AIME是选拔美国数学奥林匹克国家队选手的关键赛事，能够为学生提供进一步发展的机会。对于申请海外名校的同学来说，AIME的高分显得尤为重要。
AIME竞赛分数价值
申请美国顶尖大学
TOP 30院校：通常需要AIME成绩达到7分以上。
TOP 20院校：至少需要8分以上，部分学校甚至可能要求更高。
参加顶级数学夏令营
Ross、Susac等夏令营：
AIME成绩需要达到9分左右，这样的分数能显著提升申请者的竞争力。
晋级USAMO
要晋级至USAMO（美国数学奥林匹克竞赛），考生需至少答对8-9道题目。
这大致相当于AIME成绩的8-9分。
AIME竞赛难度分析
AIME竞赛相较于AMC10/12而言，难度确实有了显著的提升，主要原因和难点如下：
1.题型差异
选择题 vs 填空题：
- AMC10/12主要是选择题，考生有时可以通过排除法或猜测提高得分概率。
- AIME全部是填空题，要求精确计算并填写答案，没有侥幸得高分的机会。
2.计算能力的要求
- 强大的计算能力和熟练度是必不可少的。
- 在没有选项辅助的情况下，稍有不慎就可能出错，导致得分降低。
3.数论部分的深度和广度
- 数论题目数量多且难度逐渐增加。
- 前五题相对简单，类似于AMC10/12的水平。
- 后十题则需要更高的综合运用能力和大量计算。
- 考察范围扩展到了更复杂的领域，如三次型不定方程及主元法等技术性较强的内容。
4.对复数单位根的重点关注
近几年的AIME考试加大了对复数单位根概念和应用方面的考查力度。
5.新增知识点——欧拉定理
在几何部分首次加入了欧拉定理的相关内容，标志着考试难度的进一步提升和对考生知识面的拓展需求。
具体来说：
AIME的前5题大致对应AMC12的第15至17题
AIME的第6至9题大致对应AMC12的第17至20题
AIME的第10至12题大致对应AMC12的第20至23题
AIME的第13至15题大致对应AMC12的第23至25题
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residents of Aimeville, there are 
who own a diamond ring, 
who own a set of golf clubs, and 
who own a garden spade. In addition, each of the 
residents who own exactly two of these things, and 
residents who own exactly three of these things. Find the number of residents of Aimeville who own all four of these things.
The sum of the items in the list is 
.
.
, and the sum of the three numbers formed by reading top to bottom is 
. The grid below is an example of such an arrangement because 
and 
.![\[\begin{array}{|c|c|c|} \hline 0 & 0 & 8 \\ \hline 9 & 9 & 1 \\ \hline \end{array}\]](https://latex.artofproblemsolving.com/3/7/8/3789264ba7fedec68fa8488d65f286df5cdbefbc.png)
and 
be positive real numbers that satisfy the following system of equations:![\[\log_2\left({x \over yz}\right) = {1 \over 2}\]](https://latex.artofproblemsolving.com/e/6/5/e65012d6b5c3081c7d2f9ec748f09b2b5cada208.png)
![\[\log_2\left({y \over xz}\right) = {1 \over 3}\]](https://latex.artofproblemsolving.com/a/8/e/a8e021d3f2a2db4d7a423d5ed27f93c58b19ea1b.png)
![\[\log_2\left({z \over xy}\right) = {1 \over 4}\]](https://latex.artofproblemsolving.com/3/a/8/3a806f801e846b4f7311aee3f5b30971dad5310b.png)
Then the value of 
is 
where 
and 
are relatively prime positive integers. Find 
.
be a convex equilateral hexagon in which all pairs of opposite sides are parallel. The triangle whose sides are extensions of segments 
, 
, and 
has side lengths 
and 
. Find the side length of the hexagon.
of positive integers. Then Bob lists all finite nonempty sets 
of positive integers with the property that the maximum element of 
sets. Find the sum of the elements of 
be the greatest four-digit integer with the property that whenever one of its digits is changed to 
, the resulting number is divisible by 
. Let 
and 
be the quotient and remainder, respectively, when 
. Find 
.
is the surface produced by revolving a circle with radius 3 around an axis in the plane of the circle that is a distance 6 from the center of the circle (so like a donut). Let 
be a sphere with a radius 11. When 
, and when 
. The difference 
can be written as 
, where ![[asy] unitsize(0.3 inch); draw(ellipse((0,0), 3, 1.75)); draw((-1.2,0.1)..(-0.8,-0.03)..(-0.4,-0.11)..(0,-0.15)..(0.4,-0.11)..(0.8,-0.03)..(1.2,0.1)); draw((-1,0.04)..(-0.5,0.12)..(0,0.16)..(0.5,0.12)..(1,0.04)); draw((0,2.4)--(0,-0.15)); draw((0,-0.15)--(0,-1.75), dashed); draw((0,-1.75)--(0,-2.25)); draw(ellipse((2,0), 1, 0.9)); draw((2.03,-0.02)--(2.9,-0.4)); [/asy]](https://latex.artofproblemsolving.com/d/4/7/d47e4fdb4c54afd9a0eabe569e4cabb139ec600a.png)
indistinguishable white chips and 
grid such that
have incenter 
and circumcenter 
with 
, circumradius 
, and inradius 
. Find 
.
kilometers per hour, and the walk takes 
hours, including 
minutes at the coffee shop. Another morning, she walks at 
kilometers per hour, and the walk takes 
hours and 
minutes, including 
kilometers per hour, how many minutes will the walk take, including the 
and 
with 
satisfy 
What is the value of 
?
