2004 AMC amc10a 真题 答案 详解

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英文真题

2004 AMC 10A Problems
Problem 1
You and five friends need to raise $1500$ dollars in donations for a charity, dividing the fundraising equally. How many dollars will each of you need to raise?
(A) $250$ (B) $300$ (C) $1500$ (D) $7500$ (E) $9000$

Problem 2
For any three real numbers $a$, $b$, and $c$, with $b \neq c$, the operation $\otimes$ is defined by: $\otimes(a,b,c)=\frac{a}{b-c}$. What is $\otimes(\otimes(1,2,3),\otimes(2,3,1),\otimes(3,1,2))?$
(A) $-\frac12$ (B) $-\frac14$ (C) $0$ (D) $\frac14$ (E) $\frac12$

Problem 3
Alicia earns $20$ dollars 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?
(A) $0.0029$ (B) $0.029$ (C) $0.29$ (D) $2.9$ (E) $29$

Problem 4
What is the value of $x$ if $|x-1|=|x-2|$?
(A) $-\frac12$ (B) $\frac12$ (C) $1$ (D) $\frac32$ (E) $2$

Problem 5
A set of three points is randomly chosen 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?
(A) $\frac{1}{21}$ (B) $\frac{1}{14}$ (C) $\frac{2}{21}$ (D) $\frac17$ (E) $\frac27$

Problem 6
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 grand-daughters have no daughters?
(A) $22$ (B) $23$ (C) $24$ (D) $25$ (E) $26$

Problem 7
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 below. The stack is completed by a single row of oranges. How many oranges are in the stack?
(A) $96$ (B) $98$ (C) $100$ (D) $101$ (E) $134$

Problem 8
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 in the 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?

真题文字详解(英文)

Problem 1 You and five friends need to raise $1500$ dollars in donations for a charity, dividing the fundraising equally. How many dollars will each of you need to raise?
$(A)\ 250\quad(B)\ 300\quad(C)\ 1500\quad(D)\ 7500\quad(E)\ 9000$

Solution There are $6$ people to split the $1500$ dollars among, so each person must raise $\frac{1500}{6}=250$ dollars. $\Rightarrow \boxed{(A)\ 250}$

Problem 2 For any three real numbers $a$, $b$, and $c$, with $b\neq c$, the operation $\otimes$ is defined by:
$\otimes(a,b,c)=\frac{a}{b-c}$ What is $\otimes(\otimes(1,2,3),\otimes(2,3,1),\otimes(3,1,2))?$
$(A)-\frac12\quad(B)-\frac14\quad(C)0\quad(D)\frac14\quad(E)\frac12$

Solution
$\otimes\left(\frac12,\frac23,\frac13\right)=\otimes(-1,1,-3)=\frac{-1}{1+3}=-\frac14=\boxed{(B)-\frac14}$

Problem 3 The following problem is from both the 2004 AMC 12A #1 and 2004 AMC 10A #3, so both problems redirect to this page. Alicia earns $20$ dollars 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?
$(A)0.0029\quad(B)0.029\quad(C)0.29\quad(D)2.9\quad(E)29$

Solution
$20$ dollars is the same as $2000$ cents, and $1.45\%$ of $2000$ is $0.0145\times 2000=29$ cents. $\Rightarrow \boxed{(E)29}$

Alternate Solution Since there can't be decimal values of cents, the answer must be $\Rightarrow \boxed{(E)29}$.

Problem 4 What is the value of $x$ if $|x-1|=|x-2|$?
$(A)-\frac12\quad(B)\frac12\quad(C)1\quad(D)\frac32\quad(E)2$

真题文字详解(中英双语)

Problem 1
You and five friends need to raise $1500$ dollars 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，并且平均分配。你们每个人需要筹集多少钱？
(A) $250$ (B) $300$ (C) $1500$ (D) $7500$ (E) $9000$

Solution
There are $6$ people to split the $1500$ dollars among, so each person must raise $\frac{1500}{6} = 250$ dollars.
有$6$个人要平分$1500$美元，所以每个人必须筹集$\frac{1500}{6}=250$美元。
(\Rightarrow) (A) $250$

Problem 2
For any three real numbers $a$, $b$, and $c$, with $b \neq c$, the operation $\otimes$ is defined by:
$\otimes(a,b,c)=\frac{a}{b-c}$
What is $\otimes(\otimes(1,2,3),\otimes(2,3,1),\otimes(3,1,2))$?
对于任意三个实数$a$、$b$和$c$，并且满足$b \neq c$，定义了一个运算$\otimes$。求$\otimes(\otimes(1,2,3),\otimes(2,3,1),\otimes(3,1,2))$是多少？
(A) $-\frac12$ (B) $-\frac14$ (C) $0$ (D) $\frac14$ (E) $\frac12$

Solution
$\otimes\left(\frac{1}{2-3},\frac{2}{3-1},\frac{3}{1-2}\right)=\otimes(-1,1,-3)=\frac{-1}{1+3}=-\frac14\Rightarrow$ (B) $-\frac14$

Problem 3
The following problem is from both the 2004 AMC 12A #1 and 2004 AMC 10A #3, so both problems redirect to this page.
以下问题既来自2004年AMC 12A #1，也来自2004年AMC 10A #3，因此这两个问题都重定向到这个页面。
Alicia earns 20 dollars 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?
阿丽西亚每小时赚20美元，其中$1.45\%$用于支付地方税。多少分之几的阿丽西亚的工资用于支付地方税？
(A) $0.0029$ (B) $0.029$ (C) $0.29$ (D) $2.9$ (E) $29$

Solution