什么是美国数学邀请赛AIME？AIME都考哪些内容？

什么是AIME？

AIME是美国数学邀请赛(American Invitational Mathematics Examination)的首字母缩写，是美国数学竞赛AMC(American Mathematics Competition)系列赛事之一，也是美国国际数学奥林匹克(IMO)代表队系列选拔赛的第二项赛事。

绝大多数晋级AIME的选手是高中生，也有极少数顶尖的初中生可晋级AIME。从AMC 12晋级并在AIME中取得高分的选手将晋级USAMO(United States of America Mathematics Olympiad)，而从AMC 10晋级并在AIME中取得高分的选手将晋级USAJMO(United States of America Junior Mathematics Olympiad)。

怎样报名参赛

AIME是邀请赛，在当年的AMC 10竞赛中排名前2.5%左右或AMC 12竞赛中排名前5%左右才能获邀参赛。

AMC 10和12晋级AIME的分数线通常在AIME考试前3周左右公布。

近5年晋级AIME的分数线如下：

AIME考试时间？

美国官网(MAA)公布的2021年AIME考试时间为:

AIME考试形式是怎样的？

考试时长

3小时

题目数量

15题

题型

填空题，答案为000-999 (含) 之间的整数

计分规则

答对得1分，答错或不答得0分

满分

15分

计算器

不允许使用

AIME都考哪些内容？

和AMC 10、AMC 12一样，考查范围仍然是算术、代数、计数、几何、数论和概率，以及其他高中数学知识。微积分不在数学竞赛考查范围内，但允许使用微积分方法解题。

AIME难度如何？

通常前几题的难度大致相当于AMC 12的水平，而越往后题目难度越大。通常多数学生能做出第1-5题；到了第6-10题则是区分度最大的题，经过专门的训练、在AMC 12或AMC 10中排名前1%左右的选手一般能做对一部分题；而一般在考场上能做出第11-15题的都是极其顶尖的选手。

从历年受邀参加AIME并获奖的中国学生和所在学校的分布情况来看，不少都是参加过国家队数学奥林匹克集训的选手和长期培养国际数学奥林匹克竞赛(IMO)选手的学校。

试卷样题

[AIME I, 2018Q1]
Let be the number of ordered pairs of integers with and such that the polynomial can be factored into the product of two (not necessarily distinct) linear factors with integer coefficients. Find the remainder when is divided by 1000.

[AIME I, 2018Q2]
The number can be written in base 14 as , can be written in base 15 as , and can be written in base 6 as , where . Find the base-10 representation of .

[AIME I, 2018Q3]
Kathy has 5 red cards and 5 green cards. She shuffles the 10 cards and lays out 5 of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders , , or will make Kathy happy, but will not. The probability that Kathy will be happy is , where and are relatively prime positive integers. Find .

[AIME I, 2018Q4]
In , and . Point lies strictly between and on and point lies strictly between and on so that . Then can be expressed in the form , where and are relatively prime positive integers. Find .

[AIME I, 2018Q5]
For each ordered pair of real numbers satisfying

there is a real number such that

Find the product of all possible values of .

[AIME I, 2018Q6]
Let be the number of complex numbers with the properties that and  is a real number. Find the remainder when is divided by 1000.

[AIME I, 2018Q7]
A right hexagonal prism has height 2. The bases are regular hexagons with side length 1. Any 3 of the 12 vertices determine a triangle. Find the number of these triangles that are isosceles (including equilateral triangles).

[AIME I, 2018Q8]
Let be an equiangular hexagon such that , , , and . Denote the diameter of the largest circle that fits inside the hexagon. Find .

[AIME I, 2018Q9]
Find the number of four-element subsets of with the property that two distinct elements of a subset have a sum of 16, and two distinct elements of a subset have a sum of 24. For example, and are two such subsets.

[AIME I, 2018Q10]
The wheel shown below consists of two circles and five spokes, with a label at each point where a spoke meets a circle. A bug walks along the wheel, starting at point . At every step of the process, the bug walks from one labeled point to an adjacent labeled point. Along the inner circle the bug only walks in a counterclockwise direction, and along the outer circle the bug only walks in a clockwise direction. For example, the bug could travel along the path , which has 10 steps. Let be the number of paths with 15 steps that begin and end at point . Find the remainder when is divided by 1000.!

[AIME I, 2018Q11]
Find the least positive integer such that when is written in base 143, its two right-most digits in base 143 are 01.

[AIME I, 2018Q12]
For every subset of , let be the sum of the elements of , with defined to be 0. If is chosen at random among all subsets of , the probability that is divisible by 3 is , where and are relatively prime positive integers. Find .

[AIME I, 2018Q13]
Let have side lengths , , and . Point lies in the interior of , and points and are the incenters of and , respectively. Find the minimum possible area of as varies along .

[AIME I, 2018Q14]
Let be a heptagon. A frog starts jumping at vertex . From any vertex of the heptagon except , the frog may jump to either of the two adjacent vertices. When it reaches vertex , the frog stops and stays there. Find the number of distinct sequences of jumps of no more than 12 jumps that end at .

[AIME I, 2018Q15]
David found four sticks of different lengths that can be used to form three non-congruent convex cyclic quadrilaterals, , , , which can each be inscribed in a circle with radius 1. Let φ denote the measure of the acute angle made by the diagonals of quadrilateral , and define φ and φ similarly. Suppose that φ, φ, and φ. All three quadrilaterals have the same area , which can be written in the form , where and are relatively prime positive integers. Find .

晋级下一轮规则

从AIME晋级USAMO或USAJMO的规则如下：

USAMO标准成绩(index score)=

AMC 12分数+10×AIME分数

USAJMO标准成绩(index score)=

AMC 10分数+10×AIME分数

有的选手可能会同时参加AMC 12A和12B，以及AIME I和II(2020年AIME II因疫情影响变更为线上举办的AOIME，但计分规则不变)，以下举例说明该选手的成绩。

某选手的成绩示例：

AMC 12A: 87

AMC 12B: 110

AIME I: 11

AIME II: 12

根据上述成绩计算出USAMO标准成绩(index score)如下：

标准成绩1：12A+10×AIME I=87+110=197

标准成绩2：12A+10×AIME II=110+120=207

标准成绩3：12B+10×AIME I=87+110=220

标准成绩4：12B+10×AIME II=110+120=230

假设当年USAMO和USAJMO的晋级分数线公布如下(正式分数线通常在AIME考试结束后3-4周公布)：

根据此晋级分数线可知，该学生的标准成绩3(220分)超过了对应的分数线(215)，而尽管其他3个标准成绩没有达到分数线，但该学生仍晋级USAMO(需为美国公民)。

备赛建议

AIME题目难度大、考试时间长，既是对学生数学竞赛题解题技巧、思维水平的考验，同样是对学生耐力的考验。因此，对想参加AIME、并在AIME竞赛中取得优秀成绩并晋级下一轮竞赛的选手来说，需要提早准备、做好长期训练的规划。