amcdrill | AMC8 AMC10 AMC12 Practice

Q1

A scout troop buys 1000 candy bars at a price of five for \$2. They sell all the candy bars at a price of two for \$1. What was their profit, in dollars?

一个童子军部队以5条2美元的价格购买了1000条糖果。他们以2条1美元的价格卖出了所有糖果。他们的利润是多少美元？

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

Q2

A positive number $x$ has the property that $x\%$ of $x$ is 4. What is $x$?

一个正数$x$具有这样的性质：$x$的$x\%$是4。$x$是多少？

(A) 2 2 (B) 4 4 (C) 10 10 (D) 20 20 (E) 40 40

Q3

A gallon of paint is used to paint a room. One third of the paint is used on the first day. On the second day, one third of the remaining paint is used. What fraction of the original amount of paint is available to use on the third day?

一加仑油漆用于粉刷一个房间。第一天用了三分之一的油漆。第二天，用了剩余油漆的三分之一。第三天可用的油漆占原始油漆的比例是多少？

(A) $\frac{1}{10}$ $\frac{1}{10}$ (B) $\frac{1}{9}$ $\frac{1}{9}$ (C) $\frac{1}{3}$ $\frac{1}{3}$ (D) $\frac{4}{9}$ $\frac{4}{9}$ (E) $\frac{5}{9}$ $\frac{5}{9}$

Q4

For real numbers $a$ and $b$, define $a \diamond b = \sqrt{a^2 + b^2}$. What is the value of $(5 \diamond 12) \diamond ((-12) \diamond (-5))$?

对于实数$a$和$b$，定义$a \diamond b = \sqrt{a^2 + b^2}$。求$(5 \diamond 12) \diamond ((-12) \diamond (-5))$的值。

(A) 0 0 (B) $\frac{17}{2}$ $\frac{17}{2}$ (C) 13 13 (D) $13\sqrt{2}$ $13\sqrt{2}$ (E) 26 26

Q5

Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs?

Brianna用她周末工作赚的部分钱买几张同样价格的CD。她用五分之一的钱买了三分之一的CD。她买完所有CD后，还剩多少比例的钱？

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

Q6

At the beginning of the school year, Lisa's goal was to earn an A on at least 80% of her 50 quizzes for the year. She earned an A on 22 of the first 30 quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she earn a grade lower than an A?

在学年初，Lisa的目标是在全年50次小测验中至少获得80%的A。她在前30次小测验中获得了22次A。如果她要实现目标，在剩余的小测验中最多能有多少次获得低于A的成绩？

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

Q7

A circle is inscribed in a square, then a square is inscribed in this circle, and finally, a circle is inscribed in this square. What is the ratio of the area of the smaller circle to the area of the larger square?

一个圆内接于一个正方形，然后一个正方形内接于这个圆，最后一个圆内接于这个正方形。小圆的面积与大正方形的面积之比是多少？

(A) $\frac{\pi}{16}$ $\frac{\pi}{16}$ (B) $\frac{\pi}{8}$ $\frac{\pi}{8}$ (C) $\frac{3\pi}{16}$ $\frac{3\pi}{16}$ (D) $\frac{\pi}{4}$ $\frac{\pi}{4}$ (E) $\frac{\pi}{2}$ $\frac{\pi}{2}$

Q8

An 8-foot by 10-foot floor is tiled with square tiles of size 1 foot by 1 foot. Each tile has a pattern consisting of four white quarter circles of radius 1/2 foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?

一个8英尺乘10英尺的地板用1英尺乘1英尺的正方形瓷砖铺成。每块瓷砖上有四个以瓷砖每个角为中心、半径1/2英尺的白色四分之一圆。瓷砖的其余部分是遮荫的。地板的遮荫面积有多少平方英尺？

(A) $80 - 20\pi$ $80 - 20\pi$ (B) $60 - 10\pi$ $60 - 10\pi$ (C) $80 - 10\pi$ $80 - 10\pi$ (D) $60 + 10\pi$ $60 + 10\pi$ (E) $80 + 10\pi$ $80 + 10\pi$

Q9

One fair die has faces 1, 1, 2, 2, 3, 3 and another has faces 4, 4, 5, 5, 6, 6. The dice are rolled and the numbers on the top faces are added. What is the probability that the sum will be odd?

