7 sept 2025

初步确定的学习内容&顺序

• 代数：列方程解问题。
• 百分比/比例和具体数字的换算。
• 列方程解比例问题。
• 列方程解行程问题。用相对运动解行程问题。
• 几何：常见角度和比例关系。
• 数论：整除、素因子分解，分数的加减乘除。
• 几何：各种几何体的面积和周长公式，算两次的技巧。
• 几何：面积的拆补计算。
• 几何：勾股定理，相似三角形，平行线。
• 数论：多位数的加减乘除。
• 代数：数列求和
  • 配对
  • 裂项
• 计数：排列组合。
• 数论：比较数字大小。
• 几何：立体几何。

如果哪一部分掌握的比较好/比较差可以随时调节进度。

补充材料 Matériels supplémentaires

Matériels supplémentaires (en anglais) :

• 基础题分类讲解视频：https://www.omegalearn.org/amc8-fundamentals
• 提高题分类讲解视频：https://www.omegalearn.org/amc8-advanced
• 往年真题：https://artofproblemsolving.com/wiki/index.php/AMC_8_Problems_and_Solutions
• 往年真题和模拟题：https://www.isinj.com/mt-amc8/

2010年试题和相关知识点

Source: https://artofproblemsolving.com/wiki/index.php/2010_AMC_8_Problems

标亮的是目前学校还没学的部分。

Problem 1

At Euclid Middle School, the mathematics teachers are Miss Germain, Mr. Newton, and Mrs. Young. There are $11$ students in Mrs. Germain’s class, $8$ students in Mr. Newton’s class, and $9$ students in Mrs. Young’s class taking the AMC 8 this year. How many mathematics students at Euclid Middle School are taking the contest? $\textbf{(A)}\ 26 \qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 29\qquad\textbf{(E)}\ 30$

基础题。 理解问题就会做。

Problem 2

If $a @ b = \frac{a\times b}{a+b}$ for $a,b$ positive integers, then what is $5 @ 10$? $\textbf{(A)}\ \frac{3}{10} \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ \frac{10}{3} \qquad\textbf{(E)}\ 50$

定义了新的运算符号 $@$（比较抽象，以后补充）。

Problem 3

The graph shows the price of five gallons of gasoline during the first ten months of the year. By what percent is the highest price more than the lowest price?

$\textbf{(A)}\ 50 \qquad \textbf{(B)}\ 62 \qquad \textbf{(C)}\ 70 \qquad \textbf{(D)}\ 89 \qquad \textbf{(E)}\ 100$

基础题。 学会看图表。注意横坐标、纵坐标都是什么。

Problem 4

What is the sum of the mean, median, and mode of the numbers $2,3,0,3,1,4,0,3$? $\textbf{(A)}\ 6.5 \qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 7.5\qquad\textbf{(D)}\ 8.5\qquad\textbf{(E)}\ 9$

先把数字从小到大排列！

Problem 5

Alice needs to replace a light bulb located $10$ centimeters below the ceiling in her kitchen. The ceiling is $2.4$ meters above the floor. Alice is $1.5$ meters tall and can reach $46$ centimeters above the top of her head. Standing on a stool, she can just reach the light bulb. What is the height of the stool, in centimeters? $\textbf{(A)}\ 32 \qquad\textbf{(B)}\ 34\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 38\qquad\textbf{(E)}\ 40$

基础题。 画图计算。

Problem 6

Which of the following figures has the greatest number of lines of symmetry? $\textbf{(A)}\ \text{equilateral triangle}$ 等边三角形 $\textbf{(B)}\ \text{non-square rhombus}$ 非正方形的菱形 $\textbf{(C)}\ \text{non-square rectangle}$ 非正方形的长方形 $\textbf{(D)}\ \text{isosceles trapezoid}$ 等腰梯形 $\textbf{(E)}\ \text{square}$ 正方形

基础题。 注意：长方形只有两条对称轴。（不是正方形的话，斜对角线不是对称轴。）

Problem 7

Using only pennies, nickels, dimes, and quarters, what is the smallest number of coins Freddie would need so he could pay any amount of money less than a dollar? 如果只使用一分钱、五分、十分和二十五分的硬币，弗雷迪最少需要多少枚硬币才能支付任何少于一美元的金额？ $\textbf{(A)}\ 6 \qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 25\qquad\textbf{(E)}\ 99$

计数问题。待补充。

也可以自己算算。

Problem 8

As Emily is riding her bicycle on a long straight road, she spots Emerson skating in the same direction $1/2$ mile in front of her. After she passes him, she can see him in her rear mirror until he is $1/2$ mile behind her. Emily rides at a constant rate of $12$ miles per hour, and Emerson skates at a constant rate of $8$ miles per hour. For how many minutes can Emily see Emerson? Emily 在路上骑车。Emerson 往同方向（上课说错了，说成了反方向）骑。 一开始 Emily 看到 Emerson 在前方 $1/2$ 英里处。 过了一会儿，Emily 从后视镜看到 Emerson 在后方 $1/2$ 英里处。 已知 Emily 速度是 $12$ 英里/小时，Emerson 速度是 $8$ 英里/小时。 那么这前后经过了多少分钟？ $\textbf{(A)}\ 6 \qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 16$

