分类： 赛事资讯

AMC适合几年级学生参加？AMC赛事每年考什么？AMC赛事晋级获奖率高吗？AMC赛事动态栏目及时更新学生关于赛事最关心的信息。

2026年AMC8分数线出了吗？AMC8究竟适合哪些学生？

AMC8作为全球最具影响力的初中数学竞赛之一，不仅是激发数学兴趣的“启蒙赛”，更是小升初择校、国际升学、竞赛进阶的重要敲门砖。其奖项设置科学、权威，尤其全球前1%和前5%的荣誉，被国内顶尖学校和海外教育机构高度认可。

本文全面解析 AMC8奖项评定标准、近年分数线走势、2026年难度变化预判，并精准匹配不同年级、能力与升学需求的学生，助你高效规划备赛路径。

一、AMC8奖项设置与含金量

AMC8奖项分为个人奖和团体奖，其中个人奖对升学价值最大，具体如下：

奖项名称 英文缩写 获奖标准 2025年分数线 含金量
满分奖 Perfect Score 答对全部25题 25分 极高，全国仅数十人，顶尖名校“明星简历”标配
全球卓越奖 DHR
（Distinguished Honor Roll）		 全球前1% 23分 ⭐⭐⭐⭐⭐
申请上海“三公”、北京“六小强”、国际学校核心加分项
全球优秀奖 HR
（Honor Roll）		 全球前5% 19分 ⭐⭐⭐⭐
小升初/初升高简历亮点，体现扎实数学能力
全球荣誉奖 AR
（Achievement Roll）		 6年级及以下 + ≥15分 15分 ⭐⭐⭐
鼓励低年级超前学习，展示早期潜力

家长重点关注：

若目标上海三公（上实、上外、浦外）、北京人大附早培等，DHR（前1%）是黄金门槛；

若申请国际学校或美高，HR（前5%）已具备显著竞争力。

二、近5年分数线趋势 & 2026年预判

近年AMC8全球前1%与前5%分数线（满分25）：

年份 DHR（前1%） HR（前5%）
2020 21 18
2022 22 19
2023 21 17
2024 22 18
2025 23 19

趋势分析：

分数线整体稳中有升，反映参赛者水平提高；

2025年DHR达23分，为近5年最高，竞争加剧。

2026年特殊变化：题目难度提升！

中国区组委会为保障公平性，改编部分题目，个别题接近AMC10水平；

题目更强调逻辑推理与跨知识点整合，减少纯计算题。

2026年分数线预判：

奖项 预测分数线 依据
DHR（前1%） 22分左右 题目变难，高分人数减少，分数线或小幅下调
HR（前5%） 18–19分 中档题稳定性强，预计与近年持平

策略建议：

目标DHR的学生，确保基础题（1–15题）零失误，中档题（16–20题）错≤1题，难题（21–25题）争取对1–2题。

三、AMC8适合哪些学生？三大维度精准匹配

1.按年级划分

年级 适配性 备赛建议
3–4年级 黄金启蒙期 若校内数学拔尖（如奥数班前30%），可开始接触AMC8思维题，重在兴趣培养
5–6年级 核心参赛群体 系统学习AMC8四大模块（算术、代数、几何、计数），目标HR/DHR
7–8年级 冲刺+过渡期 冲刺DHR，同时衔接AMC10内容，为AIME晋级铺路

特别提醒：

上海“三公”简历投递通常在5年级上学期4月，因此最晚4年级下启动AMC8备考！

2.按能力划分

数学基础扎实：熟练掌握分数、百分比、比例、简单方程、平面几何；

逻辑思维强：能快速识别题目陷阱，构建解题路径（如分类讨论、逆向思维）；

有挑战意愿：计划未来参加AMC10/12、AIME，或走数学竞赛路线。

3.按升学需求划分

升学目标 AMC8作用 建议目标奖项
国内小升初（三公、六小强） 简历核心亮点，部分学校设“AMC8 DHR直通”通道 DHR（前1%）
国际学校/双语学校初升高 证明数学能力与国际接轨 HR（前5%）及以上
未来出国读美高/本科 积累早期学术竞赛记录 HR起步，DHR更佳