She wins a prize if at least two of her numbers were 
and 
are drawn such that 
are collinear. Also, 
all lie on a circle. If 
and 
what is the length of 
?![[asy] import graph; unitsize(0.1cm); pair A = (0,0);pair B = (70,0);pair C = (70,16);pair D = (0,16);pair E = (3,16);pair F = (90,16);pair G = (90,33);pair H = (3,33); dot(A^^B^^C^^D^^E^^F^^G^^H); label("$A$", A, S);label("$B$", B, S);label("$C$", C, N);label("$D$", D, N);label("$E$", E, S);label("$F$", F, S);label("$G$", G, N);label("$H$", H, N); draw(E--D--A--B--C--E--H--G--F--C); [/asy]](https://latex.artofproblemsolving.com/1/2/f/12f0a4224f27ffa406ded2c01cfd349ecd0c04de.png)
that follow the lines from the lower left corner to the upper right corner on an 
grid. Find the number of such paths that change direction exactly four times, like in the examples shown below.![[asy] size(7.5cm); usepackage("tikz");label("\begin{tikzpicture}[scale=.4]\draw(0,0)grid(8,8);\draw[line width=2,red](0,0)--(2,0)--(2,3)--(5,3)--(5,8)--(8,8);\end{tikzpicture}",origin); label("\begin{tikzpicture}[scale=.4]\draw(0,0)grid(8,8);\draw[line width=2,red](0,0)--(0,3)--(3,3)--(3,5)--(8,5)--(8,8);\end{tikzpicture}",E); [/asy]](https://latex.artofproblemsolving.com/f/a/2/fa2be3ff5b03203d7779a7a359fa05e2bf9d0c55.png)
![\[(75+117i)z+\frac{96+144i}{z}\]](https://latex.artofproblemsolving.com/8/9/f/89f4e07cc86e94772fd84e963c46fad30e9b51f7.png)
where 
.
can be placed tangent to 
of 
so that the circles are sequentially tangent to each other, with the first circle being tangent to 
, as shown. Similarly, ![[asy] pair A = (2,1); pair B = (0,0); pair C = (3,0); dot(A^^B^^C); label("$A$", A, N); label("$B$", B, S); label("$C$", C, S); draw(A--B--C--cycle); for(real i=0.62; i<2.7; i+=0.29){ draw(circle((i,0.145), 0.145)); } [/asy]](https://latex.artofproblemsolving.com/e/c/5/ec5d6f1f995c862b1609e69add5733cb092f1f3f.png)
and are in that order. If its diagonals intersect at the origin, find the largest number less than 
for all rhombuses 
. Let the tangents to 
intersect at point 
, and let 
intersect 
. If 
, 
, and 
, 
can be written as the form 
.
?
are deposited in 
bins, labeled 
and 
, using the following procedure. Ball 
are deposited in 
are deposited in bin 
![\[1 + 3+5+7+\cdots+97+99\]](https://latex.artofproblemsolving.com/f/b/a/fba42eee201e89836d88d1fb08c49a5c0a97c39b.png)
When the new expression was evaluated, it was negative. What is the least number of plus signs that Melanie could have changed to minus signs?
dollars by 
. How many digits does this number of dollars have when written as a numeral in base 
as 
is sufficient for this problem.)
is a rectangle with 
and 
. Point 
lies 
, point 
, and 
is a right angle. The areas of 
and 
are equal. What is the area of 
?![[asy] pair X = (0, 0); pair W = (0, 4); pair Y = (8, 0); pair Z = (8, 4); label("$X$", X, dir(180)); label("$W$", W, dir(180)); label("$Y$", Y, dir(0)); label("$Z$", Z, dir(0)); draw(W--X--Y--Z--cycle); dot(X); dot(Y); dot(W); dot(Z); pair M = (2, 0); pair A = (8, 3); label("$A$", A, dir(0)); dot(M); dot(A); draw(W--M--A--cycle); markscalefactor = 0.05; draw(rightanglemark(W, M, A)); label("$M$", M, dir(-90)); [/asy]](https://latex.artofproblemsolving.com/a/e/9/ae9f151d0a601f719d48015b0f36651f319802a3.png)
![\[\frac{\log_2x\cdot\log_3x}{\log_2x+\log_3x}=2?\]](https://latex.artofproblemsolving.com/f/f/e/ffe17a6345042ba69850f77f5df1309773bfb1be.png)
such that 
. A target 
A dart is thrown and lands at a random point in 
where 
, 
, 
, 
, and 
with 
. The range of the list is 
are possible?
