AMC & AIME 전략적 가치 분석

미국 최상위권 대학 입시, 데이터로 보는 성공 로드맵

AMC 12 응시자 중

상위 5%

만이 AIME에 진출합니다.

(AMC 10의 경우 상위 약 2.5%)

이 수치는 AIME 진출이 단순한 시험 참가를 넘어, 전국 단위에서 증명된 최상위권의 수학적 재능을 의미한다는 것을 보여줍니다. 입학사정관들은 이 지표를 통해 지원자의 학문적 깊이와 잠재력을 가늠합니다.

선발 시스템 한눈에 보기

AMC부터 미국 수학 올림피아드(USAMO)까지, 전체 선발 과정과 각 단계의 의미를 파악하세요.

AMC 10 / 12

상위 2.5% - 5% 진출

AIME

점수 합산(Index) 기준 상위권 선발

USAMO / USAJMO

최상위권 선발

국제 수학 올림피아드(IMO) 대표팀 선발

데이터로 보는 AIME의 위상

최근 3년간의 AIME 진출 자격 점수(Cutoff) 변화 추이를 통해 시험의 난이도와 경쟁률을 확인하세요.

AIME 진출 자격 점수는 매년 시험의 난이도에 따라 변동되며, 이는 지속적으로 높은 수준의 경쟁이 유지됨을 보여줍니다.

입시에서의 전략적 가치

AIME 성적은 다른 학업 지표와 비교했을 때 어떤 차별점을 가질까요?

학업 성취도 비교

정성적 평가 요소

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지적 호기심과 열정: 교과 과정을 넘어선 심화 학습 능력을 증명합니다.
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창의적 문제 해결 능력: 정형화되지 않은 문제에 대한 독창적 접근 능력을 보여줍니다.
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끈기와 성실성: 장기간의 노력이 필요한 어려운 목표를 달성하는 능력을 입증합니다.

목표 수준별 준비 로드맵

자신의 목표에 맞는 효과적인 학습 전략과 자료를 확인하세요.

기초 다지기 (AIME 진출 목표)

개념 학습: AoPS Introduction 시리즈로 대수, 기하, 정수론, 조합론 기초 확립.
유형별 문제 풀이: AoPS Alcumus 및 과거 AMC 문제로 실전 감각 익히기.
실전 모의고사: 시간 관리 능력 향상을 위한 주기적인 기출문제 풀이.

심화 학습 (AIME 고득점 목표)

심화 이론: AoPS Intermediate 시리즈 및 올림피아드 교재로 고급 이론 학습.
고난도 문제 분석: 과거 AIME 및 USAMO 기출문제 심층 분석.
커뮤니티 활용: AoPS 포럼에서 다른 학생들과 풀이법 토론 및 학습.

AMC 10/12 Sample Problems

Check out the types and difficulty of AMC problems. Click 'Show Solution' below each problem for an explanation.

AMC 10

Problem 1 (Algebra/Number Theory):

Let $N$ be a three-digit positive integer. The sum of the digits of $N$ is 11, and the sum of the squares of the digits of $N$ is 49. What is the smallest possible value of $N$?

Solution:
Let the digits of the three-digit integer $N$ be $a, b, c$ (where $a, b, c$ are integers from 0 to 9, and $a \neq 0$).
1) $a + b + c = 11$
2) $a^2 + b^2 + c^2 = 49$
From the second condition, the square of each digit must be less than 49, so each digit must be 6 or less (if a digit were 7, its square is 49, requiring the others to be 0, which contradicts the sum of 11).
Let's find a combination of three integers whose sum is 11 and whose sum of squares is 49. It's easiest to start by considering the largest possible digit. If the largest digit is 6, let's say $a=6$.
Then $b+c=5$ and $b^2+c^2 = 49 - 6^2 = 13$.
Using $(b+c)^2 = b^2+c^2+2bc$, we have $5^2 = 13+2bc \implies 25 = 13+2bc \implies 2bc = 12 \implies bc=6$.
The two integers that sum to 5 and have a product of 6 are 2 and 3.
Thus, the three digits are 2, 3, and 6. Their sum is $2+3+6=11$, and the sum of their squares is $2^2+3^2+6^2=4+9+36=49$, satisfying all conditions.
The smallest three-digit number that can be formed with these digits is made by placing the smallest digit, 2, in the hundreds place. The answer is 236.

Problem 2 (Geometry):

What is the area of an equilateral triangle inscribed in a circle of radius 3?

Solution:
Let the center of the circle be O and the radius be $R=3$. When an equilateral triangle ABC is inscribed in a circle, the center of the circle O is both the circumcenter and the centroid of the triangle.
The distance from the circumcenter to each vertex is equal to the radius R. So, $OA=OB=OC=3$.
Let the side length of the equilateral triangle be $s$.
The height $h$ of the triangle is $h = \frac{\sqrt{3}}{2}s$.
The centroid divides the median (height) in a 2:1 ratio. The distance from a vertex to the centroid is $\frac{2}{3}$ of the height. This distance is equal to the circumradius $R$.
Therefore, $R = \frac{2}{3}h = \frac{2}{3} \times \frac{\sqrt{3}}{2}s = \frac{\sqrt{3}}{3}s$.
We have $3 = \frac{\sqrt{3}}{3}s \implies s = \frac{9}{\sqrt{3}} = 3\sqrt{3}$.
The area of an equilateral triangle is given by the formula $\frac{\sqrt{3}}{4}s^2$.
Area $= \frac{\sqrt{3}}{4}(3\sqrt{3})^2 = \frac{\sqrt{3}}{4}(27) = \frac{27\sqrt{3}}{4}$.
The answer is $\frac{27\sqrt{3}}{4}$.