扫码进入AMC8专属学习社群，海量备赛资料&体验课程等你开启！

发送【年级+学校+课程体系】

2027年AMC8备考课程安排

针对于不同基础的同学，制定了系统的备考课程体系，分别为：Pre-AMC8课程、AMC8竞赛全程班、AMC8竞赛进阶强化班和AMC8考前模考班！

不知道是否适合AMC8？

扫码回复【在读年级+AMC8测试题】进行AMC8前测题/专业分析评估

基础较为薄弱的学生——Pre-AMC8课程

基础较为扎实的学生——AMC8竞赛全程班

已具备5%水平的学生——AMC8竞赛进阶强化班

知识完备考前冲刺热身的同学——AMC8考前模考班

2000年AIME II卷真题及答案

2000年AIME II 真题：

Problem 1

The number

$\frac 2{\log_4{2000^6}} + \frac 3{\log_5{2000^6}}$
can be written as $\frac mn$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Problem 2

A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola $x^2 - y^2 = 2000^2$?

Problem 3

A deck of forty cards consists of four 1's, four 2's,..., and four 10's. A matching pair (two cards with the same number) is removed from the deck. Given that these cards are not returned to the deck, let $m/n$ be the probability that two randomly selected cards also form a pair, where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Problem 4

What is the smallest positive integer with six positive odd integer divisors and twelve positive even integer divisors?

Problem 5

Given eight distinguishable rings, let $n$ be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order of rings on each finger is significant, but it is not required that each finger have a ring. Find the leftmost three nonzero digits of $n$.

Problem 6

One base of a trapezoid is $100$ units longer than the other base. The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the ratio $2: 3$. Let $x$ be the length of the segment joining the legs of the trapezoid that is parallel to the bases and that divides the trapezoid into two regions of equal area. Find the greatest integer that does not exceed $x^2/100$.

Problem 7

Given that

$\frac 1{2!17!}+\frac 1{3!16!}+\frac 1{4!15!}+\frac 1{5!14!}+\frac 1{6!13!}+\frac 1{7!12!}+\frac 1{8!11!}+\frac 1{9!10!}=\frac N{1!18!}$
find the greatest integer that is less than $\frac N{100}$.

Problem 8

In trapezoid $ABCD$, leg $\overline{BC}$ is perpendicular to bases $\overline{AB}$ and $\overline{CD}$, and diagonals $\overline{AC}$ and $\overline{BD}$ are perpendicular. Given that $AB=\sqrt{11}$ and $AD=\sqrt{1001}$, find $BC^2$.

Problem 9

Given that $z$ is a complex number such that $z+\frac 1z=2\cos 3^\circ$, find the least integer that is greater than $z^{2000}+\frac 1{z^{2000}}$.

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2000年AIME I卷真题及答案

2000年AIME I 真题：

Problem 1

Find the least positive integer $n$ such that no matter how $10^{n}$ is expressed as the product of any two positive integers, at least one of these two integers contains the digit $0$.

Problem 2

Let $u$ and $v$ be integers satisfying $0 < v < u$. Let $A = (u,v)$, let $B$ be the reflection of $A$ across the line $y = x$, let $C$ be the reflection of $B$ across the y-axis, let $D$ be the reflection of $C$ across the x-axis, and let $E$ be the reflection of $D$ across the y-axis. The area of pentagon $ABCDE$ is $451$. Find $u + v$.

Problem 3

In the expansion of $(ax + b)^{2000},$ where $a$ and $b$ are relatively prime positive integers, the coefficients of $x^{2}$ and $x^{3}$ are equal. Find $a + b$.

Problem 4

The diagram shows a rectangle that has been dissected into nine non-overlapping squares. Given that the width and the height of the rectangle are relatively prime positive integers, find the perimeter of the rectangle.

[asy]defaultpen(linewidth(0.7)); draw((0,0)--(69,0)--(69,61)--(0,61)--(0,0));draw((36,0)--(36,36)--(0,36)); draw((36,33)--(69,33));draw((41,33)--(41,61));draw((25,36)--(25,61)); draw((34,36)--(34,45)--(25,45)); draw((36,36)--(36,38)--(34,38)); draw((36,38)--(41,38)); draw((34,45)--(41,45));[/asy]

Problem 5

Each of two boxes contains both black and white marbles, and the total number of marbles in the two boxes is $25.$ One marble is taken out of each box randomly. The probability that both marbles are black is $\frac{27}{50},$ and the probability that both marbles are white is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. What is $m + n$?