行程问题。

列方程。

或者：

Problem 9

Ryan got $80\%$ of the problems correct on a $25$-problem test, $90\%$ on a $40$-problem test, and $70\%$ on a $10$-problem test. What percent of all the problems did Ryan answer correctly? Ryan 做了三套题，第一套题 $25$ 题，答对 $80\%$；第二套题 $40$ 题，答对 $90\%$；第三套题 $25$ 题，答对 $70\%$。答对的题数占总题数的百分之多少？ $\textbf{(A)}\ 64 \qquad\textbf{(B)}\ 75\qquad\textbf{(C)}\ 80\qquad\textbf{(D)}\ 84\qquad\textbf{(E)}\ 86$

比例/百分比问题。

分步考虑：

• 先算：总共有几道题？
• 算算：总共答对几道题？
• 最后再算：答对的题数占总题数的百分之多少？

Problem 10

Six pepperoni circles will exactly fit across the diameter of a $12$-inch pizza when placed. If a total of $24$ circles of pepperoni are placed on this pizza without overlap, what fraction of the pizza is covered by pepperoni? 当在一张直径为 $12$ 英寸的披萨上摆放时，$6$ 片意大利辣香肠刚好能沿着披萨的直径排下。如果在这张披萨上不重叠地放置 $24$ 片辣香肠，那么披萨被辣香肠覆盖的部分占整个披萨的几分之几？ $\textbf{(A)}\ \frac 12 \qquad\textbf{(B)}\ \frac 23 \qquad\textbf{(C)}\ \frac 34 \qquad\textbf{(D)}\ \frac 56 \qquad\textbf{(E)}\ \frac 78$

图形题。 公式：半径为 $r$ 的圆的：

• 直径 $d = 2r$
• 周长 $C = 2 \pi r$.
• 面积 $S = \pi r^2$

意大利辣香肠半径 $= \frac{12/2}{6} = 1$ 英寸。$24$ 的话面积就是 $24 \times \pi \times 1^2 = 24 \pi$。 pizza 的面积是 $\pi (12/2)^2 = 36 \pi$。 百分之几？占比是 $\frac{24 \pi}{36 \pi} = \dots$

Problem 11

The top of one tree is $16$ feet higher than the top of another tree. The heights of the two trees are in the ratio $3:4$. In feet, how tall is the taller tree? $\textbf{(A)}\ 48 \qquad\textbf{(B)}\ 64 \qquad\textbf{(C)}\ 80 \qquad\textbf{(D)}\ 96\qquad\textbf{(E)}\ 112$

比例/百分比问题。

Problem 12

Of the $500$ balls in a large bag, 80% are red and the rest are blue. How many of the red balls must be removed from the bag so that $75\%$ of the remaining balls are red? $\textbf{(A)}\ 25 \qquad\textbf{(B)}\ 50 \qquad\textbf{(C)}\ 75 \qquad\textbf{(D)}\ 100\qquad\textbf{(E)}\ 150$

比例/百分比问题。

列方程。

Problem 13

The lengths of the sides of a triangle in inches are three consecutive integers. The length of the shortest side is $30\%$ of the perimeter. What is the length of the longest side? 三角形三条边边长是连续的三个整数。最短的边长是周长的 $30\%$。那么最长的边长为多少？ $\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11$

Problem 14

What is the sum of the prime factors of $2010$? $2010$ 所有的 facteurs premiers 的 somme 是多少？ $\textbf{(A)}\ 67 \qquad\textbf{(B)}\ 75\qquad\textbf{(C)}\ 77\qquad\textbf{(D)}\ 201\qquad\textbf{(E)}\ 210$

Problem 15

A jar contains five different colors of gumdrops: $30\%$ are blue, $20\%$ are brown, $15\%$ red, $10\%$ yellow, and the other $30$ gumdrops are green. If half of the blue gumdrops are replaced with brown gumdrops, how many gumdrops will be brown? 一个罐子有五种软糖，$30\%$ 蓝，$20\%$ 棕，$15\%$ 红，$10\%$ 黄，剩下的 $30$ 个是绿色的。如果一半的蓝色换成棕色，那么棕色的比例变成了多少？ $\textbf{(A)}\ 35 \qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 64$

比例的概念。

Problem 16

A square and a circle have the same area. What is the ratio of the side length of the square to the radius of the circle? 一个正方形和圆形有相同的面积。那正方形的边长和圆的半径比例是多少？ $\textbf{(A)}\ \frac{\sqrt{\pi}}{2} \qquad\textbf{(B)}\ \sqrt{\pi} \qquad\textbf{(C)}\ \pi \qquad\textbf{(D)}\ 2\pi \qquad\textbf{(E)}\ \pi^{2}$

