amcdrill | AMC8 AMC10 AMC12 Practice

Q1

You and five friends need to raise $1500 in donations for a charity, dividing the fundraising equally. How many dollars will each of you need to raise?

你和五个朋友需要为慈善机构筹集1500美元的捐款，平均分摊筹款。每人需要筹集多少美元？

(A) 250 250 (B) 300 300 (C) 1500 1500 (D) 7500 7500 (E) 9000 9000

Q2

For any three real numbers $a$, $b$, and $c$, with $b \neq c$, the operation $\circledcirc$ is defined by $$\circledcirc(a, b, c) = \frac{a}{b - c}.$$ What is $\circledcirc (\circledcirc(1, 2, 3), \circledcirc(2, 3, 1), \circledcirc(3, 1, 2))$?

对于任意三个实数$a$、$b$和$c$，其中$b \neq c$，操作$\circledcirc$定义为$$\circledcirc(a, b, c) = \frac{a}{b - c}。$$什么是$\circledcirc (\circledcirc(1, 2, 3), \circledcirc(2, 3, 1), \circledcirc(3, 1, 2))$？

(A) -1/2 -$\frac{1}{2}$ (B) -1/4 -$\frac{1}{4}$ (C) 0 0 (D) 1/4 $\frac{1}{4}$ (E) 1/2 $\frac{1}{2}$

Q3

Alicia earns $20 per hour, of which 1.45% is deducted to pay local taxes. How many cents per hour of Alicia’s wages are used to pay local taxes?

Alicia每小时赚20美元，其中1.45%被扣除用于支付地方税。Alicia每小时的工资有多少美分用于支付地方税？

(A) 0.0029 0.0029 (B) 0.029 0.029 (C) 0.29 0.29 (D) 2.9 2.9 (E) 29 29

Q4

What is the value of $x$ if $|x-1| = |x-2|$?

如果$|x-1| = |x-2|$，$x$的值是多少？

(A) -1/2 -$\frac{1}{2}$ (B) 1/2 $\frac{1}{2}$ (C) 1 1 (D) 3/2 $\frac{3}{2}$ (E) 2 2

Q5

A set of three points is chosen randomly from the grid shown. Each three-point set has the same probability of being chosen. What is the probability that the points lie on the same straight line?

从所示网格中随机选择三个点。每个三点集被选择的概率相同。点位于同一条直线上的概率是多少？

(A) 1/21 $\frac{1}{21}$ (B) 1/14 $\frac{1}{14}$ (C) 2/21 $\frac{2}{21}$ (D) 1/7 $\frac{1}{7}$ (E) 2/7 $\frac{2}{7}$

Q6

Bertha has 6 daughters and no sons. Some of her daughters have 6 daughters, and the rest have none. Bertha has a total of 30 daughters and granddaughters, and no great-granddaughters. How many of Bertha’s daughters and granddaughters have no daughters?

Bertha 有 6 个女儿，没有儿子。她的一些女儿有 6 个女儿，其余的没有。Bertha 总共有 30 个女儿和孙女，没有曾孙女。Bertha 的女儿和孙女中，有多少人没有女儿？

(A) 22 22 (B) 23 23 (C) 24 24 (D) 25 25 (E) 26 26

Q7

A grocer stacks oranges in a pyramid-like stack whose rectangular base is 5 oranges by 8 oranges. Each orange above the first level rests in a pocket formed by four oranges in the level below. The stack is completed by a single row of oranges. How many oranges are in the stack?

一个杂货商将橙子堆成金字塔状堆，矩形底面是 5 排橙子乘 8 排橙子。每层以上的橙子都搁在下面一层四个橙子形成的凹槽中。堆栈由一行橙子完成。堆栈中共有多少个橙子？

(A) 96 96 (B) 98 98 (C) 100 100 (D) 101 101 (E) 134 134

Q8

A game is played with tokens according to the following rule. In each round, the player with the most tokens gives one token to each of the other players and also places one token into a discard pile. The game ends when some player runs out of tokens. Players A, B, and C start with 15, 14, and 13 tokens, respectively. How many rounds will there be in the game?

一个游戏使用筹码，按照以下规则进行。每轮中，筹码最多的玩家给每个其他玩家一个筹码，并将一个筹码放入弃牌堆。游戏在某个玩家筹码用尽时结束。玩家 A、B 和 C 分别起始有 15、14 和 13 个筹码。游戏将进行多少轮？

(A) 36 36 (B) 37 37 (C) 38 38 (D) 39 39 (E) 40 40

Q9

In the figure, $\angle EAB$ and $\angle ABC$ are right angles, $AB = 4$, $BC = 6$, $AE = 8$, and $AC$ and $BE$ intersect at $D$. What is the difference between the areas of $\triangle ADE$ and $\triangle BDC$?

