2007 AMC amc10b 真题 答案 详解

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英文真题

2007 AMC 10B Problems
Problem 1
Isabella's house has 3 bedrooms. Each bedroom is 12 feet long, 10 feet wide, and 8 feet high. Isabella must paint the walls of all the bedrooms. Doorways and windows, which will not be painted, occupy 60 square feet in each bedroom. How many square feet of walls must be painted?
(A) 678 (B) 768 (C) 786 (D) 867 (E) 876

Problem 2
Define the operation $a \star b = (a + b)b$. What is $(3 \star 5)-(5 \star 3)$?
(A) -16 (B) -8 (C) 0 (D) 8 (E) 16

Problem 3
A college student drove his compact car 120 miles home for the weekend and averaged 30 miles per gallon. On the return trip the student drove his parents' SUV and averaged only 20 miles per gallon. What was the average gas mileage, in miles per gallon, for the round trip?
(A) 22 (B) 24 (C) 25 (D) 26 (E) 28

Problem 4
The point O is the center of the circle circumscribed about △ABC, with ∠BOC=120° and ∠AOB=140°. What is the degree measure of ∠ABC?
(A) 35 (B) 40 (C) 45 (D) 50 (E) 60

Problem 5
In a certain land, all Arogs are Brafs, all Crups are Brafs, all Dramps are Arogs, and all Crups are Dramps. Which of the following statements is implied by these facts?
(A) All Dramps are Brafs and are Crups.
(B) All Brafs are Crups and are Dramps.
(C) All Arogs are Crups and are Dramps.
(D) All Crups are Arogs and are Brafs.
(E) All Arogs are Dramps and some Arogs may not be Crups.

Problem 6
The 2007 AMC 10 will be scored by awarding 6 points for each correct response, 0 points for each incorrect response, and 1.5 points for each problem left unanswered. After looking over the 25 problems, Sarah has decided to attempt the first 22 and leave only the last 3 unanswered. How many of the first 22 problems must she solve correctly in order to score at least 100 points?
(A) 13 (B) 14 (C) 15 (D) 16 (E) 17

Problem 7
All sides of the convex pentagon ABCDE are of equal length, and ∠A = ∠B = 90°. What is the degree measure of ∠E?
(A) 90 (B) 108 (C) 120 (D) 144 (E) 150

Problem 8
On the trip home from the meeting where this AMC10 was constructed, the Contest Chair noted that his airport parking receipt had digits of the form bbcaac, where 0 ≤ a < b < c ≤ 9, and b was the average of a and c. How many different five-digit numbers satisfy all these properties?
(A) 12 (B) 16 (C) 18 (D) 20 (E) 25

真题文字详解(英文)

Problem 1 The following problem is from both the 2007 AMC 12B #1 and 2007 AMC 10B #1, so both problems redirect to this page. Isabella's house has 3 bedrooms. Each bedroom is 12 feet long, 10 feet wide, and 8 feet high. Isabella must paint the walls of all the bedrooms. Doorways and windows, which will not be painted, occupy 60 square feet in each bedroom. How many square feet of walls must be painted? (A) 678 (B) 768 (C) 786 (D) 867 (E) 876 Solution There are four walls in each bedroom (she can't paint floors or ceilings). Therefore, we calculate the number of square feet of walls there is in one bedroom: 2 · (12 · 8 + 10 · 8) - 60 = 2 · 176 - 60 = 292 We have three bedrooms, so she must paint 292 · 3 = (E) 876 square feet of walls. Problem 2 Define the operation by a * b = (a + b)b. What is (3 * 5) - (5 * 3)? (A) -16 (B) -8 (C) 0 (D) 8 (E) 16 Solution 1 Substitute and simplify. (3 + 5)5 - (5 + 3)3 = (3 + 5)2 = 8 · 2 = (E) 16 Solution 2 Note that (a * b) - (b * a) = (a + b)b - (b + a)a = (a + b)(b - a) = b^2 - a^2. We can substitute a = 3 and b = 5 to get 5^2 - 3^2 = (E) 16. Problem 3 The following problem is from both the 2007 AMC 12B #2 and 2007 AMC 10B #3, so both problems redirect to this page. A college student drove his compact car 120 miles home for the weekend and averaged 30 miles per gallon. On the return trip the student drove his parents' SUV and averaged only 20 miles per gallon. What was the average gas mileage, in miles per gallon, for the round trip? (A) 22 (B) 24 (C) 25 (D) 26 (E) 28 Solution 1 The trip was 240 miles long and took 120/30 + 120/20 = 4 + 6 = 10 gallons. Therefore, the average mileage was 240/10 = (B) 24 Solution 2 Alternatively, we can use the harmonic mean to get 2/(1/20 + 1/30) = 2/((1)/(12)) = (B) 24

真题文字详解(中英双语)

Problem 1 The following problem is from both the 2007 AMC 12B #1 and 2007 AMC 10B #1, so both problems redirect to this page. 以下问题来自2007年AMC 12B #1和2007年AMC 10B #1，因此这两个问题都重定向到该页面。

Isabella's house has 3 bedrooms. Each bedroom is 12 feet long, 10 feet wide, and 8 feet high. Isabella must paint the walls of all the bedrooms. Doorways and windows, which will not be painted, occupy 60 square feet in each bedroom. How many square feet of walls must be painted? 伊莎贝拉的房屋有3间卧室。每间卧室长12英尺，宽10英尺，高8英尺。伊莎贝拉必须粉刷所有卧室的墙壁。门和窗户不会被粉刷，它们在每间卧室中占据60平方英尺。她必须粉刷多少平方英尺的墙壁？
(A) 678 (B) 768 (C) 786 (D) 867 (E) 876

Solution There are four walls in each bedroom (she can't paint floors or ceilings). Therefore, we calculate the number of square feet of walls there is in one bedroom: 每间卧室有四面墙（她不能粉刷地板或天花板）。因此，我们计算一间卧室中有多少平方英尺的墙壁：
$$2\cdot(12\cdot8+10\cdot8)-60=2\cdot176-60=292$$
We have three bedrooms, so she must paint $292\cdot3=\textbf{(E)}\ 876$ square feet of walls. 我们有三间卧室，因此她必须粉刷$292\cdot3=\textbf{(E)}\ 876$平方英尺的墙壁。

Problem 2 Define the operation $\star$ by $a \star b=(a+b)b$. What is $(3\star5)-(5\star3)$? 定义操作$\star$为$a \star b=(a+b)b$。那么$(3\star5)-(5\star3)$是什么？
(A) -16 (B) -8 (C) 0 (D) 8 (E) 16

Solution 1 Substitute and simplify. 代入并简化。
$$(3+5)5-(5+3)3=(3+5)2=8\cdot2=\textbf{(E)}\ 16$$

Solution 2 Note that $(a \star b)-(b \star a)=(a+b)b-(b+a)a=(a+b)(b-a)=b^2-a^2$. We can substitute $a=3$ and $b=5$ to get $5^2-3^2=\textbf{(E)}\ 16$. 注意到$(a \star b)-(b \star a)=(a+b)b-(b+a)a=(a+b)(b-a)=b^2-a^2$。我们可以代入$a=3$和$b=5$得到$5^2-3^2=\textbf{(E)}\ 16$。

Problem 3 The following problem is from both the 2007 AMC 12B #2 and 2007 AMC 10B #3, so both problems redirect to this page. 以下问题来自2007年AMC 12B #2和2007年AMC 10B #3，因此这两个问题都重定向到该页面。