什麽是美國數學邀請賽AIME？AIME都考哪些內容？

什麽(me) 是AIME？

AIME是美國數學邀請賽(American Invitational Mathematics Examination)的首字母縮寫(xie) ，是美國數學競賽AMC(American Mathematics Competition)係列賽事之一，也是美國國際數學奧林匹克(IMO)代表隊係列選拔賽的第二項賽事。

全麵解讀｜美國數學邀請賽AIME 絕大多數晉級AIME的選手是高中生，也有極少數頂尖的初中生可晉級AIME。從(cong) AMC 12晉級並在AIME中取得高分的選手將晉級USAMO(United States of America Mathematics Olympiad)，而從(cong) AMC 10晉級並在AIME中取得高分的選手將晉級USAJMO(United States of America Junior Mathematics Olympiad)。

怎樣報名參賽

AIME是邀請賽，在當年的AMC 10競賽中排名前2.5%左右或AMC 12競賽中排名前5%左右才能獲邀參賽。

AMC 10和12晉級AIME的分數線通常在AIME考試前3周左右公布。

近5年晉級AIME的分數線如下：

年份	AMC 10A	AMC 10B	AMC 12A	AMC 12B
2020	103.5	102	87	87
2019	103.5	108	84	94.5
2018	111	108	93	99
2017	112.5	120	96	100
2016	110	110	93	93

AIME考試時間？

美國官網(MAA)公布的2021年AIME考試時間為(wei) :

AIME I (AIME主賽)	AIME II (AIME替代賽)
3月10日(周三)	3月18日(周四)

AIME考試形式是怎樣的？

考試時長

3小時

題目數量

15題

題型

填空題，答案為(wei) 000-999 (含) 之間的整數

計分規則

答對得1分，答錯或不答得0分

滿分

15分

計算器

不允許使用

AIME都考哪些內(nei) 容？

和AMC 10、AMC 12一樣，考查範圍仍然是算術、代數、計數、幾何、數論和概率，以及其他高中數學知識。微積分不在數學競賽考查範圍內(nei) ，但允許使用微積分方法解題。

AIME難度如何？

通常前幾題的難度大致相當於(yu) AMC 12的水平，而越往後題目難度越大。通常多數學生能做出第1-5題；到了第6-10題則是區分度最大的題，經過專(zhuan) 門的訓練、在AMC 12或AMC 10中排名前1%左右的選手一般能做對一部分題；而一般在考場上能做出第11-15題的都是極其頂尖的選手。

從(cong) 曆年受邀參加AIME並獲獎的中國學生和所在學校的分布情況來看，不少都是參加過國家隊數學奧林匹克集訓的選手和長期培養(yang) 國際數學奧林匹克競賽(IMO)選手的學校。

試卷樣題

[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.!

全麵解讀｜美國數學邀請賽AIME

[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 .

晉級下一輪規則

從(cong) AIME晉級USAMO或USAJMO的規則如下：

USAMO標準成績(index score)=

AMC 12分數+10×AIME分數

USAJMO標準成績(index score)=

AMC 10分數+10×AIME分數

有的選手可能會(hui) 同時參加AMC 12A和12B，以及AIME I和II(2020年AIME II因疫情影響變更為(wei) 線上舉(ju) 辦的AOIME，但計分規則不變)，以下舉(ju) 例說明該選手的成績。

某選手的成績示例：

AMC 12A: 87

AMC 12B: 110

AIME I: 11

AIME II: 12