2009 AMC amc12a 真题 答案 详解

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英文真题

2009 AMC 12A Problems

Problem 1

Kim's flight took off from Newark at 10:34 AM and landed in Miami at 1:18 PM. Both cities are in the same time zone. If her flight took ( h ) hours and ( m ) minutes, with ( 0 < m < 60 ), what is ( h + m )?

((\text{A})\ 46\quad(\text{B})\ 47\quad(\text{C})\ 50\quad(\text{D})\ 53\quad(\text{E})\ 54)

Problem 2

Which of the following is equal to ( 1+\frac{1}{1+\frac{1}{1+1}} )?

((\text{A})\ \frac{5}{4}\quad(\text{B})\ \frac{3}{2}\quad(\text{C})\ \frac{5}{3}\quad(\text{D})\ 2\quad(\text{E})\ 3)

Problem 3

What number is one third of the way from ( \frac{1}{4} ) to ( \frac{3}{4} )?

((\text{A})\ \frac{1}{3}\quad(\text{B})\ \frac{5}{12}\quad(\text{C})\ \frac{1}{2}\quad(\text{D})\ \frac{7}{12}\quad(\text{E})\ \frac{2}{3})

Problem 4

Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes, and quarters. Which of the following could not be the total value of the four coins, in cents?

((\text{A})\ 15\quad(\text{B})\ 25\quad(\text{C})\ 35\quad(\text{D})\ 45\quad(\text{E})\ 55)

Problem 5

One dimension of a cube is increased by 1, another is decreased by 1, and the third is left unchanged. The volume of the new rectangular solid is 5 less than that of the cube. What was the volume of the cube?

((\text{A})\ 8\quad(\text{B})\ 27\quad(\text{C})\ 64\quad(\text{D})\ 125\quad(\text{E})\ 216)

Problem 6

Suppose that ( P = 2^m ) and ( Q = 3^n ). Which of the following is equal to ( 12^{mn} ) for every pair of integers ( (m,n) )?

((\text{A})\ P^2Q\quad(\text{B})\ P^nQ^m\quad(\text{C})\ P^nQ^{2m}\quad(\text{D})\ P^{2m}Q^n\quad(\text{E})\ P^{2n}Q^m)

Problem 7

The first three terms of an arithmetic sequence are ( 2x-3 ), ( 5x-11 ), and ( 3x+1 ) respectively. The ( n )th term of the sequence is 2009. What is ( n )?

((\text{A})\ 255\quad(\text{B})\ 502\quad(\text{C})\ 1004\quad(\text{D})\ 1506\quad(\text{E})\ 8037)

Problem 8

Four congruent rectangles are placed as shown. The area of the outer square is 4 times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side?

真题文字详解(英文)

Problem 1

Kim's flight took off from Newark at 10:34 AM and landed in Miami at 1:18 PM. Both cities are in the same time zone. If her flight took ( h ) hours and ( m ) minutes, with ( 0 < m < 60 ), what is ( h + m )?

((A) 46 \quad (B) 47 \quad (C) 50 \quad (D) 53 \quad (E) 54)

Solution

There is 1 hour and ( 60 - 34 = 26 ) minutes between 10:34 AM and noon; and there is 1 hour and 18 minutes between noon and 1:18 PM. Hence the flight took 2 hours and ( 26 + 18 = 44 ) minutes, and ( h + m = 46 ) (A).

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Problem 2

The following problem is from both the 2009 AMC 12A #2 and 2009 AMC 10A #3, so both problems redirect to this page.

Which of the following is equal to ( 1 + \frac{1}{1 + \frac{1}{1+1}} )?

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

Solution

We compute:

( 1 + \frac{1}{1 + \frac{1}{1+1}} = 1 + \frac{1}{1 + \frac{1}{2}} = 1 + \frac{1}{\frac{3}{2}} = 1 + \frac{2}{3} = \frac{5}{3} )

This is choice C.

Interesting sidenote: The continued fraction ( 1 + \frac{1}{1 + \frac{1}{1+1...}} ) is equal to the golden ratio, or ( \frac{1+\sqrt{5}}{2} ).

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Problem 3

What number is one third of the way from ( \frac{1}{4} ) to ( \frac{3}{4} )?

((A) \frac{1}{3} \quad (B) \frac{5}{12} \quad (C) \frac{1}{2} \quad (D) \frac{7}{12} \quad (E) \frac{2}{3})

Solution

真题文字详解(中英双语)

Problem 1 Kim's flight took off from Newark at 10:34 AM and landed in Miami at 1:18 PM. Both cities are in the same time zone. If her flight took h hours and m minutes, with 0 < m < 60, what is h + m? 金姆的航班于上午10:34从纽瓦克起飞，下午1:18在迈阿密降落。这两个城市在同一时区。如果她的航班持续了h小时和m分钟，并且0 < m < 60，那么h+m是多少？ (A) 46 (B) 47 (C) 50 (D) 53 (E) 54 Solution There is 1 hour and 60 - 34 = 26 minutes between 10:34 AM and noon; and there is 1 hour and 18 minutes between noon and 1:18 PM. Hence the flight took 2 hours and 26 + 18 = 44 minutes, and h + m = 46 (A). 从上午10:34到中午有1小时和60-34=26分钟；从中午到下午1:18有1小时和18分钟。因此，航班持续了2小时和26+18=44分钟，并且h+m=46(A)。 Problem 2 The following problem is from both the 2009 AMC 12A #2 and 2009 AMC 10A #3, so both problems redirect to this page. 这个问题既来自2009年AMC 12A第2题，也来自2009年AMC 10A第3题，因此两个问题都重定向到这个页面。 Which of the following is equal to 1 + \frac{1}{1+\frac{1}{1+1}}? 以下哪一项等于1 + \frac{1}{1+\frac{1}{1+1}}? (A) \frac{5}{4} (B) \frac{3}{2} (C) \frac{5}{3} (D) 2 (E) 3 Solution We compute: 1 + \frac{1}{1+\frac{1}{1+1}} = 1 + \frac{1}{1+\frac{1}{1+1}} = 1 + \frac{1}{1+\frac{1}{\frac{3}{2}}} = 1 + \frac{1}{\frac{5}{3}} = \frac{5}{3} This is choice C. 这是选项C。