2008 AMC amc10a 真题 答案 详解

英文真题

2008 AMC 10A Problems Problem 1 A bakery owner turns on his doughnut machine at 8:30 AM. At 11:10 AM the machine has completed one third of the day's job. At what time will the doughnut machine complete the job? (A) 1:50 PM (B) 3:00 PM (C) 3:30 PM (D) 4:30 PM (E) 5:50 PM Problem 2 A square is drawn inside a rectangle. The ratio of the width of the rectangle to a side of the square is 2 : 1. The ratio of the rectangle's length to its width is 2 : 1. What percent of the rectangle's area is in the square? (A) 12.5 (B) 25 (C) 50 (D) 75 (E) 87.5 Problem 3 For the positive integer n, let (\langle n \rangle) denote the sum of all the positive divisors of n with the exception of n itself. For example, (\langle 4 \rangle = 1 + 2 = 3) and (\langle 12 \rangle = 1 + 2 + 3 + 4 + 6 = 16). What is (\langle \langle 6 \rangle \rangle)? (A) 6 (B) 12 (C) 24 (D) 32 (E) 36 Problem 4 Suppose that (\frac{2}{3}) of 10 bananas are worth as much as 8 oranges. How many oranges are worth as much as (\frac{1}{2}) of 5 bananas? (A) 2 (B) (\frac{5}{2}) (C) 3 (D) (\frac{7}{2}) (E) 4 Problem 5 Which of the following is equal to the product (\frac{8}{4} \cdot \frac{12}{8} \cdot \frac{16}{12} \cdots \frac{4n+4}{4n} \cdots \frac{2008}{2004})? (A) 251 (B) 502 (C) 1004 (D) 2008 (E) 4016 Problem 6 A triathlete competes in a triathlon in which the swimming, biking, and running segments are all of the same length. The triathlete swims at a rate of 3 kilometers per hour, bikes at a rate of 20 kilometers per hour, and runs at a rate of 10 kilometers per hour. Which of the following is closest to the triathlete's average speed, in kilometers per hour, for the entire race? (A) 3 (B) 4 (C) 5 (D) 6 (E) 7 Problem 7 The fraction (\frac{(3^{2008})^2 - (3^{2006})^2}{(3^{2007})^2 - (3^{2005})^2}) simplifies to which of the following? (A) 1 (B) (\frac{9}{4}) (C) 3 (D) (\frac{9}{2}) (E) 9 Problem 8 Heather compares the price of a new computer at two different stores. Store A offers 15% off the sticker price followed by a $90 rebate, and store B offers 25% off the same sticker price with no rebate. Heather saves $15 by buying the computer at store A instead of store B. What is the sticker price of the computer, in dollars?

真题文字详解(英文)

Problem 1 The following problem is from both the 2008 AMC 12A #1 and 2008 AMC 10A #1, so both problems redirect to this page. A bakery owner turns on his doughnut machine at 8:30 AM. At 11:10 AM the machine has completed one third of the day's job. At what time will the doughnut machine complete the job? (A) 1:50 PM (B) 3:00 PM (C) 3:30 PM (D) 4:30 PM (E) 5:50 PM Solution The machine completes one-third of the job in $11:10 - 8:30 = 2:40$ hours. Thus, the entire job is completed in $3 \cdot (2:40) = 8:00$ hours. Since the machine was started at 8:30 AM, the job will be finished 8 hours later, at 4:30 PM. The answer is (D). Note: $2:40$ means 2 hours and 40 minutes. $3$ multiplied by this time interval is 8 hours.

Problem 2 A square is drawn inside a rectangle. The ratio of the width of the rectangle to a side of the square is $2:1$. The ratio of the rectangle's length to its width is $2:1$. What percent of the rectangle's area is inside the square? (A) 12.5 (B) 25 (C) 50 (D) 75 (E) 87.5 Solution 1 Since they are asking for the "ratio" of two things, we can say that the side of the square is anything that we want. So if we say that it is $1$, then the width of the rectangle is $2$, and the length is $4$, thus making the total area of the rectangle $8$. The area of the square is just $1$. So the percent of the rectangle's area inside the square is $\frac{1}{8} \times 100 = 12.5 \rightarrow \boxed{\text{A}}$