Problem 6

For how many ordered pairs $(x,y)$ of integers is it true that $0 < x < y < 10^{6}$ and that the arithmetic mean of $x$ and $y$ is exactly $2$ more than the geometric mean of $x$ and $y$?

Problem 7

Suppose that $x,$ $y,$ and $z$ are three positive numbers that satisfy the equations $xyz = 1,$ $x + \frac {1}{z} = 5,$ and $y + \frac {1}{x} = 29.$ Then $z + \frac {1}{y} = \frac {m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Problem 8

A container in the shape of a right circular cone is 12 inches tall and its base has a 5-inch radius. The liquid that is sealed inside is 9 inches deep when the cone is held with its point down and its base horizontal. When the liquid is held with its point up and its base horizontal, the height of the liquid is $m - n\sqrt [3]{p},$ where $m,$ $n,$ and $p$ are positive integers and $p$ is not divisible by the cube of any prime number. Find $m + n + p$.

Problem 9

The system of equations\begin{eqnarray*}\log_{10}(2000xy) - (\log_{10}x)(\log_{10}y) & = & 4 \\ \log_{10}(2yz) - (\log_{10}y)(\log_{10}z) & = & 1 \\ \log_{10}(zx) - (\log_{10}z)(\log_{10}x) & = & 0 \\ \end{eqnarray*}

has two solutions $(x_{1},y_{1},z_{1})$ and $(x_{2},y_{2},z_{2})$. Find $y_{1} + y_{2}$.

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2001年AIME II卷真题及答案

2001年AIME II 真题：

Problem 1

Let $N$ be the largest positive integer with the following property: reading from left to right, each pair of consecutive digits of $N$ forms a perfect square. What are the leftmost three digits of $N$?

Problem 2

Each of the 2001 students at a high school studies either Spanish or French, and some study both. The number who study Spanish is between 80 percent and 85 percent of the school population, and the number who study French is between 30 percent and 40 percent. Let $m$ be the smallest number of students who could study both languages, and let $M$ be the largest number of students who could study both languages. Find $M-m$.

Problem 3

Given that\begin{align*}x_{1}&=211,\\ x_{2}&=375,\\ x_{3}&=420,\\ x_{4}&=523,\ \text{and}\\ x_{n}&=x_{n-1}-x_{n-2}+x_{n-3}-x_{n-4}\ \text{when}\ n\geq5, \end{align*}find the value of $x_{531}+x_{753}+x_{975}$.

Problem 4

Let $R = (8,6)$. The lines whose equations are $8y = 15x$ and $10y = 3x$ contain points $P$ and $Q$, respectively, such that $R$ is the midpoint of $\overline{PQ}$. The length of $PQ$ equals $\frac {m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Problem 5

A set of positive numbers has the $triangle~property$ if it has three distinct elements that are the lengths of the sides of a triangle whose area is positive. Consider sets $\{4, 5, 6, \ldots, n\}$ of consecutive positive integers, all of whose ten-element subsets have the triangle property. What is the largest possible value of $n$?

Problem 6

Square $ABCD$ is inscribed in a circle. Square $EFGH$ has vertices $E$ and $F$ on $\overline{CD}$ and vertices $G$ and $H$ on the circle. The ratio of the area of square $EFGH$ to the area of square $ABCD$ can be expressed as $\frac {m}{n}$ where $m$ and $n$ are relatively prime positive integers and $m < n$. Find $10n + m$.

Problem 7

Let $\triangle{PQR}$ be a right triangle with $PQ = 90$, $PR = 120$, and $QR = 150$. Let $C_{1}$ be the inscribed circle. Construct $\overline{ST}$ with $S$ on $\overline{PR}$ and $T$ on $\overline{QR}$, such that $\overline{ST}$ is perpendicular to $\overline{PR}$ and tangent to $C_{1}$. Construct $\overline{UV}$ with $U$ on $\overline{PQ}$ and $V$ on $\overline{QR}$ such that $\overline{UV}$ is perpendicular to $\overline{PQ}$ and tangent to $C_{1}$. Let $C_{2}$ be the inscribed circle of $\triangle{RST}$ and $C_{3}$ the inscribed circle of $\triangle{QUV}$. The distance between the centers of $C_{2}$ and $C_{3}$ can be written as $\sqrt {10n}$. What is $n$?