一个公平骰子有面1, 1, 2, 2, 3, 3，另一个有面4, 4, 5, 5, 6, 6。掷这两个骰子，将上面面的数字相加。和为奇数的概率是多少？

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

Q10

In $\triangle ABC$, we have $AC = BC = 7$ and $AB = 2$. Suppose that $D$ is a point on line $AB$ such that $B$ lies between $A$ and $D$ and $CD = 8$. What is $BD$?

在$\triangle ABC$中，$AC = BC = 7$且$AB = 2$。设$D$是线$AB$上的一点，使得$B$在$A$和$D$之间，且$CD = 8$。$BD$是多少？

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

Q11

The first term of a sequence is 2005. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the 2005th term of the sequence?

一个数列的第一项是 2005。每项之后的项是前一项各位数字立方的和。这个数列的第 2005 项是多少？

(A) 29 29 (B) 55 55 (C) 85 85 (D) 133 133 (E) 250 250

Q12

Twelve fair dice are rolled. What is the probability that the product of the numbers on the top faces is prime?

掷 12 个公平的骰子。顶面数字的乘积是质数的概率是多少？

(A) $(\frac{1}{12})^{12}$ $(\frac{1}{12})^{12}$ (B) $(\frac{1}{6})^{12}$ $(\frac{1}{6})^{12}$ (C) $2(\frac{1}{6})^{11}$ $2(\frac{1}{6})^{11}$ (D) $\frac{5}{2}(\frac{1}{6})^{11}$ $\frac{5}{2}(\frac{1}{6})^{11}$ (E) $(\frac{1}{6})^{10}$ $(\frac{1}{6})^{10}$

Q13

How many numbers between 1 and 2005 are integer multiples of 3 or 4 but not 12?

1 到 2005 之间有多少个数是 3 或 4 的整数倍但不是 12 的整数倍？

(A) 501 501 (B) 668 668 (C) 835 835 (D) 1002 1002 (E) 1169 1169

Q14

Equilateral $\triangle ABC$ has side length 2, $M$ is the midpoint of $\overline{AC}$, and $C$ is the midpoint of $\overline{BD}$. What is the area of $\triangle CDM$?

等边 $\triangle ABC$ 边长为 2，$M$ 是 $\overline{AC}$ 的中点，$C$ 是 $\overline{BD}$ 的中点。求 $\triangle CDM$ 的面积。

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

Q15

An envelope contains eight bills: 2 ones, 2 fives, 2 tens, and 2 twenties. Two bills are drawn at random without replacement. What is the probability that their sum is $20$ or more?

一个信封里有 8 张钞票：2 张 1 元，2 张 5 元，2 张 10 元，2 张 20 元。随机不放回抽取 2 张钞票。它们的和为 20 元或更多的概率是多少？

(A) $\frac{1}{4}$ $\frac{1}{4}$ (B) $\frac{2}{5}$ $\frac{2}{5}$ (C) $\frac{3}{7}$ $\frac{3}{7}$ (D) $\frac{1}{2}$ $\frac{1}{2}$ (E) $\frac{2}{3}$ $\frac{2}{3}$

Q16

The quadratic equation $x^2 + mx + n = 0$ has roots that are twice those of $x^2 + px + m = 0$, and none of $m$, $n$, and $p$ is zero. What is the value of $n/p$?

二次方程 $x^2 + mx + n = 0$ 的根是 $x^2 + px + m = 0$ 的根的两倍，且 $m$、$n$ 和 $p$ 均不为零。$n/p$ 的值是多少？

(A) 1 1 (B) 2 2 (C) 4 4 (D) 8 8 (E) 16 16

Q17

Suppose that $4^a = 5$, $5^b = 6$, $6^c = 7$, and $7^d = 8$. What is $a \cdot b \cdot c \cdot d$?

