494b169.png)
. The probability that the roots of the polynomial![\[x^4 + 2ax^3 + (2a - 2)x^2 + (-4a + 3)x - 2\]](https://latex.artofproblemsolving.com/3/8/a/38a3a5e9eaaa5a957b5f806c1179ce9a38a46d20.png)
, where 
and 
are relatively prime positive integers. Find 
.
where 
and ![\[\sqrt{\log_b n} = \log_b \sqrt{n} \qquad \text{and} \qquad b \cdot \log_b n = \log_b (bn).\]](https://latex.artofproblemsolving.com/e/1/8/e188ed1d68418cfaed35d91cc7b0f333a84f6b1b.png)
The value of 
where 
and 
are relatively prime positive integers. Find 
lines, no 
of which are parallel. Suppose that there are 
points where exactly 
points where exactly 
points where exactly 
points where exactly 
is a perfect square can be written as 
where 
and 
are positive integers. Find 
be a point on the circle circumscribing square 
that satisfies 
and 
Find the area of 
and 
have leading coefficients 
respectively. The graphs of both polynomials pass through the two points 
and 
Find 
whose representation in base nine is 
where 
and 
are (not necessarily distinct) digits.
parallel bases 
and 
have lengths 
and 
respectively, and 
The angle bisectors of 
and 
meet at 
and the angle bisectors of 
and 
meet at 
Find 
and 
where 
Find the number of ordered pairs 
of positive integers not exceeding 
that satisfy the equation 
meters wide flows from west to east at a rate of 
meters per minute. Melanie and Sherry sit on the south bank of the river with Melanie a distance of 
meters downstream from Sherry. Relative to the water, Melanie swims at 
meters per minute, and Sherry swims at 
meters per minute. At the same time, Melanie and Sherry begin swimming in straight lines to a point on the north bank of the river that is equidistant from their starting positions. The two women arrive at this point simultaneously. Find 
such that the sequence![\[3,4,5,a,b,30,40,50\]](https://latex.artofproblemsolving.com/1/0/f/10f3618dca767ca85f66cb8fd4b671468c6e2317.png)
is strictly increasing and no set of four (not necessarily consecutive) terms forms an arithmetic progression.