方程 + 比例的概念。

Problem 17

The diagram shows an octagon consisting of $10$ unit squares. The portion below $\overline{PQ}$ is a unit square and a triangle with base $5$. If $\overline{PQ}$ bisects the area of the octagon, what is the ratio $\frac{XQ}{QY}$? $PQ$ 把十个方块分成两个面积相等的部分。

$\textbf{(A)}\ \frac 25 \qquad \textbf{(B)}\ \frac 12 \qquad \textbf{(C)}\ \frac 35 \qquad \textbf{(D)}\ \frac 23 \qquad \textbf{(E)}\ \frac 34$

面积问题 + 比例的概念。

方法：分步考虑：

• 面积应该是多少？
• 面积和 $YQ$ 长度的关系？
• 求出 $YQ$ 的长度。
• 求出比例 $\frac{XQ}{QY}$。

Problem 18

A decorative window is made up of a rectangle with semicircles on either end. The ratio of $AD$ to $AB$ is $3:2$, and $AB$ is 30 inches. What is the ratio of the area of the rectangle to the combined areas of the semicircles? 长度比例 $\frac{AD}{AB} = \frac{3}{2}$，且 $AB = 30$ 英尺。求长方形的面积和两个半圆形面积总和的比例。

$\textbf{(A)}\ 2:3 \qquad\textbf{(B)}\ 3:2\qquad\textbf{(C)}\ 6:\pi \qquad\textbf{(D)}\ 9: \pi \qquad\textbf{(E)}\ 30 : \pi$

面积公式 + 比例的概念。

Problem 19

The two circles pictured have the same center $C$. Chord $\overline{AD}$ is tangent to the inner circle at $B$, $AC$ is $10$, and chord $\overline{AD}$ has length $16$. What is the area between the two circles?

$\textbf{(A)}\ 36 \pi \qquad\textbf{(B)}\ 49 \pi\qquad\textbf{(C)}\ 64 \pi\qquad\textbf{(D)}\ 81 \pi\qquad\textbf{(E)}\ 100 \pi$

Problem 20

In a room, $2/5$ of the people are wearing gloves, and $3/4$ of the people are wearing hats. What is the minimum number of people in the room wearing both a hat and a glove? 一个房间里 $2/5$ 的人戴手套，$3/4$ 的人戴帽子。房间里至少多少人同时戴帽子和手套？ $\textbf{(A)}\ 3 \qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 20$

比例问题。较难。 提示：房间里的人数被 $5$ 整除，被 $4$ 整除。

Problem 21

Hui is an avid reader. She bought a copy of the best seller Math is Beautiful. On the first day, Hui read $1/5$ of the pages plus $12$ more, and on the second day she read $1/4$ of the remaining pages plus $15$ pages. On the third day she read $1/3$ of the remaining pages plus $18$ pages. She then realized that there were only $62$ pages left to read, which she read the next day. How many pages are in this book? $\textbf{(A)}\ 120 \qquad\textbf{(B)}\ 180\qquad\textbf{(C)}\ 240\qquad\textbf{(D)}\ 300\qquad\textbf{(E)}\ 360$

比例问题。

Problem 22

The hundreds digit of a three-digit number is $2$ more than the units digit. The digits of the three-digit number are reversed, and the result is subtracted from the original three-digit number. What is the units digit of the result? 有个三位数 $abc$，百位数比个位数多 $2$。把三位数倒过来写成 $cba$，那么 $abc - cba$ 的个位数是多少？ $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 8$

考察对多位数整数加减法的掌握。

Problem 23

Semicircles $POQ$ and $ROS$ pass through the center of circle $O$. What is the ratio of the combined areas of the two semicircles to the area of circle $O$?

$\textbf{(A)}\ \frac{\sqrt 2}{4}\qquad\textbf{(B)}\ \frac{1}{2}\qquad\textbf{(C)}\ \frac{2}{\pi}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{\sqrt 2}{2}$

Problem 24

What is the correct ordering of the three numbers, $10^8$, $5^{12}$, and $2^{24}$? $\textbf{(A)}\ 2{}^2{}^4<10^8<5{}^1{}^2$ $\textbf{(B)}\ 2{}^2{}^4<5{}^1{}^2<10^8$ $\textbf{(C)}\ 5{}^1{}^2<2{}^2{}^4<10^8$ $\textbf{(D)}\ 10^8<5{}^1{}^2<2{}^2{}^4$ $\textbf{(E)}\ 10^8<2{}^2{}^4<5{}^1{}^2$

$5^9$ 读作“$5$ puissance $9$”（$5$ 的 $9$ 次方），表示 $9$ 个 $5$ 相乘（即 $5 \times 5 \times 5 \times \cdots \times 5$ 总共 $9$ 个 $5$）。

Problem 25

Everyday at school, Jo climbs a flight of $6$ stairs. Jo can take the stairs $1$, $2$, or $3$ at a time. For example, Jo could climb $3$, then $1$, then $2$. In how many ways can Jo climb the stairs? $\textbf{(A)}\ 13 \qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 22\qquad\textbf{(E)}\ 24$