假设 $4^a = 5$，$5^b = 6$，$6^c = 7$，且 $7^d = 8$。$a \cdot b \cdot c \cdot d$ 是多少？

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

Q18

All of David’s telephone numbers have the form 555–abc–defg, where a, b, c, d, e, f, and g are distinct digits and in increasing order, and none is either 0 or 1. How many different telephone numbers can David have?

David 的所有电话号码形式为 555–abc–defg，其中 a, b, c, d, e, f, g 是不同的数字且按升序排列，且都不为 0 或 1。David 可以有多少个不同的电话号码？

(A) 1 1 (B) 2 2 (C) 7 7 (D) 8 8 (E) 9 9

Q19

On a certain math exam, 10% of the students got 70 points, 25% got 80 points, 20% got 85 points, 15% got 90 points, and the rest got 95 points. What is the difference between the mean and the median score on this exam?

在某次数学考试中，10% 的学生得 70 分，25% 得 80 分，20% 得 85 分，15% 得 90 分，其余得 95 分。这次考试的平均分与中位数的差是多少？

(A) 0 0 (B) 1 1 (C) 2 2 (D) 4 4 (E) 5 5

Q20

What is the average (mean) of all 5-digit numbers that can be formed by using each of the digits 1, 3, 5, 7, and 8 exactly once?

使用数字 1、3、5、7、8 各一次可以形成的全部五位数的平均数（均值）是多少？

(A) 48000 48000 (B) 49999.5 49999.5 (C) 53332.8 53332.8 (D) 55555 55555 (E) 56432.8 56432.8

Q21

Forty slips are placed into a hat, each bearing a number 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10, with each number entered on four slips. Four slips are drawn from the hat at random and without replacement. Let $p$ be the probability that all four slips bear the same number. Let $q$ be the probability that two of the slips bear a number $a$ and the other two bear a number $b \ne a$. What is the value of $q/p$?

帽子里放入40张纸条，每张纸条上写着数字1、2、3、4、5、6、7、8、9或10，每个数字有四张纸条。从帽子里随机无放回抽取四张纸条。设$p$为四张纸条上数字都相同的概率。设$q$为其中两张纸条上数字为$a$，另外两张为$b\ne a$的概率。$q/p$的值是多少？

(A) 162 162 (B) 180 180 (C) 324 324 (D) 360 360 (E) 720 720

Q22

For how many positive integers $n$ less than or equal to 24 is $n!$ evenly divisible by $1 + 2 + \cdots + n$?

对于多少个正整数$n\le 24$，$n!$能被$1 + 2 + \cdots + n$整除？

(A) 8 8 (B) 12 12 (C) 16 16 (D) 17 17 (E) 21 21

Q23

In trapezoid ABCD we have AB parallel to DC, E as the midpoint of BC, and F as the midpoint of DA. The area of ABEF is twice the area of FECD. What is AB/DC?

梯形ABCD中，AB平行的DC，E为BC中点，F为DA中点。区域ABEF的面积是FECD面积的两倍。AB/DC是多少？

(A) 2 2 (B) 3 3 (C) 5 5 (D) 6 6 (E) 8 8

Q24

Let x and y be two-digit integers such that y is obtained by reversing the digits of x. The integers x and y satisfy $x^2 - y^2 = m^2$ for some positive integer m. What is x + y + m?

设$x$和$y$是两位整数，$y$由$x$的数字反转得到。整数$x$和$y$满足$x^2 - y^2 = m^2$，其中$m$为正整数。$x + y + m$是多少？

(A) 88 88 (B) 112 112 (C) 116 116 (D) 144 144 (E) 154 154

Q25

A subset B of the set of integers from 1 to 100, inclusive, has the property that no two elements of B sum to 125. What is the maximum possible number of elements in B?

{1,2,…,100}的一个子集B，具有B中无两个元素和为125的性质。B的最大可能元素个数是多少？

(A) 50 50 (B) 51 51 (C) 62 62 (D) 65 65 (E) 68 68

AMC10 2005 B