amcdrill | AMC8 AMC10 AMC12 Practice

Q1

While eating out, Mike and Joe each tipped their server $2. Mike tipped 10% of his bill and Joe tipped 20% of his bill. What was the difference, in dollars, between their bills?

外出就餐时，Mike 和 Joe 各给了服务员 2 美元小费。Mike 的小费是他账单的 10%，Joe 的小费是他账单的 20%。他们的账单差额是多少美元？

(A) 2 2 (B) 4 4 (C) 5 5 (D) 10 10 (E) 20 20

Q2

For each pair of real numbers $a \neq b$, define the operation $\star$ as $(a \star b) = \frac{a+b}{a-b}$. What is the value of $((1 \star 2) \star 3)$?

对于每对实数 $a \neq b$，定义操作 $\star$ 为 $(a \star b) = \frac{a+b}{a-b}$。求 $((1 \star 2) \star 3)$ 的值。

(A) $-\frac{2}{3}$ $-\frac{2}{3}$ (B) $-\frac{1}{5}$ $-\frac{1}{5}$ (C) 0 0 (D) $\frac{1}{2}$ $\frac{1}{2}$ (E) This value is not defined. 此值未定义。

Q3

The equations $2x + 7 = 3$ and $bx -10 = -2$ have the same solution $x$. What is the value of $b$?

方程 $2x + 7 = 3$ 和 $bx -10 = -2$ 有相同的解 $x$。求 $b$ 的值。

(A) -8 -8 (B) -4 -4 (C) -2 -2 (D) 4 4 (E) 8 8

Q4

A rectangle with a diagonal of length $x$ is twice as long as it is wide. What is the area of the rectangle?

一个对角线长度为 $x$ 的矩形，长是宽的两倍。求该矩形的面积。

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

Q5

A store normally sells windows at $100 each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately?

一家商店正常出售窗户每个 100 美元。本周商店提供每购买四个窗户赠送一个免费窗户。Dave 需要七个窗户，Doug 需要八个窗户。如果他们一起购买而不是单独购买，能节省多少美元？

(A) 100 100 (B) 200 200 (C) 300 300 (D) 400 400 (E) 500 500

Q6

The average (mean) of 20 numbers is 30, and the average of 30 other numbers is 20. What is the average of all 50 numbers?

20 个数的平均数（均值）是 30，另外 30 个数的平均数是 20。这 50 个数的平均数是多少？

(A) 23 23 (B) 24 24 (C) 25 25 (D) 26 26 (E) 27 27

Q7

Josh and Mike live 13 miles apart. Yesterday Josh started to ride his bicycle toward Mike’s house. A little later Mike started to ride his bicycle toward Josh’s house. When they met, Josh had ridden for twice the length of time as Mike and at four-fifths of Mike’s rate. How many miles had Mike ridden when they met?

Josh 和 Mike 相距 13 英里。昨天 Josh 开始骑自行车向 Mike 家骑去。稍后 Mike 开始骑自行车向 Josh 家骑去。他们相遇时，Josh 的骑行时间是 Mike 的两倍，且速度是 Mike 速度的四分之五。他们相遇时 Mike 骑了多少英里？

(A) 4 4 (B) 5 5 (C) 6 6 (D) 7 7 (E) 8 8

Q8

In the figure, the length of side AB of square ABCD is $\sqrt{50}$, E is between B and H, and BE = 1. What is the area of the inner square EFGH?

在图中，正方形 ABCD 的边 AB 长为 $\sqrt{50}$，E 在 B 和 H 之间，且 BE = 1。内正方形 EFGH 的面积是多少？

(A) 25 25 (B) 32 32 (C) 36 36 (D) 40 40 (E) 42 42

Q9

Three tiles are marked X and two other tiles are marked O. The five tiles are randomly arranged in a row. What is the probability that the arrangement reads XOXOX?