在图中，$\angle EAB$ 和 $\angle ABC$ 是直角，$AB = 4$，$BC = 6$，$AE = 8$，且 $AC$ 和 $BE$ 相交于 $D$。$\triangle ADE$ 和 $\triangle BDC$ 的面积差是多少？

(A) 2 2 (B) 4 4 (C) 5 5 (D) 8 8 (E) 9 9

Q10

Coin A is flipped three times and coin B is flipped four times. What is the probability that the number of heads obtained from flipping the two fair coins is the same?

硬币 A 抛三次，硬币 B 抛四次。抛两个公平硬币得到的正面次数相同的概率是多少？

(A) 19/128 $\frac{19}{128}$ (B) 23/128 $\frac{23}{128}$ (C) 1/4 $\frac{1}{4}$ (D) 35/128 $\frac{35}{128}$ (E) 1/2 $\frac{1}{2}$

Q11

A company sells peanut butter in cylindrical jars. Marketing research suggests that using wider jars will increase sales. If the diameter of the jars is increased by 25% without altering the volume, by what percent must the height be decreased?

一家公司用圆柱形罐子出售花生酱。市场研究表明，使用更宽的罐子会增加销量。如果罐子的直径增加25%，体积不变，则高度必须减少百分之多少？

(A) 10 10 (B) 25 25 (C) 36 36 (D) 50 50 (E) 60 60

Q12

Henry’s Hamburger Heaven offers its hamburgers with the following condiments: ketchup, mustard, mayonnaise, tomato, lettuce, pickles, cheese, and onions. A customer can choose one, two, or three meat patties, and any collection of condiments. How many different kinds of hamburgers can be ordered?

Henry的汉堡天堂提供汉堡时有以下调味品：番茄酱、芥末、美乃滋、西红柿、生菜、泡菜、奶酪和洋葱。顾客可以选择一个、两个或三个肉饼，以及任意组合的调味品。有多少种不同的汉堡可以点？

(A) 24 24 (B) 256 256 (C) 768 768 (D) 40320 40320 (E) 120960 120960

Q13

At a party, each man danced with exactly three women and each woman danced with exactly two men. Twelve men attended the party. How many women attended the party?

在一次派对上，每个男人与恰好三个女人跳舞，每个女人与恰好两个男人跳舞。有12个男人参加派对。有多少女人参加派对？

(A) 8 8 (B) 12 12 (C) 16 16 (D) 18 18 (E) 24 24

Q14

The average value of all the pennies, nickels, dimes, and quarters in Paula’s purse is 20 cents. If she had one more quarter, the average value would be 21 cents. How many dimes does she have in her purse?

Paula钱包里所有一分币、五分币、十分币和二十五分币的平均价值是20美分。如果她再多一枚二十五分币，平均价值将变为21美分。她钱包里有多少十分币？

(A) 0 0 (B) 1 1 (C) 2 2 (D) 3 3 (E) 4 4

Q15

Given that $-4 \leq x \leq -2$ and $2 \leq y \leq 4$, what is the largest possible value of $\frac{x + y}{x}$?

已知$-4\leq x\leq -2$且$2\leq y\leq 4$，求$\frac{x+y}{x}$的最大可能值？

(A) -1 -1 (B) -1/2 -$\frac{1}{2}$ (C) 0 0 (D) 1/2 $\frac{1}{2}$ (E) 1 1

Q16

The $5 \times 5$ grid shown contains a collection of squares with sizes from $1 \times 1$ to $5 \times 5$. How many of these squares contain the black center square?

如图所示的 $5 \times 5$ 网格包含从 $1 \times 1$ 到 $5 \times 5$ 各种大小的正方形。其中有多少个正方形包含黑色中心方块？

(A) 12 12 (B) 15 15 (C) 17 17 (D) 19 19 (E) 20 20

Q17

Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run 100 meters. They next meet after Sally has run 150 meters past their first meeting point. Each girl runs at a constant speed. What is the length of the track in meters?

Brenda 和 Sally 在一个圆形跑道上朝相反方向跑步，从直径对称的两个点开始。她们第一次相遇时，Brenda 已跑了 100 米。她们第二次相遇时，Sally 已从第一次相遇点多跑了 150 米。每个女孩都以恒定速度跑步。跑道的长度是多少米？

(A) 250 250 (B) 300 300 (C) 350 350 (D) 400 400 (E) 500 500

Q18

A sequence of three real numbers forms an arithmetic progression with a first term of 9. If 2 is added to the second term and 20 is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term of the geometric progression?

