2004 AMC amc10b 真题 答案 详解

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英文真题

2004 AMC 10B Problems

Problem 1
Each row of the Misty Moon Amphitheater has 33 seats. Rows 12 through 22 are reserved for a youth club. How many seats are reserved for this club?
(A) 297 (B) 330 (C) 363 (D) 396 (E) 726

Problem 2
How many two-digit positive integers have at least one 7 as a digit?
(A) 10 (B) 18 (C) 19 (D) 20 (E) 30

Problem 3
At each basketball practice last week, Jenny made twice as many free throws as she made at the previous practice. At her fifth practice she made 48 free throws. How many free throws did she make at the first practice?
(A) 3 (B) 6 (C) 9 (D) 12 (E) 15

Problem 4
A standard six-sided die is rolled, and ( P ) is the product of the five numbers that are visible. What is the largest number that is certain to divide ( P )?
(A) 6 (B) 12 (C) 24 (D) 144 (E) 720

Problem 5
In the expression ( c \cdot a^b - d ), the values of ( a, b, c ), and ( d ) are 0, 1, 2, and 3, although not necessarily in that order. What is the maximum possible value of the result?
(A) 5 (B) 6 (C) 8 (D) 9 (E) 10

Problem 6
Which of the following numbers is a perfect square?
(A) (98! \cdot 99!) (B) (98! \cdot 100!) (C) (99! \cdot 100!) (D) (99! \cdot 101!) (E) (100! \cdot 101!)

Problem 7
On a trip from the United States to Canada, Isabella took ( d ) U.S. dollars. At the border she exchanged them all, receiving 10 Canadian dollars for every 7 U.S. dollars. After spending 60 Canadian dollars, she had ( d ) Canadian dollars left. What is the sum of the digits of ( d )?
(A) 5 (B) 6 (C) 7 (D) 8 (E) 9

Problem 8
Minneapolis-St. Paul International Airport is 8 miles southwest of downtown St. Paul and 10 miles southeast of downtown Minneapolis. Which of the following is closest to the number of miles between downtown St. Paul and downtown Minneapolis?
(A) 13 (B) 14 (C) 15 (D) 16 (E) 17

Problem 9
A square has sides of length 10, and a circle centered at one of its vertices has radius 10. What is the area of the union of the regions enclosed by the square and the circle?
(A) (200 + 25\pi) (B) (100 + 75\pi) (C) (75 + 100\pi) (D) (100 + 100\pi) (E) (100 + 125\pi)

Problem 10
A grocer makes a display of cans in which the top row has one can and each lower row has two more cans than the row above it. If the display contains 100 cans, how many rows does it contain?

真题文字详解(英文)

Problem 1 Each row of the Misty Moon Amphitheater has 33 seats. Rows 12 through 22 are reserved for a youth club. How many seats are reserved for this club?
(A) 297 (B) 330 (C) 363 (D) 396 (E) 726

Solution There are (22 - 12 + 1 = 11) rows of 33 seats, giving (11 \times 33 = (C) 363) seats.

See also Problem 2 How many two-digit positive integers have at least one 7 as a digit?
(A) 10 (B) 18 (C) 19 (D) 20 (E) 30

Solution 1 Ten numbers (70, 71, ..., 79) have 7 as the tens digit. Nine numbers (17, 27, ..., 97) have it as the ones digit. Number 77 is in both sets. Thus the result is (10 + 9 - 1 = 18 \Rightarrow (B) 18).

Solution 2 We use complementary counting. The complement of having at least one 7 as a digit is having no 7s as a digit. We have 9 digits to choose from for the first digit and 10 digits for the second. This gives a total of (9 \times 10 = 90) two-digit numbers. But since we cannot have 7 as a digit, we have 8 first digits and 9 second digits to choose from. Thus there are (8 \times 9 = 72) two-digit numbers without a 7 as a digit. (90) (The total number of two-digit numbers) − (72) (The number of two-digit numbers without a 7) = (18 \Rightarrow (B) 18).

See also Problem 3 At each basketball practice last week, Jenny made twice as many free throws as she made at the previous practice. At her fifth practice she made 48 free throws. How many free throws did she make at the first practice?
(A) 3 (B) 6 (C) 9 (D) 12 (E) 15

Solution At the fourth practice she made (48/2 = 24) throws, at the third one it was (24/2 = 12), then we get (12/2 = 6) throws for the second practice, and finally (6/2 = 3 \Rightarrow (A) 3) throws at the first one.

See also Problem 4

真题文字详解(中英双语)

Problem 1 Each row of the Misty Moon Amphitheater has 33 seats. Rows 12 through 22 are reserved for a youth club. How many seats are reserved for this club? Misty Moon 歌剧院的每一排有 33 个座位。第 12 排到第 22 排是为一个青年俱乐部保留的。这个俱乐部有多少个座位被保留？ (A) 297 (B) 330 (C) 363 (D) 396 (E) 726 Solution There are 22 - 12 + 1 = 11 rows of 33 seats, giving 11 × 33 = (C) 363 seats. 一共有 22 - 12 + 1 = 11 排，每排有 33 个座位，总共有 11 × 33 = (C) 363 个座位。 See also Problem 2 How many two-digit positive integers have at least one 7 as a digit? 有多少个两位数的正整数至少包含一个 7 作为数字？ (A) 10 (B) 18 (C) 19 (D) 20 (E) 30 Solution 1 Ten numbers (70, 71, ..., 79) have 7 as the tens digit. Nine numbers (17, 27, ..., 97) have it as the ones digit. Number 77 is in both sets. 十个数 (70, 71, ..., 79) 有 7 作为十位数。九个数 (17, 27, ..., 97) 有它作为个位数。数 77 同时属于这两个集合。 Thus the result is 10 + 9 - 1 = 18 → (B) 18 因此结果是 10 + 9 - 1 = 18 → (B) 18 Solution 2 We use complementary counting. The complement of having at least one 7 as a digit is having no 7s as a digit. 我们使用补数法。至少有一个 7 作为数字的补集是没有任何 7 作为数字。 We have 9 digits to choose from for the first digit and 10 digits for the second. This gives a total of 9 × 10 = 90 two-digit numbers. 我们有 9 位可以选择作为第一个数字，有 10 位可以选择作为第二个数字。这给出了总共 9 × 10 = 90 两个数字。 But since we cannot have 7 as a digit, we have 8 first digits and 9 second digits to choose from. 但是我们不能把 7 当作数字，所以我们有 8 个第一位数字和 9 个第二位数字可以选择。 Thus there are 8 × 9 = 72 two-digit numbers without a 7 as a digit. 因此，有 8 × 9 = 72 个没有 7 的两位数。