amcdrill | AMC8 AMC10 AMC12 Practice

Q1

At Euclid Middle School the mathematics teachers are Miss Germain, Mr. Newton, and Mrs. Young. There are 11 students in Miss Germain's class, 8 students in Mr. Newton's class, and 9 students in Mrs. Young's class taking the AMC 8 Contest this year. How many mathematics students at Euclid Middle School are taking the contest?

在欧几里得中学，数学老师有Germain小姐、Newton先生和Young太太。Germain小姐的班级有11名学生，Newton先生的班级有8名学生，Young太太的班级有9名学生参加今年的AMC 8竞赛。欧几里得中学有多少数学学生参加竞赛？

(A) 26 26 (B) 27 27 (C) 28 28 (D) 29 29 (E) 30 30

Q2

If $a * b = \frac{a \times b}{a+b}$ for $a, b$ positive integers, then what is $5 * 10$?

如果对于正整数$a, b$，定义$a * b = \frac{a \times b}{a+b}$，那么$5 * 10$等于多少？

(A) $\frac{3}{10}$ $\frac{3}{10}$ (B) 1 1 (C) 2 2 (D) $\frac{10}{3}$ $\frac{10}{3}$ (E) 50 50

Q3

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?

图表显示了今年前十个月五加仑汽油的价格。最高价格比最低价格高百分之多少？

(A) 50 50 (B) 62 62 (C) 70 70 (D) 89 89 (E) 100 100

Q4

What is the sum of the mean, median, and mode of the numbers 2, 3, 0, 3, 1, 4, 0, 3?

数字2, 3, 0, 3, 1, 4, 0, 3的均值、中位数和众数的和是多少？

(A) 6.5 6.5 (B) 7 7 (C) 7.5 7.5 (D) 8.5 8.5 (E) 9 9

Q5

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?

Alice需要更换厨房天花板下方10厘米处的灯泡。天花板离地面2.4米。Alice身高1.5米，伸直手臂可达头顶上方46厘米。站在凳子上，她刚好能碰到灯泡。凳子的高度是多少厘米？

(A) 32 32 (B) 34 34 (C) 36 36 (D) 38 38 (E) 40 40

Q6

Which of the following figures has the greatest number of lines of symmetry?

下列图形中，哪一个有最多的对称轴？

(A) equilateral triangle 等边三角形 (B) non-square rhombus 非正方形菱形 (C) non-square rectangle 非正方形矩形 (D) isosceles trapezoid 等腰梯形 (E) square 正方形

Q7

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?

仅使用便士、五分币、十分币和二十五分币，弗雷迪需要最少的硬币数量，才能支付不到一美元的任意金额？

(A) 6 6 (B) 10 10 (C) 15 15 (D) 25 25 (E) 99 99

Q8

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 view 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?

艾米莉骑自行车在一条长直路上行驶时，发现前方半英里处有埃默森在同方向滑旱冰。她超过他后，通过后视镜能看到他，直到他落后她半英里。艾米莉以恒定速度12英里/小时骑行，埃默森以恒定速度8英里/小时滑行。艾米莉能看到埃默森多少分钟？

(A) 6 6 (B) 8 8 (C) 12 12 (D) 15 15 (E) 16 16

Q9

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?

瑞安在一份25题的测试中答对了80%，在一份40题的测试中答对了90%，在一份10题的测试中答对了70%。瑞安全部题目答对的百分比是多少？

(A) 63 63 (B) 75 75 (C) 80 80 (D) 84 84 (E) 86 86

Q10

Six pepperoni circles will exactly fit across the diameter of a 12-inch pizza when placed as shown. If a total of 24 circles of pepperoni are placed on this pizza without overlap, what fraction of the pizza is covered by pepperoni?

六个意大利辣香肠圆片正好能如图所示放置在12英寸比萨的直径上。如果总共放置24个不重叠的辣香肠圆片，比萨被辣香肠覆盖的分数是多少？

(A) 1/2 1/2 (B) 2/3 2/3 (C) 3/4 3/4 (D) 5/6 5/6 (E) 7/8 7/8

Q11

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?

一棵树的顶部比另一棵树的顶部高16英尺。两棵树的高度之比为3:4。较高的树有多高（英尺）？

(A) 48 48 (B) 64 64 (C) 80 80 (D) 96 96 (E) 112 112

Q12

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?

一个大袋子里有500个球，其中80%是红色的，其余是蓝色的。需要移除多少个红球，使得剩下球中75%是红色的？

(A) 25 25 (B) 50 50 (C) 75 75 (D) 100 100 (E) 150 150

Q13

The lengths of the sides of a triangle measured 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?

一个三角形的边长（英寸）是三个连续整数。最短边长是周长的30%。最长边有多长？

(A) 7 7 (B) 8 8 (C) 9 9 (D) 10 10 (E) 11 11

Q14

What is the sum of the prime factors of 2010?

2010的质因数之和是多少？

(A) 67 67 (B) 75 75 (C) 77 77 (D) 201 201 (E) 210 210

Q15

A jar contains five different colors of gum drops: 30% are blue, 20% are brown, 15% are red, 10% are yellow, and the other 30 gum drops are green. If half of the blue gum drops are replaced by brown gum drops, how many of the gum drops will be brown?

