对国际学校的学生而言，AMC成绩不仅是一项学术成就，更是申请顶尖大学的重要参考。在很多知名高等学府，如麻省理工学院（MIT）、斯坦福大学、哈佛大学等，学生的数学能力被视为一个不可忽视的评价维度。AMC10/12的优异成绩可以让学生在激烈的申请中脱颖而出，帮助他们获得进入这些全球顶尖学府的机会。
AIME含金量有多高？
1.申请名校的重要加分项
顶尖大学的认可：许多顶尖大学，如斯坦福大学、麻省理工学院（MIT）、卡内基梅隆大学等，在申请表格中都专门设有填写AMC/AIME成绩的栏目。
竞争优势：在标化成绩相同的情况下，AIME高分可以让学生在众多申请者中脱颖而出。特别是在STEM专业申请中，AIME成绩是一个非常重要的参考指标。
2.多样化的学术背景
AIME高分不仅展示了学生的数学能力，还体现了他们的逻辑思维、分析能力和解决问题的能力。这些能力在各种学术和职业领域都非常宝贵。
3.对数学能力的极高肯定
AIME的题目难度远高于AMC10/12，涉及的知识点更广泛，解题过程更复杂。能够在AIME中取得高分，表明学生具备扎实的数学基础和出色的解题能力。
AMC10/12哪个更容易晋级AIME？怎么选？
1.目标导向的选择
如果目标是进入AIME：
AMC12：由于AMC12的晋级AIME分数线通常比AMC10低15分左右，因此对于希望进入AIME的学生来说，AMC12是一个更为容易的选择。此外，AMC12的申请价值更大，对于大学申请有额外的优势。
如果目标是更高阶的竞赛（如USAMO或USAJMO）：
AMC10：虽然AMC10的晋级AIME分数线较高，但AMC10的高分相对容易取得，因此通过AMC10路径进入USAJMO的机会更大。USAJMO的标准成绩是AMC10分数加上10倍的AIME分数，因此AMC10的高分可以显著提高总分。
2.学生的基础和年级
10年级及以下：对于基础较好的10年级及以下学生，可以选择同时报考AMC10和AMC12。这样可以增加晋级AIME的机会，并且在两个竞赛中都有表现的机会。
11年级及以上：由于11年级及以上的学生不能再参加AMC10，因此只能选择AMC12。此时，重点应该是提高AMC12的成绩，争取进入AIME并获得高分。
3.具体建议
基础较好、目标明确：如果学生数学基础扎实，目标明确（例如，希望进入USAJMO），可以优先选择AMC10，因为AMC10的高分更容易取得。
基础一般、希望稳妥晋级AIME：如果学生数学基础一般，希望稳妥晋级AIME，可以优先选择AMC12，因为晋级AIME的分数线较低。
双保险策略：对于基础较好的10年级及以下学生，建议同时报考AMC10和AMC12，这样可以在两个竞赛中都有表现的机会，增加晋级AIME的概率。
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AIME教材“宝藏书”在手，备赛无忧
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?
?
are deposited in 
are deposited in bin 
![\[1+3+5+7+...+97+99\]](https://latex.artofproblemsolving.com/a/3/0/a307bbc8aeccf32d5dea44474103639a57f21408.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?
is divided by 
?
. What is the units digit of 
and 
have arithmetic mean 
. The arithmetic mean of 
and 
is 
. What is the arithmetic mean of 
and 
?
be the intersection of lines 
and 
. What is the ratio of the area of quadrilateral 
to the area of 
?