有 3 块标有 X 的瓦片和 2 块标有 O 的瓦片。5 块瓦片随机排成一排。排列为 XOXOX 的概率是多少？

(A) $\frac{1}{12}$ $\frac{1}{12}$ (B) $\frac{1}{10}$ $\frac{1}{10}$ (C) $\frac{1}{6}$ $\frac{1}{6}$ (D) $\frac{1}{4}$ $\frac{1}{4}$ (E) $\frac{1}{3}$ $\frac{1}{3}$

Q10

There are two values of $a$ for which the equation $4x^2 + ax + 8x + 9 = 0$ has only one solution for $x$. What is the sum of those values of $a$?

方程 $4x^2 + ax + 8x + 9 = 0$ 有唯一 x 解的 a 有两个值。这些 a 的值之和是多少？

(A) -16 -16 (B) -8 -8 (C) 0 0 (D) 8 8 (E) 20 20

Q11

A wooden cube $n$ units on a side is painted red on all six faces and then cut into $n^3$ unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is $n$?

一个边长 $n$ 单位的木制立方体所有六个面都被涂成红色，然后被切成 $n^3$ 个单位立方体。单位立方体总面数中有恰好四分之一是红色的。$n$ 是多少？

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

Q12

The figure shown is called a trefoil and is constructed by drawing circular sectors about sides of the congruent equilateral triangles. What is the area of a trefoil whose horizontal base has length 2?

图中所示图形称为三叶草，由在全等等边三角形的边上绘制圆扇形构成。其水平底边长度为 2 的三叶草的面积是多少？

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

Q13

How many positive integers $n$ satisfy the following condition: $(130n)^{50} > n^{100} > 2^{200}$?

有多少正整数 $n$ 满足条件：$(130n)^{50} > n^{100} > 2^{200}$？

(A) 0 0 (B) 7 7 (C) 12 12 (D) 65 65 (E) 125 125

Q14

How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits?

有多少个三位数满足中间数字是首位和末位数字平均数的性质？

(A) 41 41 (B) 42 42 (C) 43 43 (D) 44 44 (E) 45 45

Q15

How many positive cubes divide $3! \cdot 5! \cdot 7!$?

有多少个正立方数整除 $3! \cdot 5! \cdot 7!$？

(A) 2 2 (B) 3 3 (C) 4 4 (D) 5 5 (E) 6 6

Q16

The sum of the digits of a two-digit number is subtracted from the number. The units digit of the result is 6. How many two-digit numbers have this property?

一个两位数的各位数字之和从该数中减去，结果的个位数是6。有多少个两位数具有这个性质？

(A) 5 5 (B) 7 7 (C) 9 9 (D) 10 10 (E) 19 19

Q17

In the five-sided star shown, the letters A, B, C, D, and E are replaced by the numbers 3, 5, 6, 7, and 9, although not necessarily in this order. The sums of the numbers at the ends of the line segments AB, BC, CD, DE, and EA form an arithmetic sequence, although not necessarily in this order. What is the middle term of the arithmetic sequence?

在所示的五角星中，字母 A、B、C、D 和 E 被数字 3、5、6、7 和 9 替换，虽然不一定按此顺序。线段 AB、BC、CD、DE 和 EA 两端数字之和形成一个等差数列，虽然不一定按此顺序。等差数列的中间项是多少？

(A) 9 9 (B) 10 10 (C) 11 11 (D) 12 12 (E) 13 13

Q18

Team A and team B play a series. The first team to win three games wins the series. Each team is equally likely to win each game, there are no ties, and the outcomes of the individual games are independent. If team B wins the second game and team A wins the series, what is the probability that team B wins the first game?

A队和B队进行一系列比赛，先赢三场的队伍赢得系列赛。每场比赛两队获胜概率相等，无平局，各场比赛结果独立。如果B队赢得第二场比赛且A队赢得系列赛，则B队赢得第一场比赛的概率是多少？

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

Q19

Three one-inch squares are placed with their bases on a line. The center square is lifted out and rotated 45°, as shown. Then it is centered and lowered into its original location until it touches both of the adjoining squares. How many inches is the point B from the line on which the bases of the original squares were placed?