See also Problem 3 For the positive integer $n$, let $(n)$ denote the sum of all the positive divisors of $n$ with the exception of $n$ itself. For example, $(4)=1+2=3$ and $(12)=1+2+3+4+6=16$ What is $(\langle 6\rangle)$? (A) 6 (B) 12 (C) 24 (D) 32 (E) 36 Solution 1 $\langle\langle 6\rangle\rangle=\langle\langle 6\rangle\rangle=\langle 6\rangle=6 \Rightarrow \text{(A)}$ Solution 2 Since $6$ is a perfect number, any such operation where $n=6$ will yield $6$ as the answer. Note: A perfect number is defined as a number that equals the sum of its positive divisors excluding itself. See also Problem 4 The following problem is from both the 2008 AMC 12A #3 and 2008 AMC 10A #4, so both problems redirect to this page. Suppose that $\frac{2}{3}$ of $10$ bananas are worth as much as $8$ oranges. How many oranges are worth as much as $\frac{1}{2}$ of $5$ bananas? (A) 2 (B) $\frac{5}{2}$ (C) 3 (D) $\frac{7}{2}$ (E) 4 Solution If $\frac{2}{3}\cdot 10$ bananas $= 8$ oranges, then $\frac{1}{2}\cdot 5$ bananas $= \left(\frac{1}{2}\cdot 5$ bananas$\right)\cdot \left(\frac{8\text{ oranges}}{\frac{2}{3}\cdot 10\text{ bananas}}\right)=3$ oranges $\Rightarrow \text{(C)}$ Problem 5 The following problem is from both the 2008 AMC 12A #4 and 2008 AMC 10A #5, so both problems redirect to this page. Which of the following is equal to the product $$\frac{8}{4}\cdot\frac{12}{8}\cdot\frac{16}{12}\cdots\frac{4n+4}{4n}\cdots\frac{2008}{2004}$$ ? (A) 251 (B) 502 (C) 1004 (D) 2008 (E) 4016 Solution 1 $$\frac{8}{4}\cdot\frac{12}{8}\cdot\frac{16}{12}\cdots\frac{4n+4}{4n}\cdots\frac

真题文字详解(中英双语)

Problem 1 The following problem is from both the 2008 AMC 12A #1 and 2008 AMC 10A #1, so both problems redirect to this page. A bakery owner turns on his doughnut machine at 8:30 AM. At 11:10 AM the machine has completed one third of the day's job. At what time will the doughnut machine complete the job? (A) 1:50 PM (B) 3:00 PM (C) 3:30 PM (D) 4:30 PM (E) 5:50 PM Solution The machine completes one-third of the job in $11:10 - 8:30 = 2:40$ hours. Thus, the entire job is completed in $3 \cdot (2:40) = 8:00$ hours. Since the machine was started at 8:30 AM, the job will be finished 8 hours later, at 4:30 PM. The answer is (D). Note: 2:40 means 2 hours and 40 minutes. 3 multiplied by this time interval is 8 hours. Problem 2 A square is drawn inside a rectangle. The ratio of the width of the rectangle to a side of the square is $2:1$. The ratio of the rectangle's length to its width is $2:1$. What percent of the rectangle's area is inside the square? (A) 12.5 (B) 25 (C) 50 (D) 75 (E) 87.5 Solution 1 Since they are asking for the "ratio" of two things, we can say that the side of the square is anything that we want. So if we say that it is $1$, then the width of the rectangle is $2$, and the length is $4$, thus making the total area of the rectangle $8$. The area of the square is just $1$. So the percent of the rectangle's area inside the square is $\frac{1}{8} \times 100 = 12.5 \longrightarrow \boxed{\text{A}}$ See also Problem 3 For the positive integer $n$, let $\langle n\rangle$ denote the sum of all the positive divisors of $n$ with the exception of $n$ itself. For example, $\langle 4\rangle=1+2=3$ and $\langle 12\rangle=1+2+3+4+6=16$ What is $\langle\langle 6\rangle\rangle$? (A) 6 (B) 12 (C) 24 (D) 32 (E) 36 Solution 1 $\langle\langle 6\rangle\rangle=\langle\langle 6\rangle\rangle=\langle 6\rangle=6 \longrightarrow \boxed{\text{A}}$ Solution 2 Since $6$ is a perfect number, any such operation where $n=6$ will yield $6$ as the answer. Note: A perfect number is defined as a number that equals the sum of its positive divisors excluding itself. See also Problem 4 The following problem is from both the 2008 AMC 12A #3 and 2008 AMC 10A #4, so both problems redirect to this page. Suppose that $\frac{2}{3}$ of $10$ bananas are worth as much as $8$ oranges. How many oranges are worth as much as $\frac{1}{2}$ of $5$ bananas? (A) 2 (B) $\frac{5}{2}$ (C) 3 (D) $\frac{7}{2}$ (E) 4 Solution If $\frac{2}{3}\cdot 10$ bananas $=8$ oranges then $\frac{1}{2}\cdot 5$ bananas $=\left(\frac{1}{2}\cdot 5\text{ bananas}\right)\cdot \left(\frac{8\text{ oranges}}{\frac{2}{3}\cdot 10\text{ bananas}}\right)=3\text{ oranges} \longrightarrow \boxed{\text{C}}$ Problem 5 The following problem is from both the 2008 AMC 12A #4 and 2008 AMC 10A #5, so both problems redirect to this page. Which of the following is equal to the product $$\frac{8}{4}\cdot \frac{12}{8}\cdot \frac{16}{12}\cdots \frac{4n+4}{4n}\cdots \frac{2008}{2004}$$ ? (A) 251 (B) 502 (C) 1004 (D) 2008 (E) 4016 Solution 1