Problem 8

A certain function $f$ has the properties that $f(3x) = 3f(x)$ for all positive real values of $x$, and that $f(x) = 1 - |x - 2|$ for $1\leq x \leq 3$. Find the smallest $x$ for which $f(x) = f(2001)$.

Problem 9

Each unit square of a 3-by-3 unit-square grid is to be colored either blue or red. For each square, either color is equally likely to be used. The probability of obtaining a grid that does not have a 2-by-2 red square is $\frac {m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

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2001年AIME I卷真题及答案

2001年AIME I 真题：

Problem 1

Find the sum of all positive two-digit integers that are divisible by each of their digits.

Problem 2

A finite set $\mathcal{S}$ of distinct real numbers has the following properties: the mean of $\mathcal{S}\cup\{1\}$ is $13$ less than the mean of $\mathcal{S}$, and the mean of $\mathcal{S}\cup\{2001\}$ is $27$ more than the mean of $\mathcal{S}$. Find the mean of $\mathcal{S}$.

Problem 3

Find the sum of the roots, real and non-real, of the equation $x^{2001}+\left(\frac 12-x\right)^{2001}=0$, given that there are no multiple roots.

Problem 4

In triangle $ABC$, angles $A$ and $B$ measure $60$ degrees and $45$ degrees, respectively. The bisector of angle $A$ intersects $\overline{BC}$ at $T$, and $AT=24$. The area of triangle $ABC$ can be written in the form $a+b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Find $a+b+c$.

Problem 5

An equilateral triangle is inscribed in the ellipse whose equation is $x^2+4y^2=4$. One vertex of the triangle is $(0,1)$, one altitude is contained in the y-axis, and the length of each side is $\sqrt{\frac mn}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Problem 6

A fair die is rolled four times. The probability that each of the final three rolls is at least as large as the roll preceding it may be expressed in the form $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Problem 7

Triangle $ABC$ has $AB=21$, $AC=22$ and $BC=20$. Points $D$ and $E$ are located on $\overline{AB}$ and $\overline{AC}$, respectively, such that $\overline{DE}$ is parallel to $\overline{BC}$ and contains the center of the inscribed circle of triangle $ABC$. Then $DE=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Problem 8

Call a positive integer $N$ a $\textit{7-10 double}$ if the digits of the base-7 representation of $N$ form a base-10 number that is twice $N$. For example, $51$ is a 7-10 double because its base-7 representation is $102$. What is the largest 7-10 double?

Problem 9

In triangle $ABC$, $AB=13$, $BC=15$ and $CA=17$. Point $D$ is on $\overline{AB}$, $E$ is on $\overline{BC}$, and $F$ is on $\overline{CA}$. Let $AD=p\cdot AB$, $BE=q\cdot BC$, and $CF=r\cdot CA$, where $p$, $q$, and $r$ are positive and satisfy $p+q+r=2/3$ and $p^2+q^2+r^2=2/5$. The ratio of the area of triangle $DEF$ to the area of triangle $ABC$ can be written in the form $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

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2002年AIME II卷真题及答案

2002年AIME II 真题：

Problem 1

Given that $x$ and $y$ are both integers between $100$ and $999$, inclusive; $y$ is the number formed by reversing the digits of $x$; and $z=|x-y|$. How many distinct values of $z$ are possible?

Problem 2

Three vertices of a cube are $P=(7,12,10)$, $Q=(8,8,1)$, and $R=(11,3,9)$. What is the surface area of the cube?

Problem 3

It is given that $\log_{6}a + \log_{6}b + \log_{6}c = 6$, where $a$, $b$, and $c$ are positive integers that form an increasing geometric sequence and $b - a$ is the square of an integer. Find $a + b + c$.

Problem 4

Patio blocks that are hexagons $1$ unit on a side are used to outline a garden by placing the blocks edge to edge with $n$ on each side. The diagram indicates the path of blocks around the garden when $n=5$.

If $n=202$, then the area of the garden enclosed by the path, not including the path itself, is $m\left(\sqrt3/2\right)$ square units, where $m$ is a positive integer. Find the remainder when $m$ is divided by $1000$.

Problem 5

Find the sum of all positive integers $a=2^n3^m$ where $n$ and $m$ are non-negative integers, for which $a^6$ is not a divisor of $6^a$.