一个罐子里有五种不同颜色的软糖：30%是蓝色的，20%是棕色的，15%是红色的，10%是黄色的，其余30个软糖是绿色的。如果将一半的蓝色软糖替换为棕色软糖，那么有多少个软糖是棕色的？

(A) 35 35 (B) 36 36 (C) 42 42 (D) 48 48 (E) 64 64

Q16

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?

一个正方形和一个圆的面积相等。正方形边长与圆半径的比值为多少？

(A) $\sqrt{\frac{\pi}{2}}$ $\sqrt{\frac{\pi}{2}}$ (B) $\sqrt{\pi}$ $\sqrt{\pi}$ (C) $\pi$ $\pi$ (D) $2\pi$ $2\pi$ (E) $\pi^2$ $\pi^2$

Q17

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 $\frac{XQ}{QY}$?

图中所示八边形由 10 个单位正方形组成。$\overline{PQ}$ 下方的部分是一个单位正方形和一个底边长为 5 的三角形。如果 $\overline{PQ}$ 将八边形的面积二等分，则比值 $\frac{XQ}{QY}$ 为多少？

(A) $\frac{2}{5}$ $\frac{2}{5}$ (B) $\frac{1}{2}$ $\frac{1}{2}$ (C) $\frac{3}{5}$ $\frac{3}{5}$ (D) $\frac{2}{3}$ $\frac{2}{3}$ (E) $\frac{3}{4}$ $\frac{3}{4}$

Q18

A decorative window is made up of a rectangle with semicircles on either end. The ratio of $AD$ to $AB$ is $3:2$ and $AB = 30$ inches. What is the ratio of the area of the rectangle to the combined areas of the semicircles?

一个装饰窗户由一个矩形两端各有一个半圆组成。$AD$ 与 $AB$ 的比值为 $3:2$，且 $AB = 30$ 英寸。矩形面积与两个半圆面积之和的比值为多少？

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

Q19

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?

图中两个圆心相同，为点 $C$。弦 $\overline{AD}$ 在点 $B$ 处与内圆相切，$AC = 10$，弦 $\overline{AD}$ 长为 16。两个圆之间的面积是多少？

(A) $36\pi$ $36\pi$ (B) $49\pi$ $49\pi$ (C) $64\pi$ $64\pi$ (D) $81\pi$ $81\pi$ (E) $100\pi$ $100\pi$

Q20

In a room, $\frac{2}{5}$ of all the people are wearing gloves, and $\frac{3}{4}$ of the people are wearing hats. What is the minimum number of people in the room wearing both a hat and gloves?

一间屋子里，$\frac{2}{5}$ 的人戴着手套，$\frac{3}{4}$ 的人戴着帽子。屋子里至少有多少人同时戴着帽子和手套？

(A) 3 3 (B) 5 5 (C) 8 8 (D) 15 15 (E) 20 20

Q21

Hui is an avid reader. She bought a copy of the best-seller *Math is Beautiful*. On the first day, Hui read $\frac{1}{5}$ of the pages plus 12 more, and on the second day she read $\frac{1}{4}$ of the remaining pages, plus 15 pages. On the third day, she read $\frac{1}{3}$ 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?

Hui 是一个狂热的读者。她买了一本畅销书《数学很美》。第一天，Hui 阅读了全书页数的 \frac{1}{5} 加上另外 12 页，第二天她阅读了剩余页数的 \frac{1}{4} 加上 15 页。第三天，她阅读了剩余页数的 \frac{1}{3} 加上 18 页。然后她发现只剩下 62 页没读了，她第二天就把这些读完了。这本书总共有多少页？

(A) 120 120 (B) 180 180 (C) 240 240 (D) 300 300 (E) 360 360

Q22

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?

一个三位数的百位数字比个位数字大 2。把这个三位数的各位数字倒过来组成的新数减去原数，结果的个位数字是多少？

(A) 0 0 (B) 2 2 (C) 4 4 (D) 6 6 (E) 8 8

Q23

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 the circle $O$?

半圆 $POQ$ 和 $ROS$ 都经过圆心 $O$ 的圆 $O$ 的圆心。两个半圆的总面积与圆 $O$ 的面积之比是多少？

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

Q24

What is the correct ordering of the three numbers $10^8$, $5^{12}$, and $2^{24}$?

三个数 $10^8$、$5^{12}$ 和 $2^{24}$ 正确的顺序是？

(A) $2^{24} < 10^8 < 5^{12}$ $2^{24} < 10^8 < 5^{12}$ (B) $2^{24} < 5^{12} < 10^8$ $2^{24} < 5^{12} < 10^8$ (C) $5^{12} < 2^{24} < 10^8$ $5^{12} < 2^{24} < 10^8$ (D) $10^8 < 5^{12} < 2^{24}$ $10^8 < 5^{12} < 2^{24}$ (E) $10^8 < 2^{24} < 5^{12}$ $10^8 < 2^{24} < 5^{12}$

Q25

Every day at school, Jo climbs a flight of 6 stairs. Jo can take stairs 1, 2, or 3 at a time. For example, Jo could climb 3, then 1, then 2 stairs. In how many ways can Jo climb the stairs?

Jo 每天上学要爬一段 6 级楼梯。Jo 可以一次迈 1、2 或 3 级台阶。例如，Jo 可以爬 3 级，然后 1 级，然后 2 级。Jo 有多少种方法爬楼梯？

(A) 13 13 (B) 18 18 (C) 20 20 (D) 22 22 (E) 24 24

AMC8 2010