Problem 3 (Combinatorics):

Five students, A, B, C, D, and E, are arranged in a line. How many arrangements are there such that A and B are not adjacent?

Solution:
This problem can be solved using complementary counting or by direct counting.
Method 1: Complementary Counting
1. Total arrangements: The number of ways to arrange 5 students in a line is $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$.
2. Arrangements where A and B are adjacent: Treat A and B as a single block (AB). We are now arranging 4 items: (AB), C, D, E. This can be done in $4! = 24$ ways. Within the block, A and B can switch places (AB or BA), so we multiply by 2. Total adjacent arrangements = $24 \times 2 = 48$.
3. Arrangements where A and B are not adjacent: (Total arrangements) - (Adjacent arrangements) = $120 - 48 = 72$.
Method 2: Direct Counting
1. Arrange the other 3 students (C, D, E) first. This can be done in $3! = 6$ ways.
2. This creates 4 possible slots where A and B can be placed (_ C _ D _ E _).
3. Choose 2 of these 4 slots for A and B. The number of ways to place A and B in these slots is $P(4,2) = \frac{4!}{(4-2)!} = 4 \times 3 = 12$.
4. Total non-adjacent arrangements = (Arrangements of C,D,E) $\times$ (Ways to place A,B) = $6 \times 12 = 72$.
The answer is 72.

AMC 12

Problem 1 (Logarithms):

What is the real value of $x$ that satisfies the equation $\log_3(x-1) + \log_3(x+1) = \log_9(64)$?

Solution:
1. For the logarithms to be defined, the arguments must be positive: $x-1 > 0$ and $x+1 > 0$. This implies $x > 1$.
2. Use the logarithm property to combine the terms on the left side: $\log_3((x-1)(x+1)) = \log_3(x^2-1)$.
3. Use the change of base formula for the term on the right side: $\log_9(64) = \frac{\log_3(64)}{\log_3(9)} = \frac{\log_3(8^2)}{\log_3(3^2)} = \frac{2\log_3(8)}{2} = \log_3(8)$.
4. The equation now becomes $\log_3(x^2-1) = \log_3(8)$.
5. Since the bases are the same, the arguments must be equal: $x^2-1 = 8 \implies x^2 = 9$.
6. Therefore, $x = 3$ or $x = -3$.
7. From the initial condition, we must have $x > 1$. The only valid solution is 3.

Problem 2 (Complex Numbers):

If a complex number $z$ satisfies $z + \frac{1}{z} = 2\cos(15^\circ)$, what is the value of $z^{12} + \frac{1}{z^{12}}$?

Solution:
1. Let's analyze the given equation. Multiply by $z$ to get a quadratic equation: $z^2 - (2\cos(15^\circ))z + 1 = 0$.
Using the quadratic formula, the solutions for $z$ are: $z = \frac{2\cos(15^\circ) \pm \sqrt{4\cos^2(15^\circ)-4}}{2} = \cos(15^\circ) \pm i\sqrt{1-\cos^2(15^\circ)} = \cos(15^\circ) \pm i\sin(15^\circ)$.
This means $z$ is a complex number on the unit circle with an angle of $\pm 15^\circ$. Let's take $z = \cos(15^\circ) + i\sin(15^\circ)$. In polar form, this is $e^{i15^\circ}$.
2. We need to find $z^{12} + \frac{1}{z^{12}}$. By De Moivre's Theorem, if $z = \cos\theta + i\sin\theta$, then $z^n = \cos(n\theta) + i\sin(n\theta)$.
Also, $\frac{1}{z^n} = z^{-n} = \cos(-n\theta) + i\sin(-n\theta) = \cos(n\theta) - i\sin(n\theta)$.
Therefore, $z^n + \frac{1}{z^n} = 2\cos(n\theta)$.
3. In our case, $n=12$ and $\theta=15^\circ$. So, $n\theta = 12 \times 15^\circ = 180^\circ$.
4. The value is $z^{12} + \frac{1}{z^{12}} = 2\cos(180^\circ) = 2 \times (-1) = -2$.
The answer is -2.

Problem 3 (Geometry):

A triangle ABC has vertices A(0,0), B(4,0), and C(2, $2\sqrt{3}$). What are the coordinates of the incenter of this triangle?

Solution:
1. First, find the lengths of the sides of the triangle.
$a = BC = \sqrt{(4-2)^2 + (0-2\sqrt{3})^2} = \sqrt{2^2 + (-2\sqrt{3})^2} = \sqrt{4+12} = \sqrt{16} = 4$.
$b = AC = \sqrt{(2-0)^2 + (2\sqrt{3}-0)^2} = \sqrt{2^2 + (2\sqrt{3})^2} = \sqrt{4+12} = \sqrt{16} = 4$.
$c = AB = \sqrt{(4-0)^2 + (0-0)^2} = \sqrt{4^2} = 4$.
2. Since all three sides have a length of 4, triangle ABC is an equilateral triangle.
3. For an equilateral triangle, the incenter, circumcenter, centroid, and orthocenter are all the same point.
4. The coordinates of the centroid are the average of the coordinates of the vertices.
Incenter x-coordinate: $\frac{0+4+2}{3} = \frac{6}{3} = 2$.
Incenter y-coordinate: $\frac{0+0+2\sqrt{3}}{3} = \frac{2\sqrt{3}}{3}$.
Thus, the coordinates of the incenter are $(2, \frac{2\sqrt{3}}{3})$.