Problem 6

Find the integer that is closest to $1000\sum_{n=3}^{10000}\frac1{n^2-4}$.

Problem 7

It is known that, for all positive integers $k$,

$1^2+2^2+3^2+\ldots+k^{2}=\frac{k(k+1)(2k+1)}6$.
Find the smallest positive integer $k$ such that $1^2+2^2+3^2+\ldots+k^2$ is a multiple of $200$.

Problem 8

Find the least positive integer $k$ for which the equation $\left\lfloor\frac{2002}{n}\right\rfloor=k$ has no integer solutions for $n$. (The notation $\lfloor x\rfloor$ means the greatest integer less than or equal to $x$.)

Problem 9

Let $\mathcal{S}$ be the set $\lbrace1,2,3,\ldots,10\rbrace$. Let $n$ be the number of sets of two non-empty disjoint subsets of $\mathcal{S}$. (Disjoint sets are defined as sets that have no common elements.) Find the remainder obtained when $n$ is divided by $1000$.

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2002年AIME I卷真题及答案

2002年AIME I 真题：

Problem 1

Many states use a sequence of three letters followed by a sequence of three digits as their standard license-plate pattern. Given that each three-letter three-digit arrangement is equally likely, the probability that such a license plate will contain at least one palindrome (a three-letter arrangement or a three-digit arrangement that reads the same left-to-right as it does right-to-left) is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Problem 2

The diagram shows twenty congruent circles arranged in three rows and enclosed in a rectangle. The circles are tangent to one another and to the sides of the rectangle as shown in the diagram. The ratio of the longer dimension of the rectangle to the shorter dimension can be written as $\frac{1}{2}\left(\sqrt{p}-q\right)$, where $p$ and $q$ are positive integers. Find $p+q$.

[asy] size(250);real x=sqrt(3); int i; draw(origin--(14,0)--(14,2+2x)--(0,2+2x)--cycle); for(i=0; i<7; i=i+1) { draw(Circle((2*i+1,1), 1)^^Circle((2*i+1,1+2x), 1)); } for(i=0; i<6; i=i+1) { draw(Circle((2*i+2,1+x), 1)); } [/asy]

Problem 3

Jane is 25 years old. Dick is older than Jane. In $n$ years, where $n$ is a positive integer, Dick's age and Jane's age will both be two-digit numbers and will have the property that Jane's age is obtained by interchanging the digits of Dick's age. Let $d$ be Dick's present age. How many ordered pairs of positive integers $(d,n)$ are possible?

Problem 4

Consider the sequence defined by $a_k=\frac 1{k^2+k}$ for $k\ge 1$. Given that $a_m+a_{m+1}+\cdots+a_{n-1}=1/29$, for positive integers $m$ and $n$ with $m<n$, find $m+n$.

Problem 5

Let $A_1, A_2, A_3, \ldots, A_{12}$ be the vertices of a regular dodecagon. How many distinct squares in the plane of the dodecagon have at least two vertices in the set $\{A_1,A_2,A_3,\ldots,A_{12}\}$?

Problem 6

The solutions to the system of equations

\begin{align*} \log_{225}{x}+\log_{64}{y} = 4\\ \log_{x}{225}- \log_{y}{64} = 1 \end{align*}are $(x_1,y_1)$ and $(x_2, y_2)$. Find $\log_{30}{(x_1y_1x_2y_2)}$.

Problem 7

The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers $x$, $y$, and $r$ with $|x|>|y|$,

$(x+y)^r=x^r+rx^{r-1}y^1+\frac{r(r-1)}2x^{r-2}y^2+\frac{r(r-1)(r-2)}{3!}x^{r-3}y^3+\cdots$
What are the first three digits to the right of the decimal point in the decimal representation of $\left(10^{2002}+1\right)^{10/7}$?

Problem 8

Find the smallest integer $k$ for which the conditions

(1) $a_1, a_2, a_3, \ldots$ is a nondecreasing sequence of positive integers

(2) $a_n=a_{n-1}+a_{n-2}$ for all $n>2$

(3) $a_9=k$

are satisfied by more than one sequence.