三个一英寸正方形放置，其底边在一根直线上。中间的正方形被取出并旋转45°，如图所示。然后将其置中并降回原位置，直到它接触到相邻的两个正方形。点B离原始正方形底边所在直线的距离有多少英寸？

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

Q20

An equiangular octagon has four sides of length 1 and four sides of length $\sqrt{2}/2$, arranged so that no two consecutive sides have the same length. What is the area of the octagon?

一个等角八边形有四条边长为1，四条边长为 $\sqrt{2}/2$，排列使得没有两条连续边等长。这个八边形的面积是多少？

(A) $\frac{7}{2}$ $\frac{7}{2}$ (B) $\frac{7\sqrt{2}}{2}$ $\frac{7\sqrt{2}}{2}$ (C) $\frac{5+4\sqrt{2}}{2}$ $\frac{5+4\sqrt{2}}{2}$ (D) $\frac{4+5\sqrt{2}}{2}$ $\frac{4+5\sqrt{2}}{2}$ (E) 7 7

Q21

For how many positive integers $n$ does $1 + 2 + \cdots + n$ evenly divide $6n$?

有且仅有几个正整数$n$使得$1 + 2 + \cdots + n$能整除$6n$？

(A) 3 3 (B) 5 5 (C) 7 7 (D) 9 9 (E) 11 11

Q22

Let $S$ be the set of the 2005 smallest positive multiples of 4, and let $T$ be the set of the 2005 smallest positive multiples of 6. How many elements are common to $S$ and $T$?

令$S$为2005个最小的正4的倍数的集合，$T$为2005个最小的正6的倍数的集合。$S$与$T$有多少个公共元素？

(A) 166 166 (B) 333 333 (C) 500 500 (D) 668 668 (E) 1001 1001

Q23

Let $\overline{AB}$ be a diameter of a circle and $C$ be a point on $\overline{AB}$ with $2 \cdot AC = BC$. Let $D$ and $E$ be points on the circle such that $DC \perp AB$ and $DE$ is a second diameter. What is the ratio of the area of $\triangle DCE$ to the area of $\triangle ABD$?

令$\overline{AB}$为圆的直径，$C$为$\overline{AB}$上的点且$2 \cdot AC = BC$。$D,E$为圆上的点使得$DC \perp AB$且$DE$为另一条直径。$ riangle DCE$的面积与$ riangle ABD$的面积之比是多少？

(A) $\frac{1}{6}$ $\frac{1}{6}$ (B) $\frac{1}{4}$ $\frac{1}{4}$ (C) $\frac{1}{3}$ $\frac{1}{3}$ (D) $\frac{1}{2}$ $\frac{1}{2}$ (E) $\frac{2}{3}$ $\frac{2}{3}$

Q24

For each positive integer $m > 1$, let $P(m)$ denote the greatest prime factor of $m$. For how many positive integers $n$ is it true that both $P(n) = \sqrt{n}$ and $P(n + 48) = \sqrt{n} + 48$?

对于每个正整数$m > 1$，令$P(m)$表示$m$的最大质因数。有且仅有几个正整数$n$满足$P(n) = \sqrt{n}$且$P(n + 48) = \sqrt{n} + 48$？

(A) 0 0 (B) 1 1 (C) 3 3 (D) 4 4 (E) 5 5

Q25

In $\triangle ABC$ we have AB = 25, BC = 39, and AC = 42. Points D and E are on AB and AC respectively, with AD = 19 and AE = 14. What is the ratio of the area of triangle ADE to the area of the quadrilateral BCED?

在$ riangle ABC$中，$AB = 25$，$BC = 39$，$AC = 42$。点$D,E$分别在$AB,AC$上，$AD = 19$，$AE = 14$。$ riangle ADE$的面积与四边形$BCED$的面积之比是多少？

(A) $\frac{266}{1521}$ $\frac{266}{1521}$ (B) $\frac{19}{75}$ $\frac{19}{75}$ (C) $\frac{1}{3}$ $\frac{1}{3}$ (D) $\frac{19}{56}$ $\frac{19}{56}$ (E) 1 1

AMC10 2005 A