amcdrill | AMC8 AMC10 AMC12 Practice

Q1

The ratio $\frac{2^{2001} \cdot 3^{2003}}{6^{2002}}$ is

比例 $\frac{2^{2001} \cdot 3^{2003}}{6^{2002}}$ 是

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

Q2

For the nonzero numbers $a$, $b$, and $c$, define $(a, b, c) = \frac{abc}{a + b + c}$. Find $(2, 4, 6)$.

对于非零数 $a$、$b$ 和 $c$，定义 $(a, b, c) = \frac{abc}{a + b + c}$。求 $(2, 4, 6)$。

(A) 1 1 (B) 2 2 (C) 4 4 (D) 6 6 (E) 24 24

Q3

The arithmetic mean of the nine numbers in the set $\{9,99,999,9999,\dots ,999999999\}$ is a 9-digit number $M$, all of whose digits are distinct. The number $M$ does not contain the digit

集合 $\{9,99,999,9999,\dots ,999999999\}$ 中九个数的算术平均数是一个 9 位数 $M$，其所有数字均不同。数 $M$ 不包含数字

(A) 0 0 (B) 2 2 (C) 4 4 (D) 6 6 (E) 8 8

Q4

What is the value of $(3x -2)(4x + 1) - \frac{3x -2}{4x + 1}$ when $x = 4$?

当 $x = 4$ 时，$(3x -2)(4x + 1) - \frac{3x -2}{4x + 1}$ 的值是多少？

(A) 0 0 (B) 1 1 (C) 10 10 (D) 11 11 (E) 12 12

Q5

Circles of radius 2 and 3 are externally tangent and are circumscribed by a third circle, as shown in the figure. Find the area of the shaded region.

半径为 2 和 3 的圆外部相切，并被第三个圆外接，如图所示。求阴影区域的面积。

(A) $3\pi$ $3\pi$ (B) $4\pi$ $4\pi$ (C) $6\pi$ $6\pi$ (D) $9\pi$ $9\pi$ (E) $12\pi$ $12\pi$

Q6

For how many positive integers $n$ is $n^2 -3n + 2$ a prime number?

有且仅有几个正整数 $n$ 使得 $n^2 -3n + 2$ 是质数？

(A) none 无 (B) one 一个 (C) two 两个 (D) more than two, but finitely many 有限多个但超过两个 (E) infinitely many 无限多个

Q7

Let $n$ be a positive integer such that $\frac{1}{2} + \frac{1}{3} + \frac{1}{7} + \frac{1}{n}$ is an integer. Which of the following statements is not true:

设 $n$ 是正整数，使得 $\frac{1}{2} + \frac{1}{3} + \frac{1}{7} + \frac{1}{n}$ 是整数。以下哪个陈述不正确：

(A) 2 divides $n$ 2 整除 $n$ (B) 3 divides $n$ 3 整除 $n$ (C) 6 divides $n$ 6 整除 $n$ (D) 7 divides $n$ 7 整除 $n$ (E) $n > 84$ $n > 84$

Q8

Suppose July of year $N$ has five Mondays. Which of the following must occur five times in August of year $N$? (Note: Both months have 31 days.)

假设 $N$ 年的 7 月有五个星期一。那么 $N$ 年的 8 月中以下哪个一定出现五次？（注：两个月都有 31 天。）

(A) Monday 星期一 (B) Tuesday 星期二 (C) Wednesday 星期三 (D) Thursday 星期四 (E) Friday 星期五

Q9

Using the letters A, M, O, S, and U, we can form 120 five-letter “words”. If these “words” are arranged in alphabetical order, then the “word” USAMO occupies position

使用字母 A, M, O, S, 和 U，可以形成 120 个五字母“单词”。如果这些“单词”按字母顺序排列，那么“单词” USAMO 占据的位置是

(A) 112 112 (B) 113 113 (C) 114 114 (D) 115 115 (E) 116 116

Q10

Suppose that $a$ and $b$ are nonzero real numbers, and that the equation $x^2 + ax + b = 0$ has solutions $a$ and $b$. Then the pair $(a, b)$ is

假设 $a$ 和 $b$ 是非零实数，且方程 $x^2 + ax + b = 0$ 的解是 $a$ 和 $b$。则对 $(a, b)$ 是

(A) (-2, 1) (-2, 1) (B) (-1, 2) (-1, 2) (C) (1, -2) (1, -2) (D) (2, -1) (2, -1) (E) (4, 4) (4, 4)

Q11

The product of three consecutive positive integers is 8 times their sum. What is the sum of their squares?

三个连续正整数的乘积是它们和的8倍。它们平方和是多少？

(A) 50 50 (B) 77 77 (C) 110 110 (D) 149 149 (E) 194 194

Q12

For which of the following values of $k$ does the equation $\frac{x-1}{x-2} = \frac{x-k}{x-6}$ have no solution for $x$?

对于下列哪个$k$值，方程$\frac{x-1}{x-2}=\frac{x-k}{x-6}$没有$x$的解？

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

Q13

Find the value(s) of $x$ such that $8xy -12y + 2x -3 = 0$ is true for all values of $y$.