三个实数的等差数列，第一项为 9。如果向第二项加 2，向第三项加 20，则所得三个数形成等比数列。几何数列第三项的最小可能值为多少？

(A) 1 1 (B) 4 4 (C) 36 36 (D) 49 49 (E) 81 81

Q19

A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet?

一个白色圆柱形筒仓直径为 30 英尺，高为 80 英尺。如图所示，筒仓上画有一条水平宽度为 3 英尺的红色条纹，绕筒仓绕了两整圈。条纹的面积是多少平方英尺？

(A) 120 120 (B) 180 180 (C) 240 240 (D) 360 360 (E) 480 480

Q20

Points E and F are located on square ABCD so that $\triangle BEF$ is equilateral. What is the ratio of the area of $\triangle DEF$ to that of $\triangle ABE$?

点 E 和 F 位于正方形 ABCD 上，使得 $\triangle BEF$ 为等边三角形。$\triangle DEF$ 的面积与 $\triangle ABE$ 的面积之比是多少？

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

Q21

Two distinct lines pass through the center of three concentric circles of radii 3, 2, and 1. The area of the shaded region in the diagram is $8/13$ of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines? (Note: $\pi$ radians is 180 degrees.)

两条不同的直线穿过三个半径分别为3、2和1的同心圆的圆心。图中阴影区域的面积是不阴影区域面积的$8/13$。这两条直线形成的锐角的弧度量是多少？（注：$\pi$弧度等于180度。）

(A) $\pi/8$ $\pi/8$ (B) $\pi/7$ $\pi/7$ (C) $\pi/6$ $\pi/6$ (D) $\pi/5$ $\pi/5$ (E) $\pi/4$ $\pi/4$

Q22

Square ABCD has side length 2. A semicircle with diameter AB is constructed inside the square, and the tangent to the semicircle from C intersects side AD at E. What is the length of CE?

正方形ABCD边长为2。以AB为直径在正方形内作半圆，从C引半圆的切线与边AD交于点E。求CE的长度。

(A) $(2 + \sqrt{5})/2$ $\frac{2 + \sqrt{5}}{2}$ (B) $\sqrt{5}$ $\sqrt{5}$ (C) $\sqrt{6}$ $\sqrt{6}$ (D) 5/2 $\frac{5}{2}$ (E) $5 - \sqrt{5}$ $5 - \sqrt{5}$

Q23

Circles A, B, and C are externally tangent to each other and internally tangent to circle D. Circles B and C are congruent. Circle A has radius 1 and passes through the center of D. What is the radius of circle B?

圆A、B、C两两外部相切，且都与圆D内部相切。圆B与圆C全等。圆A半径为1且经过圆D的圆心。求圆B的半径。

(A) 2/3 $\frac{2}{3}$ (B) $\sqrt{3}/2$ $\frac{\sqrt{3}}{2}$ (C) 7/8 $\frac{7}{8}$ (D) 8/9 $\frac{8}{9}$ (E) $(1 + \sqrt{3})/3$ $\frac{1 + \sqrt{3}}{3}$

Q24

Let $a_1, a_2, \cdots $, be a sequence with the following properties. (i) $a_1 = 1$, and (ii) $a_{2n} = n \cdot a_n$ for any positive integer $n$. What is the value of $a_{2100}$?

设序列$a_1, a_2, \cdots$满足：(i) $a_1 = 1$，(ii) 对任意正整数$n$，$a_{2n} = n \cdot a_n$。求$a_{2100}$的值。

(A) 1 1 (B) 2^{99} 2^{99} (C) 2^{100} 2^{100} (D) 24950 24950 (E) 2^{9999} 2^{9999}

Q25

Three mutually tangent spheres of radius 1 rest on a horizontal plane. A sphere of radius 2 rests on them. What is the distance from the plane to the top of the larger sphere?

三个半径为1的互切球体放置在水平平面上。一个半径为2的球体放置在它们上面。求该大球顶点到平面的距离。

(A) $3 + \sqrt{30}/2$ $3 + \frac{\sqrt{30}}{2}$ (B) $3 + \sqrt{69}/3$ $3 + \frac{\sqrt{69}}{3}$ (C) $3 + \sqrt{123}/4$ $3 + \frac{\sqrt{123}}{4}$ (D) $52/9$ $\frac{52}{9}$ (E) $3 + 2\sqrt{2}$ $3 + 2\sqrt{2}$

AMC10 2004 A