amcdrill | AMC8 AMC10 AMC12 Practice

Q1

Mary thought of a positive two-digit number. She multiplied it by 3 and added 11. Then she switched the digits of the result, obtaining a number between 71 and 75, inclusive. What was Mary's number?

玛丽想了一个两位正整数。她将其乘以3并加11。然后她交换结果的各位数字，得到一个71到75（包含两端）之间的数。玛丽的数是多少？

(A) 11 11 (B) 12 12 (C) 13 13 (D) 14 14 (E) 15 15

Q2

Sofia ran 5 laps around the 400-meter track at her school. For each lap, she ran the first 100 meters at an average speed of 4 meters per second and the remaining 300 meters at an average speed of 5 meters per second. How much time did Sofia take running the 5 laps?

索菲亚在学校400米跑道上跑了5圈。每圈她前100米以平均速度4米/秒跑，其余300米以平均速度5米/秒跑。索菲亚跑5圈总共用了多少时间？

(A) 5 minutes and 35 seconds 5 分 35 秒 (B) 6 minutes and 40 seconds 6 分 40 秒 (C) 7 minutes and 5 seconds 7 分 5 秒 (D) 7 minutes and 25 seconds 7 分 25 秒 (E) 8 minutes and 10 seconds 8 分 10 秒

Q3

Real numbers $x$, $y$, and $z$ satisfy the inequalities $0 < x < 1$, $-1 < y < 0$, and $1 < z < 2$. Which of the following numbers is necessarily positive?

实数$x$、$y$和$z$满足不等式$0 < x < 1$、$-1 < y < 0$和$1 < z < 2$。下面哪个数必然为正？

(A) $y + x^2$ $y + x^2$ (B) $y + xz$ $y + xz$ (C) $y + y^2$ $y + y^2$ (D) $y + 2y^2$ $y + 2y^2$ (E) $y + z$ $y + z$

Q4

Suppose that $x$ and $y$ are nonzero real numbers such that $$\frac{3x + y}{x - 3y} = -2.$$ What is the value of $\frac{x + 3y}{3x - y}$?

假设$x$和$y$是非零实数，使得$$\frac{3x + y}{x - 3y} = -2.$$$\frac{x + 3y}{3x - y}$的值是多少？

(A) -3 -3 (B) -1 -1 (C) 1 1 (D) 2 2 (E) 3 3

Q5

Camilla had twice as many blueberry jelly beans as cherry jelly beans. After eating 10 pieces of each kind, she now has three times as many blueberry jelly beans as cherry jelly beans. How many blueberry jelly beans did she originally have?

卡米拉蓝莓果冻豆的数量是樱桃果冻豆的两倍。吃掉每种10颗后，她现在蓝莓果冻豆的数量是樱桃果冻豆的三倍。她原来有多少蓝莓果冻豆？

(A) 10 10 (B) 20 20 (C) 30 30 (D) 40 40 (E) 50 50

Q6

What is the largest number of solid 2-in × 2-in × 1-in blocks that can fit in a 3-in × 2-in × 3-in box?

一个 $3$-英寸 $ imes 2$-英寸 $ imes 3$-英寸的盒子中，能放入的最大个数 $2$-英寸 $ imes 2$-英寸 $ imes 1$-英寸的实心方块是多少？

(A) 3 3 (B) 4 4 (C) 5 5 (D) 6 6 (E) 7 7

Q7

Samia set off on her bicycle to visit her friend, traveling at an average speed of 17 kilometers per hour. When she had gone half the distance to her friend’s house, a tire went flat, and she walked the rest of the way at 5 kilometers per hour. In all it took her 44 minutes to reach her friend’s house. In kilometers rounded to the nearest tenth, how far did Samia walk?

Samia 骑自行车以平均时速 17 公里的速度去朋友家。当她走了到朋友家一半的距离时，轮胎爆了，她步行剩余距离，时速 5 公里。总共用了 44 分钟到达朋友家。四舍五入到十分位，Samia 步行了多少公里？

(A) 2.0 2.0 (B) 2.2 2.2 (C) 2.8 2.8 (D) 3.4 3.4 (E) 4.4 4.4

Q8

Points $A(11, 9)$ and $B(2, -3)$ are vertices of $\triangle ABC$ with $AB = AC$. The altitude from $A$ meets the opposite side at $D(-1, 3)$. What are the coordinates of point $C$?

点 $A(11, 9)$ 和 $B(2, -3)$ 是 $\triangle ABC$ 的顶点，且 $AB = AC$。从 $A$ 垂至对边的高与对边交于 $D(-1, 3)$。点 $C$ 的坐标是什么？

(A) (-8, 9) (-8, 9) (B) (-4, 8) (-4, 8) (C) (-4, 9) (-4, 9) (D) (-2, 3) (-2, 3) (E) (-1, 0) (-1, 0)

Q9

A radio program has a quiz consisting of 3 multiple-choice questions, each with 3 choices. A contestant wins if he or she gets 2 or more of the questions right. The contestant answers randomly to each question. What is the probability of winning?