Problem 9

Harold, Tanya, and Ulysses paint a very long picket fence. Harold starts with the first picket and paints every $h$th picket; Tanya starts with the second picket and paints every $t$th picket; and Ulysses starts with the third picket and paints every $u$th picket. Call the positive integer $100h+10t+u$ $\textit{paintable}$ when the triple $(h,t,u)$ of positive integers results in every picket being painted exactly once. Find the sum of all the paintable integers.

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2003年AIME II卷真题及答案

2003年AIME II 真题：

Problem 1

The product $N$ of three positive integers is 6 times their sum, and one of the integers is the sum of the other two. Find the sum of all possible values of $N$.

Problem 2

Let $N$ be the greatest integer multiple of 8, whose digits are all different. What is the remainder when $N$ is divided by 1000?

Problem 3

Define a $\text{good word}$ as a sequence of letters that consists only of the letters $A$, $B$, and $C$ - some of these letters may not appear in the sequence - and in which $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, and $C$ is never immediately followed by $A$. How many seven-letter good words are there?

Problem 4

In a regular tetrahedron, the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to that of the larger is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Problem 5

A cylindrical log has diameter $12$ inches. A wedge is cut from the log by making two planar cuts that go entirely through the log. The first is perpendicular to the axis of the cylinder, and the plane of the second cut forms a $45^\circ$ angle with the plane of the first cut. The intersection of these two planes has exactly one point in common with the log. The number of cubic inches in the wedge can be expressed as $n\pi$, where n is a positive integer. Find $n$.

Problem 6

In triangle $ABC,$ $AB = 13,$ $BC = 14,$ $AC = 15,$ and point $G$ is the intersection of the medians. Points $A',$ $B',$ and $C',$ are the images of $A,$ $B,$ and $C,$ respectively, after a $180^\circ$ rotation about $G.$ What is the area of the union of the two regions enclosed by the triangles $ABC$ and $A'B'C'?$

Problem 7

Find the area of rhombus $ABCD$ given that the radii of the circles circumscribed around triangles $ABD$ and $ACD$ are $12.5$ and $25$, respectively.

Problem 8

Find the eighth term of the sequence $1440,$ $1716,$ $1848,\ldots,$ whose terms are formed by multiplying the corresponding terms of two arithmetic sequences.

Problem 9

Consider the polynomials $P(x) = x^{6} - x^{5} - x^{3} - x^{2} - x$ and $Q(x) = x^{4} - x^{3} - x^{2} - 1.$ Given that $z_{1},z_{2},z_{3},$ and $z_{4}$ are the roots of $Q(x) = 0,$ find $P(z_{1}) + P(z_{2}) + P(z_{3}) + P(z_{4}).$

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2003年AIME I卷真题及答案

2003年AIME I 真题：

Problem 1

Given that

$\frac{((3!)!)!}{3!} = k \cdot n!,$
where $k$ and $n$ are positive integers and $n$ is as large as possible, find $k + n.$

Problem 2

One hundred concentric circles with radii $1, 2, 3, \dots, 100$ are drawn in a plane. The interior of the circle of radius 1 is colored red, and each region bounded by consecutive circles is colored either red or green, with no two adjacent regions the same color. The ratio of the total area of the green regions to the area of the circle of radius 100 can be expressed as $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Problem 3

Let the set $\mathcal{S} = \{8, 5, 1, 13, 34, 3, 21, 2\}.$ Susan makes a list as follows: for each two-element subset of $\mathcal{S},$ she writes on her list the greater of the set's two elements. Find the sum of the numbers on the list.

Problem 4

Given that $\log_{10} \sin x + \log_{10} \cos x = -1$ and that $\log_{10} (\sin x + \cos x) = \frac{1}{2} (\log_{10} n - 1),$ find $n.$

Problem 5

Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures 3 by 4 by 5 units. Given that the volume of this set is $\frac{m + n \pi}{p},$ where $m, n,$ and $p$ are positive integers, and $n$ and $p$ are relatively prime, find $m + n + p.$

Problem 6

The sum of the areas of all triangles whose vertices are also vertices of a 1 by 1 by 1 cube is $m + \sqrt{n} + \sqrt{p},$ where $m, n,$ and $p$ are integers. Find $m + n + p.$

Problem 7

Point $B$ is on $\overline{AC}$ with $AB = 9$ and $BC = 21.$ Point $D$ is not on $\overline{AC}$ so that $AD = CD,$ and $AD$ and $BD$ are integers. Let $s$ be the sum of all possible perimeters of $\triangle ACD.$ Find $s.$

Problem 8

In an increasing sequence of four positive integers, the first three terms form an arithmetic progression, the last three terms form a geometric progression, and the first and fourth terms differ by $30.$ Find the sum of the four terms.