求使得$8xy-12y+2x-3=0$对所有$y$值都成立的$x$值。

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

Q14

The number $256^4 \cdot 64^{25}$ is the square of a positive integer $N$. In decimal representation, the sum of the digits of $N$ is

数$256^4\cdot64^{25}$是一个正整数$N$的平方。在十进制表示中，$N$的各位数字之和是

(A) 7 7 (B) 14 14 (C) 21 21 (D) 28 28 (E) 35 35

Q15

The positive integers $A$, $B$, $A - B$, and $A + B$ are all prime numbers. The sum of these four primes is

正整数$A$、$B$、$A-B$和$A+B$都是素数。这四个素数的和是

(A) even 偶数 (B) divisible by 3 被3整除 (C) divisible by 5 被5整除 (D) divisible by 7 被7整除 (E) prime 素数

Q16

For how many integers $n$ is $\frac{n}{20-n}$ the square of an integer?

有几个整数 $n$ 使得 $\frac{n}{20-n}$ 是一个整数的平方？

(A) 1 1 (B) 2 2 (C) 3 3 (D) 4 4 (E) 10 10

Q17

A regular octagon ABCDEFGH has sides of length two. Find the area of $\triangle ADG$.

一个边长为二的正八边形 ABCDEFGH，求 $\triangle ADG$ 的面积。

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

Q18

Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect?

平面内画四个不同的圆，最多有多少个至少有两个圆相交的点？

(A) 8 8 (B) 9 9 (C) 10 10 (D) 12 12 (E) 16 16

Q19

Suppose that $\{a_n\}$ is an arithmetic sequence with $a_1 + a_2 + \cdots + a_{100} = 100$ and $a_{101} + a_{102} + \cdots + a_{200} = 200$. What is the value of $a_2 - a_1$?

设 $\{a_n\}$ 是等差数列，且 $a_1 + a_2 + \cdots + a_{100} = 100$，$a_{101} + a_{102} + \cdots + a_{200} = 200$。求 $a_2 - a_1$ 的值。

(A) 0.0001 0.0001 (B) 0.001 0.001 (C) 0.01 0.01 (D) 0.1 0.1 (E) 1 1

Q20

Let $a$, $b$, and $c$ be real numbers such that $a -7b + 8c = 4$ and $8a + 4b -c = 7$. Then $a^2 - b^2 + c^2$ is

设 $a$、$b$ 和 $c$ 是实数，使得 $a -7b + 8c = 4$ 和 $8a + 4b -c = 7$。则 $a^2 - b^2 + c^2$ 是

(A) 0 0 (B) 1 1 (C) 4 4 (D) 7 7 (E) 8 8

Q21

Andy’s lawn has twice as much area as Beth’s lawn and three times as much area as Carlos’ lawn. Carlos’ lawn mower cuts half as fast as Beth’s mower and one third as fast as Andy’s mower. If they all start to mow their lawns at the same time, who will finish first?

安迪的草坪面积是贝丝草坪面积的两倍，是卡洛斯草坪面积的三倍。卡洛斯的割草机割草速度是贝丝割草机的一半，是安迪割草机三分之一。他们同时开始修剪草坪，谁会最先完成？

(A) Andy Andy (B) Beth Beth (C) Carlos Carlos (D) Andy and Carlos tie for first. Andy和Carlos并列第一 (E) All three tie. 三者并列

Q22

Let $\triangle XOY$ be a right-angled triangle with $\angle XOY = 90^\circ$. Let $M$ and $N$ be the midpoints of legs $OX$ and $OY$, respectively. Given that $XN = 19$ and $YM = 22$, find $XY$.

设 $\triangle XOY$ 为直角三角形，$\angle XOY = 90^\circ$。$M$ 和 $N$ 分别为腿 $OX$ 和 $OY$ 的中点。已知 $XN = 19$ 和 $YM = 22$，求 $XY$。

(A) 24 24 (B) 26 26 (C) 28 28 (D) 30 30 (E) 32 32

Q23

Let $\{a_k\}$ be a sequence of integers such that $a_1 = 1$ and $a_{m+n} = a_m + a_n + mn$, for all positive integers $m$ and $n$. Then $a_{12}$ is

设 $\{a_k\}$ 为整数序列，满足 $a_1 = 1$ 且 $a_{m+n} = a_m + a_n + mn$，对所有正整数 $m$ 和 $n$ 成立。则 $a_{12}$ 是

(A) 45 45 (B) 56 56 (C) 67 67 (D) 78 78 (E) 89 89

Q24

Riders on a Ferris wheel travel in a circle in a vertical plane. A particular wheel has radius 20 feet and revolves at the constant rate of one revolution per minute. How many seconds does it take a rider to travel from the bottom of the wheel to a point 10 vertical feet above the bottom?

摩天轮上的乘客在垂直平面内沿圆周运动。该摩天轮半径为 20 英尺，每分钟匀速转一圈。乘客从轮底到轮底上方 10 英尺垂直高度处需要多少秒？

(A) 5 5 (B) 6 6 (C) 7.5 7.5 (D) 10 10 (E) 15 15

Q25

When 15 is appended to a list of integers, the mean is increased by 2. When 1 is appended to the enlarged list, the mean of the enlarged list is decreased by 1. How many integers were in the original list?

将 15 添加到整数列表后，均值增加 2。将 1 添加到扩大列表后，扩大列表的均值减少 1。原列表中有多少个整数？

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

AMC10 2002 B