一个广播节目有一个测验，包括 3 个多选题，每个题有 3 个选项。参赛者答对 2 个或更多题则获胜。参赛者对每个题随机作答。获胜的概率是多少？

(A) \dfrac{1}{27} \dfrac{1}{27} (B) \dfrac{1}{9} \dfrac{1}{9} (C) \dfrac{2}{9} \dfrac{2}{9} (D) \dfrac{7}{27} \dfrac{7}{27} (E) \dfrac{1}{2} \dfrac{1}{2}

Q10

The lines with equations $ax - 2y = c$ and $2x + by = -c$ are perpendicular and intersect at $(1, -5)$. What is $c$?

方程 $ax - 2y = c$ 和 $2x + by = -c$ 表示两条垂直且相交于点 $(1, -5)$ 的直线。$c$ 的值是多少？

(A) -13 -13 (B) -8 -8 (C) 2 2 (D) 8 8 (E) 13 13

Q11

At Typico High School, 60% of the students like dancing, and the rest dislike it. Of those who like dancing, 80% say that they like it, and the rest say that they dislike it. Of those who dislike dancing, 90% say that they dislike it, and the rest say that they like it. What fraction of students who say they dislike dancing actually like it?

在Typico高中，60%的学生喜欢跳舞，其余的不喜欢。喜欢跳舞的学生中，80%说他们喜欢，其余说他们不喜欢。不喜欢跳舞的学生中，90%说他们不喜欢，其余说他们喜欢。说不喜欢跳舞的学生中，实际喜欢跳舞的学生分数是多少？

(A) 10% 10% (B) 12% 12% (C) 20% 20% (D) 25% 25% (E) $33\dfrac{1}{3}\%$ $33\dfrac{1}{3}\%$

Q12

Elmer’s new car gets 50% better fuel efficiency, measured in kilometers per liter, than his old car. However, his new car uses diesel fuel, which is 20% more expensive per liter than the gasoline his old car uses. By what percent will Elmer save money if he uses his new car instead of his old car for a long trip?

Elmer的新车燃油效率比旧车高50%，以千米每升计。然而，新车使用柴油，每升价格比旧车用的汽油贵20%。Elmer用新车代替旧车进行长途旅行，能节省多少钱的百分比？

(A) 20% 20% (B) $26 \frac{2}{3}\%$ $26 \frac{2}{3}\%$ (C) $27 \frac{7}{9}\%$ $27 \frac{7}{9}\%$ (D) $33 \frac{1}{3}\%$ $33 \frac{1}{3}\%$ (E) $41 \frac{2}{3}\%$ $41 \frac{2}{3}\%$

Q13

There are 20 students participating in an after-school program offering classes in yoga, bridge, and painting. Each student must take at least one of these three classes, but may take two or all three. There are 10 students taking yoga, 13 taking bridge, and 9 taking painting. There are 9 students taking at least two classes. How many students are taking all three classes?

有20名学生参加课后项目，提供瑜伽、桥牌和绘画课程。每名学生至少选一门课，但可选两门或三门。有10名学生选瑜伽，13名选桥牌，9名选绘画。有9名学生至少选两门课。选三门课的学生有多少名？

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

Q14

An integer $N$ is selected at random in the range $1 \leq N \leq 2020$. What is the probability that the remainder when $N^{16}$ is divided by 5 is 1?

随机选取整数$N$，范围$1 \leq N \leq 2020$。$N^{16}$除以5的余数为1的概率是多少？

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

Q15

Rectangle ABCD has $AB = 3$ and $BC = 4$. Point E is the foot of the perpendicular from B to diagonal AC. What is the area of $\triangle ADE$?

矩形ABCD有$AB = 3$，$BC = 4$。点E是从B到对角线AC的垂足。$ riangle ADE$的面积是多少？

(A) 1 1 (B) \frac{42}{25} \frac{42}{25} (C) \frac{28}{15} \frac{28}{15} (D) 2 2 (E) \frac{54}{25} \frac{54}{25}

Q16

How many of the base-ten numerals for the positive integers less than or equal to 2017 contain the digit 0?

在小于等于2017的正整数的十进制表示中，有多少个包含数字0？

(A) 469 469 (B) 471 471 (C) 475 475 (D) 478 478 (E) 481 481

Q17

Call a positive integer monotonous if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, 3, 23578, and 987620 are monotonous, but 88, 7434, and 23557 are not. How many monotonous positive integers are there?

