amcdrill | AMC8 AMC10 AMC12 Practice

Q1

An amusement park has a collection of scale models, with ratio 1:20, of buildings and other sights from around the country. The height of the United States Capitol is 289 feet. What is the height in feet of its replica at this park, rounded to the nearest whole number?

一个游乐园有一系列比例为 1:20 的建筑物和其他景点的比例模型。美国国会大厦的高度是 289 英尺。这个公园里的复制品高度是多少英尺，四舍五入到最近的整数？

(A) 14 14 (B) 15 15 (C) 16 16 (D) 18 18 (E) 20 20

Q2

What is the value of the product $(1 + \frac{1}{1}) \cdot (1 + \frac{1}{2}) \cdot (1 + \frac{1}{3}) \cdot (1 + \frac{1}{4}) \cdot (1 + \frac{1}{5}) \cdot (1 + \frac{1}{6})$?

计算乘积的值 $(1 + \frac{1}{1}) \cdot (1 + \frac{1}{2}) \cdot (1 + \frac{1}{3}) \cdot (1 + \frac{1}{4}) \cdot (1 + \frac{1}{5}) \cdot (1 + \frac{1}{6})$？

(A) $\frac{7}{6}$ $\frac{7}{6}$ (B) $\frac{4}{3}$ $\frac{4}{3}$ (C) $\frac{7}{2}$ $\frac{7}{2}$ (D) 7 7 (E) 8 8

Q3

Students Arn, Bob, Cyd, Dan, Eve, and Fon are arranged in that order in a circle. They start counting: Arn first, then Bob, and so forth. When the number contains a 7 as a digit (such as 47) or is a multiple of 7 that person leaves the circle and the counting continues. Who is the last one present in the circle?

学生 Arn、Bob、Cyd、Dan、Eve 和 Fon 按此顺序围成一圈。他们开始计数：Arn 先，然后 Bob，依此类推。当数字包含数字 7（如 47）或是 7 的倍数时，那个人离开圈子，计数继续。谁是圈子里最后剩下的一个人？

(A) Arn Arn (B) Bob Bob (C) Cyd Cyd (D) Dan Dan (E) Eve Eve

Q4

The twelve-sided figure shown has been drawn on 1 cm × 1 cm graph paper. What is the area of the figure in cm²?

图中所示的十二边形是在 1 cm × 1 cm 方格纸上绘制的。该图形的面积是多少平方厘米？

(A) 12 12 (B) 12.5 12.5 (C) 13 13 (D) 13.5 13.5 (E) 14 14

Q5

What is the value of $1+3+5+\dots+2017+2019-2-4-6-\dots-2016-2018$?

计算 $1+3+5+\dots+2017+2019-2-4-6-\dots-2016-2018$ 的值？

(A) -1010 -1010 (B) -1009 -1009 (C) 1008 1008 (D) 1009 1009 (E) 1010 1010

Q6

On a trip to the beach, Anh traveled 50 miles on the highway and 10 miles on a coastal access road. He drove three times as fast on the highway as on the coastal road. If Anh spent 30 minutes driving on the coastal road, how many minutes did his entire trip take?

在去海滩的旅行中，Anh 在高速公路上行驶了50英里，在沿海通道上行驶了10英里。他在高速公路上的速度是沿海道路速度的三倍。如果Anh在沿海道路上行驶了30分钟，整个旅行总共用了多少分钟？

(A) 50 50 (B) 70 70 (C) 80 80 (D) 90 90 (E) 100 100

Q7

The 5-digit number $2\ 0\ 1\ 8\ 7$ is divisible by 9. What is the remainder when this number is divided by 8?

五位数 $2\ 0\ 1\ 8\ 7$ 能被9整除。这个数除以8的余数是多少？

(A) 1 1 (B) 3 3 (C) 5 5 (D) 6 6 (E) 7 7

Q8

Mr. Garcia asked the members of his health class how many days last week they exercised for at least 30 minutes. The results are summarized in the following bar graph, where the heights of the bars represent the number of students. What was the mean number of days of exercise last week, rounded to the nearest hundredth, reported by the students in Mr. Garcia's class?

Garcia先生询问他的健康课成员上周锻炼至少30分钟的天数。结果总结在下面的条形图中，条的高度表示学生人数。学生们报告的上周锻炼天数的平均数，四舍五入到最近的百分位，是多少？

(A) 3.50 3.50 (B) 3.57 3.57 (C) 4.36 4.36 (D) 4.50 4.50 (E) 5.00 5.00

Q9

Tyler is tiling the floor of his 12 foot by 16 foot living room. He plans to place one-foot by one-foot square tiles to form a border along the edges of the room and to fill in the rest of the floor with two-foot by two-foot square tiles. How many tiles will he use?

