2010 AMC amc12a 真题 答案 详解

---

中英双语真题

2010 AMC12A 2010 AMC12A Problem 1 What is (20 - (2010 - 201) + (2010 - (201 - 20)))? 表达式(20-(2010-201)+(2010-(201-20)))的值是多少? (A) -4020 (B) 0 (C) 40 (D) 401 (E) 4020 Problem 2 A ferry boat shuttles tourists to an island every hour starting at 10 AM until its last trip, which starts at 3 PM. One day the boat captain notes that on the 10 AM trip there were 100 tourists on the ferry boat, and that on each successive trip, the number of tourists was 1 fewer than on the previous trip. How many tourists did the ferry take to the island that day? 一艘渡轮从上午10点开始，每隔一小时就会把游客送到一个岛上，直到最后一班（最后一班下午3点出发）。一天，船长注意到，上午10点的这一班，渡轮上有100名游客，而且在后续的每一班，游客人数都比前一班车少1人。问那一天渡轮总共运了多少游客到岛上？ (A) 585 (B) 594 (C) 672 (D) 679 (E) 694 Problem 3 Rectangle ABCD, pictured below, shares 50% of its area with square EFGH. Square EFGH shares 20% of its area with rectangle ABCD. What is (\frac{AB}{AD})? 矩形ABCD如下图所示，和正方形EFGH重叠部分的面积占了它总面积的50%。而重叠部分的面积占了正方形EFGH总面积的20%。问(\frac{AB}{AD})是多少?

英文真题

2010 AMC 12A Problems

Problem 1
What is (20 - (2010 - 201) + (2010 - (201 - 20))?) ((A)) (-4020) ((B)) (0) ((C)) (40) ((D)) (401) ((E)) (4020)

Problem 2
A ferry boat shuttles tourists to an island every hour starting at 10 AM until its last trip, which starts at 3 PM. One day the boat captain notes that on the 10 AM trip there were 100 tourists on the ferry boat, and that on each successive trip, the number of tourists was 1 fewer than on the previous trip. How many tourists did the ferry take to the island that day? ((A)) (585) ((B)) (594) ((C)) (672) ((D)) (679) ((E)) (694)

Problem 3
Rectangle (ABCD), pictured below, shares 50% of its area with square (EFGH). Square (EFGH) shares 20% of its area with rectangle (ABCD). What is (\frac{AB}{AD})? ((A)) (4) ((B)) (5) ((C)) (6) ((D)) (8) ((E)) (10)

Problem 4
If (x < 0), then which of the following must be positive?
((A)) (\frac{x}{|x|}) ((B)) (-x^2) ((C)) (-2^x) ((D)) (-x^{-1}) ((E)) (\sqrt[3]{x})

Problem 5
Halfway through a 100-shot archery tournament, Chelsea leads by 50 points. For each shot a bullseye scores 10 points, with other possible scores being 8, 4, 2, and 0 points. Chelsea always scores at least 4 points on each shot. If Chelsea's next (n) shots are bullseyes she will be guaranteed victory. What is the minimum value for (n)? ((A)) (38) ((B)) (40) ((C)) (42) ((D)) (44) ((E)) (46)

Problem 6
A palindrome, such as 83438, is a number that remains the same when its digits are reversed. The numbers (x) and (x+32) are three-digit and four-digit palindromes, respectively. What is the sum of the digits of (x)? ((A)) (20) ((B)) (21) ((C)) (22) ((D)) (23) ((E)) (24)

Problem 7
Logan is constructing a scaled model of his town. The city's water tower stands 40 meters high, and the top portion is a sphere that holds 100,000 liters of water. Logan's miniature water tower holds 0.1 liters. How tall, in meters, should Logan make his tower? ((A)) (0.04) ((B)) (\frac{0.4}{\pi}) ((C)) (0.4) ((D)) (\frac{4}{\pi}) ((E)) (4)

真题文字详解(英文)

Problem 1 What is $(20-(2010-201))+(2010-(201-20))? \n(A)-4020 (B) 0 (C) 40 (D) 401 (E) 4020$ Solution $20 - 2010 + 201 + 2010 - 201 + 20 = 20 + 20 = (C) 40$ Problem 2 A ferry boat shuttles tourists to an island every hour starting at 10 AM until its last trip, which starts at 3 PM. One day the boat captain notes that on the 10 AM trip there were 100 tourists on the ferry boat, and that on each successive trip, the number of tourists was 1 fewer than on the previous trip. How many tourists did the ferry take to the island that day? (A) 585 (B) 594 (C) 672 (D) 679 (E) 694 Solution 1 It is easy to see that the ferry boat takes 6 trips total. The total number of people taken to the island is: $100+99+98+97+96+95=6(100)-(1+2+3+4+5)=600-15=585 (A)$ Solution 2 You can notice that: $100+95=99+96=98+97=195$ This means that the sum is: $100+99+98+97+96+95=195*3=585 (A)$ Problem 3 Rectangle ABCD, pictured below, shares 50% of its area with square EFGH. Square EFGH shares 20% of its area with rectangle ABCD. What is $\frac{AB}{AD}$ ? (A) 4 (B) 5 (C) 6 (D) 8 (E) 10 Solution 1 If we shift A to coincide with E, and add new horizontal lines to divide EFGH into five equal parts: This helps us to see that AD=a/5 and AB=2a, where a=EF. Hence $\frac{AB}{AD}=\frac{2a}{a/5}=10.$ Solution 2

真题文字详解(中英双语)

Problem 1 What is $(20-(2010-201))+(2010-(201-20))?
(A)-4020 (B) 0 (C) 40 (D) 401 (E) 4020
Solution
$20 - 2010 + 201 + 2010 - 201 + 20 = 20 + 20 = \boxed{(C) 40}$

Problem 2 A ferry boat shuttles tourists to an island every hour starting at 10 AM until its last trip, which starts at 3 PM. One day the boat captain notes that on the 10 AM trip there were 100 tourists on the ferry boat, and that on each successive trip, the number of tourists was 1 fewer than on the previous trip. How many tourists did the ferry take to the island that day?
一艘渡轮从上午10点开始每小时一班，直到下午3点最后一次航行，运送游客到一座岛屿上。一天，船长注意到上午10点的航程上有100名游客，而且之后每一班船上的游客都比前一班车少1人。那天渡轮一共运送了多少名游客到岛上？
(A) 585 (B) 594 (C) 672 (D) 679 (E) 694
Solution 1
It is easy to see that the ferry boat takes 6 trips total. The total number of people taken to the island is:
很容易看出，渡轮总共航行了6次。运送到岛上的总人数是：
$100+99+98+97+96+95=6(100)-(1+2+3+4+5)=600-15=\boxed{585\text{ (A)}}$

Solution 2
You can notice that:
你可以注意到：
$100+95=99+96=98+97=195$
This means that the sum is:
这意味着总和是：
$100+99+98+97+96+95=195*3=\boxed{585\text{ (A)}}$

Problem 3 Rectangle ABCD, pictured below, shares 50% of its area with square EFGH. Square EFGH shares 20% of its area with rectangle ABCD. What is $\frac{AB}{AD}$?
矩形ABCD，如下图所示，与正方形EFGH共享了50%的面积。正方形EFGH与矩形ABCD共享了20%的面积。(\frac{AB}{AD})是多少？
(A) 4 (B) 5 (C) 6 (D) 8 (E) 10
Solution 1
If we shift A to coincide with E, and add new horizontal lines to divide EFGH into five equal parts:
如果我们把A移动到与E重合，并添加新的水平线将EFGH分成五个相等的部分：