称一个正整数为单调的，如果它是一位数，或者其数字从左到右阅读时形成严格递增或严格递减序列。例如，3、23578和987620是单调的，但88、7434和23557不是。有多少个单调正整数？

(A) 1024 1024 (B) 1524 1524 (C) 1533 1533 (D) 1536 1536 (E) 2048 2048

Q18

In the figure below, 3 of the 6 disks are to be painted blue, 2 are to be painted red, and 1 is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible?

在下图中，6个圆盘中有3个涂蓝色，2个涂红色，1个涂绿色。通过整个图形的旋转或反射可以得到的两种涂色视为相同。有多少种不同的涂色可能？

(A) 6 6 (B) 8 8 (C) 9 9 (D) 12 12 (E) 15 15

Q19

Let ABC be an equilateral triangle. Extend side AB beyond B to a point B′ so that BB′ = 3AB. Similarly, extend side BC beyond C to a point C′ so that CC′ = 3BC, and extend side CA beyond A to a point A′ so that AA′ = 3CA. What is the ratio of the area of △A′B′C′ to the area of △ABC ?

设ABC为正三角形。将边AB向B外延至点B′，使BB′=3AB。类似地，将边BC向C外延至点C′，使CC′=3BC，将边CA向A外延至点A′，使AA′=3CA。△A′B′C′的面积与△ABC的面积之比是多少？

(A) 9 : 1 9 : 1 (B) 16 : 1 16 : 1 (C) 25 : 1 25 : 1 (D) 36 : 1 36 : 1 (E) 37 : 1 37 : 1

Q20

The number 21! = 51,090,942,171,709,440,000 has over 60,000 positive integer divisors. One of them is chosen at random. What is the probability that it is odd?

数21!=51,090,942,171,709,440,000有超过60,000个正整数因数。其中随机选一个。因数为奇数的概率是多少？

(A) \dfrac{1}{21} \dfrac{1}{21} (B) \dfrac{1}{19} \dfrac{1}{19} (C) \dfrac{1}{18} \dfrac{1}{18} (D) \dfrac{1}{2} \dfrac{1}{2} (E) \dfrac{11}{21} \dfrac{11}{21}

Q21

In △ABC, AB = 6, AC = 8, BC = 10, and D is the midpoint of BC. What is the sum of the radii of the circles inscribed in △ADB and △ADC ?

在△ABC中，AB=6，AC=8，BC=10，D是BC的中点。△ADB和△ADC的内切圆半径之和是多少？

(A) \sqrt{5} \sqrt{5} (B) \dfrac{11}{4} \dfrac{11}{4} (C) 2\sqrt{2} 2\sqrt{2} (D) \dfrac{17}{6} \dfrac{17}{6} (E) 3 3

Q22

The diameter AB of a circle of radius 2 is extended to a point D outside the circle so that BD = 3. Point E is chosen so that ED = 5 and line ED is perpendicular to line AD. Segment AE intersects the circle at a point C between A and E. What is the area of △ABC ?

半径为2的圆的直径AB延长到圆外一点D，使得BD=3。选择点E，使得ED=5且直线ED垂直于直线AD。线段AE与圆相交于A和E之间的点C。△ABC的面积是多少？

(A) \dfrac{120}{37} \dfrac{120}{37} (B) \dfrac{140}{39} \dfrac{140}{39} (C) \dfrac{145}{39} \dfrac{145}{39} (D) \dfrac{140}{37} \dfrac{140}{37} (E) \dfrac{120}{31} \dfrac{120}{31}

Q23

Let $N = 123456789101112 \dots 4344$ be the 79-digit number that is formed by writing the integers from 1 to 44 in order, one after the other. What is the remainder when $N$ is divided by 45?

设$N=123456789101112\dots4344$是由1到44的整数依次写成的79位数。$N$除以45的余数是多少？

(A) 1 1 (B) 4 4 (C) 9 9 (D) 18 18 (E) 44 44

Q24

The vertices of an equilateral triangle lie on the hyperbola $xy = 1$, and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle?

一个正三角形的顶点位于双曲线$xy=1$上，且该双曲线的一个顶点是该三角形的质心。求该三角形面积的平方。

(A) 48 48 (B) 60 60 (C) 108 108 (D) 120 120 (E) 169 169

Q25

Last year Isabella took 7 math tests and received 7 different scores, each an integer between 91 and 100, inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was 95. What was her score on the sixth test?

去年Isabella参加了7次数学测验，得到7个不同的分数，每个分数是91到100之间的整数。每次测验后她注意到其测验平均分是整数。第七次测验的分数是95。她的第六次测验分数是多少？

(A) 92 92 (B) 94 94 (C) 96 96 (D) 98 98 (E) 100 100

AMC10 2017 B