Tyler正在为他12英尺乘16英尺的客厅铺瓷砖。他计划沿房间边缘放置一英尺乘一英尺的方形瓷砖作为边框，并用两英尺乘两英尺的方形瓷砖填充其余地板。他将使用多少块瓷砖？

(A) 48 48 (B) 87 87 (C) 91 91 (D) 96 96 (E) 120 120

Q10

The harmonic mean of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic mean of 1, 2, and 4?

一组非零数的调和平均数是这些数的倒数的平均数的倒数。1、2和4的调和平均数是多少？

(A) 3 3 (B) $\frac{7}{12}$ $\frac{7}{12}$ (C) $\frac{12}{7}$ $\frac{12}{7}$ (D) $\frac{7}{4}$ $\frac{7}{4}$ (E) $\frac{7}{3}$ $\frac{7}{3}$

Q11

Abby, Bridget, and four of their classmates will be seated in two rows of three for a group picture, as shown. X X X X X X If the seating positions are assigned randomly, what is the probability that Abby and Bridget are adjacent to each other in the same row or the same column?

Abby、Bridget 和他们的四位同学将坐在两排三列的位置上拍团体照，如图所示。 X X X X X X 如果座位位置是随机分配的，那么Abby和Bridget在同一行或同一列相邻的概率是多少？

(A) $\frac{1}{3}$ $\frac{1}{3}$ (B) $\frac{2}{5}$ $\frac{2}{5}$ (C) $\frac{7}{15}$ $\frac{7}{15}$ (D) $\frac{1}{2}$ $\frac{1}{2}$ (E) $\frac{2}{3}$ $\frac{2}{3}$

Q12

The clock in Sri’s car, which is not accurate, gains time at a constant rate. One day as he begins shopping he notes that his car clock and his watch (which is accurate) both say 12:00 noon. When he is done shopping, his watch says 12:30 and his car clock says 12:35. Later that day, Sri loses his watch. He looks at his car clock and it says 7:00. What is the actual time?

Sri车上的钟不准确，以恒定速率快走。有一天他开始购物时，注意到他的车钟和他的手表（准确的）都显示12:00中午。购物结束后，他的手表显示12:30，车钟显示12:35。那天晚些时候，Sri丢了手表。他看车钟显示7:00，实际时间是几点？

(A) 5:50 5:50 (B) 6:00 6:00 (C) 6:30 6:30 (D) 6:55 6:55 (E) 8:10 8:10

Q13

Laila took five math tests, each worth a maximum of 100 points. Laila’s score on each test was an integer between 0 and 100, inclusive. Laila received the same score on the first four tests, and she received a higher score on the last test. Her average score on the five tests was 82. How many values are possible for Laila’s score on the last test?

Laila参加了五次数学测验，每次满分100分。Laila每次测验的得分是0到100之间的整数。前四次测验得分相同，最后一次得分更高。五次测验的平均分是82。最后一次测验的得分有多少种可能值？

(A) 4 4 (B) 5 5 (C) 9 9 (D) 10 10 (E) 18 18

Q14

Let $N$ be the greatest five-digit number whose digits have a product of 120. What is the sum of the digits of $N$?

设 $N$ 是各位数字乘积为120的最大五位数。$N$ 的各位数字之和是多少？

(A) 15 15 (B) 16 16 (C) 17 17 (D) 18 18 (E) 20 20

Q15

In the diagram below, a diameter of each of the two smaller circles is a radius of the larger circle. If the two smaller circles have a combined area of 1 square unit, then what is the area of the shaded region, in square units?

下图中，两个小圆的直径均为大圆的半径。如果两个小圆的总面积为1平方单位，则阴影区域的面积是多少平方单位？

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

Q16

Professor Chang has nine different language books lined up on a bookshelf: two Arabic, three German, and four Spanish. How many ways are there to arrange the nine books on the shelf keeping the Arabic books together and keeping the Spanish books together?

张教授有九本不同的语言书籍排放在书架上：两本阿拉伯语，三本德语，四本西班牙语。保持阿拉伯语书籍在一起且西班牙语书籍在一起，将这九本书排放在书架上的方法有多少种？

(A) 1440 1440 (B) 2880 2880 (C) 5760 5760 (D) 182,440 182,440 (E) 362,880 362,880

Q17

Bella begins to walk from her house toward her friend Ella's house. At the same time, Ella begins to ride her bicycle toward Bella's house. They each maintain a constant speed, and Ella rides 5 times as fast as Bella walks. The distance between their houses is 2 miles, which is 10,560 feet, and Bella covers $2\frac{1}{2}$ feet with each step. How many steps will Bella take by the time she meets Ella?