Problem 9

An integer between $1000$ and $9999,$ inclusive, is called balanced if the sum of its two leftmost digits equals the sum of its two rightmost digits. How many balanced integers are there?

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2004年AIME II卷真题及答案

2004年AIME II 真题：

Problem 1

A chord of a circle is perpendicular to a radius at the midpoint of the radius. The ratio of the area of the larger of the two regions into which the chord divides the circle to the smaller can be expressed in the form $\frac{a\pi+b\sqrt{c}}{d\pi-e\sqrt{f}},$ where $a, b, c, d, e,$ and $f$ are positive integers, $a$ and $e$ are relatively prime, and neither $c$ nor $f$ is divisible by the square of any prime. Find the remainder when the product $abcdef$ is divided by 1000.

Problem 2

A jar has 10 red candies and 10 blue candies. Terry picks two candies at random, then Mary picks two of the remaining candies at random. Given that the probability that they get the same color combination, irrespective of order, is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$

Problem 3

A solid rectangular block is formed by gluing together $N$ congruent 1-cm cubes face to face. When the block is viewed so that three of its faces are visible, exactly 231 of the 1-cm cubes cannot be seen. Find the smallest possible value of $N.$

Problem 4

How many positive integers less than 10,000 have at most two different digits?

Problem 5

In order to complete a large job, 1000 workers were hired, just enough to complete the job on schedule. All the workers stayed on the job while the first quarter of the work was done, so the first quarter of the work was completed on schedule. Then 100 workers were laid off, so the second quarter of the work was completed behind schedule. Then an additional 100 workers were laid off, so the third quarter of the work was completed still further behind schedule. Given that all workers work at the same rate, what is the minimum number of additional workers, beyond the 800 workers still on the job at the end of the third quarter, that must be hired after three-quarters of the work has been completed so that the entire project can be completed on schedule or before?

Problem 6

Three clever monkeys divide a pile of bananas. The first monkey takes some bananas from the pile, keeps three-fourths of them, and divides the rest equally between the other two. The second monkey takes some bananas from the pile, keeps one-fourth of them, and divides the rest equally between the other two. The third monkey takes the remaining bananas from the pile, keeps one-twelfth of them, and divides the rest equally between the other two. Given that each monkey receives a whole number of bananas whenever the bananas are divided, and the numbers of bananas the first, second, and third monkeys have at the end of the process are in the ratio $3: 2: 1,$ what is the least possible total for the number of bananas?

Problem 7

$ABCD$ is a rectangular sheet of paper that has been folded so that corner $B$ is matched with point $B'$ on edge $AD.$ The crease is $EF,$ where $E$ is on $AB$ and $F$ is on $CD.$ The dimensions $AE=8, BE=17,$ and $CF=3$ are given. The perimeter of rectangle $ABCD$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

[asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, B=(25,0), C=(25,70/3), D=(0,70/3), E=(8,0), F=(22,70/3), Bp=reflect(E,F)*B, Cp=reflect(E,F)*C; draw(F--D--A--E); draw(E--B--C--F, linetype("4 4")); filldraw(E--F--Cp--Bp--cycle, white, black); pair point=( 12.5, 35/3 ); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F)); label("$B^\prime$", Bp, dir(point--Bp)); label("$C^\prime$", Cp, dir(point--Cp));[/asy]

Problem 8

How many positive integer divisors of $2004^{2004}$ are divisible by exactly 2004 positive integers?

Problem 9

A sequence of positive integers with $a_1=1$ and $a_9+a_{10}=646$ is formed so that the first three terms are in geometric progression, the second, third, and fourth terms are in arithmetic progression, and, in general, for all $n\ge1,$ the terms $a_{2n-1}, a_{2n}, a_{2n+1}$ are in geometric progression, and the terms $a_{2n}, a_{2n+1},$ and $a_{2n+2}$ are in arithmetic progression. Let $a_n$ be the greatest term in this sequence that is less than 1000. Find $n+a_n.$

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