贝拉从她家开始向朋友艾拉的家步行。与此同时，艾拉开始骑自行车向贝拉家骑行。她们各自保持恒定速度，艾拉骑行的速度是贝拉步行速度的5倍。两家之间的距离是2英里，即10,560英尺，贝拉每步走 $2\frac{1}{2}$ 英尺。到她与艾拉相遇时，贝拉会走多少步？

(A) 704 704 (B) 845 845 (C) 1056 1056 (D) 1760 1760 (E) 3520 3520

Q18

How many positive factors does 23,232 have?

23,232 有多少个正因数？

(A) 9 9 (B) 12 12 (C) 28 28 (D) 36 36 (E) 42 42

Q19

In a sign pyramid a cell gets a “+” if the two cells below it have the same sign, and it gets a “-” if the two cells below it have different signs. The diagram below illustrates a sign pyramid with four levels. How many possible ways are there to fill the four cells in the bottom row to produce a “+” at the top of the pyramid?

在一个符号金字塔中，如果下方两个单元格符号相同，则上方的单元格为“+”，如果不同则为“-”。下图展示了一个四层的符号金字塔。填充底行四个单元格有多少种可能的方式，使得金字塔顶端为“+”？

(A) 2 2 (B) 4 4 (C) 8 8 (D) 12 12 (E) 16 16

Q20

In $\triangle ABC$, point $E$ is on $\overline{AB}$ with $AE = 1$ and $EB = 2$. Point $D$ is on $\overline{AC}$ so that $\overline{DE} \parallel \overline{BC}$ and point $F$ is on $\overline{BC}$ so that $\overline{EF} \parallel \overline{AC}$. What is the ratio of the area of $\triangle CDEF$ to the area of $\triangle ABC$?

在 $\triangle ABC$ 中，点 $E$ 在 $\overline{AB}$ 上，$AE = 1$，$EB = 2$。点 $D$ 在 $\overline{AC}$ 上，使得 $\overline{DE} \parallel \overline{BC}$，点 $F$ 在 $\overline{BC}$ 上，使得 $\overline{EF} \parallel \overline{AC}$。$ riangle CDEF$ 的面积与 $\triangle ABC$ 的面积之比是多少？

(A) $\frac{4}{9}$ $\frac{4}{9}$ (B) $\frac{1}{2}$ $\frac{1}{2}$ (C) $\frac{5}{9}$ $\frac{5}{9}$ (D) $\frac{3}{5}$ $\frac{3}{5}$ (E) $\frac{2}{3}$ $\frac{2}{3}$

Q21

How many positive three-digit integers have a remainder of 2 when divided by 6, a remainder of 5 when divided by 9, and a remainder of 7 when divided by 11?

有有多少个正的三位数，除以6余2，除以9余5，除以11余7？

(A) 1 1 (B) 2 2 (C) 3 3 (D) 4 4 (E) 5 5

Q22

Point E is the midpoint of side CD in square ABCD, and BE meets diagonal AC at F. The area of quadrilateral AFED is 45. What is the area of ABCD?

在正方形ABCD中，点E是边CD的中点，BE与对角线AC相交于F。四边形AFED的面积是45。ABCD的面积是多少？

(A) 100 100 (B) 108 108 (C) 120 120 (D) 135 135 (E) 144 144

Q23

From a regular octagon, a triangle is formed by connecting three randomly chosen vertices of the octagon. What is the probability that at least one of the sides of the triangle is also a side of the octagon?

从一个正八边形中，通过连接随机选择的三个顶点形成一个三角形。三角形至少有一条边也是八边形的边的概率是多少？

(A) $\frac{2}{7}$ $\frac{2}{7}$ (B) $\frac{5}{42}$ $\frac{5}{42}$ (C) $\frac{11}{14}$ $\frac{11}{14}$ (D) $\frac{5}{7}$ $\frac{5}{7}$ (E) $\frac{6}{7}$ $\frac{6}{7}$

Q24

In the cube ABCDEFGH with opposite vertices C and E, J and I are the midpoints of edges $\overline{FB}$ and $\overline{HD}$, respectively. Let $R$ be the ratio of the area of the cross-section EJCI to the area of one of the faces of the cube. What is $R^2$?

在立方体ABCDEFGH中，相对顶点为C和E，J和I分别是边$\overline{FB}$和$\overline{HD}$的中点。设$R$为截面EJCI的面积与立方体一个面的面积之比。$R^2$是多少？

(A) $\frac{5}{4}$ $\frac{5}{4}$ (B) $\frac{4}{3}$ $\frac{4}{3}$ (C) $\frac{3}{2}$ $\frac{3}{2}$ (D) $\frac{25}{16}$ $\frac{25}{16}$ (E) $\frac{9}{4}$ $\frac{9}{4}$

Q25

How many perfect cubes lie between $2^8 + 1$ and $2^{18} + 1$, inclusive?

有多少个完全立方数位于$2^8 + 1$和$2^{18} + 1$之间（包含边界）？

(A) 4 4 (B) 9 9 (C) 10 10 (D) 57 57 (E) 58 58

AMC8 2018