2004 AMC amc12b 真题 答案 详解

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英文真题

2004 AMC 12B Problems

Problem 1

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 2

In the expression c · 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 3

If x and y are positive integers for which 2^x * 3^y = 1296, what is the value of x + y?

(A)8 (B)9 (C)10 (D)11 (E)12

Problem 4

An integer x, with 10 ≤ x ≤ 99, is to be chosen. If all choices are equally likely, what is the probability that at least one digit of x is a 7?

(A)1/9 (B)1/5 (C)19/90 (D)2/9 (E)1/3

Problem 5

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 6

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 7

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+25π (B)100+75π (C)75+100π (D)100+100π (E)100+125π

Problem 8

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?

(A)5 (B)8 (C)9 (D)10 (E)11

Problem 9

The point (-3, 2) is rotated 90° clockwise around the origin to point B. Point B is then reflected over the line x = y to point C. What are the coordinates of C?

(A)(-3,-2) (B)(-2,-3) (C)(2,-3) (D)(2,3) (E)(3,2)

真题文字详解(英文)

Problem 1

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

Each day Jenny makes half as many free throws as she does at the next practice. Hence on the fourth day she made (\frac{1}{2} \cdot 48 = 24) free throws, on the third (12), on the second (6), and on the first (3 \Rightarrow (A)).

Because there are five days, or four transformations between days (day 1 → day 2 → day 3 → day 4 → day 5), she makes (48 \cdot \frac{1}{2^4} = (A) 3)

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Problem 2

The following problem is from both the 2004 AMC 12B #2 and 2004 AMC 10B #5, so both problems redirect to this page.

Problem 2

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

Solution

If (a = 0) or (c = 0), the expression evaluates to (-d < 0). If (b = 0), the expression evaluates to (c - d \leq 2). Case (d = 0) remains. In that case, we want to maximize (c \cdot a^{b}) where ({a,b,c} = {1,2,3}). Trying out the six possibilities we get that the greatest is ((a,b,c) = (3,2,1)), where (c \cdot a^{b} = 1 \cdot 3^{2} = (D) 9).

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Problem 3

If (x) and (y) are positive integers for which (2^{x}3^{y}=1296), what is the value of (x+y)?

(A) 8 (B) 9 (C) 10 (D) 11 (E) 12

Solution

(1296=2^{4}3^{4}) and (4+4=\boxed{8} \Rightarrow (A)).

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Problem 4

An integer (x), with (10 \leq x \leq 99), is to be chosen. If all choices are equally likely, what is the probability that at least one digit of (x) is a (7)?

(A)(\frac{1}{9}) (B)(\frac{1}{5}) (C)(\frac{19}{90}) (D)(\frac{2}{9}) (E)(\frac{1}{3})

Solution

真题文字详解(中英双语)

Problem 1

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

Each day Jenny makes half as many free throws as she does at the next practice. Hence on the fourth day she made (\frac{1}{2} \cdot 48 = 24) free throws, on the third (12), on the second (6), and on the first (3 \Rightarrow (A)).

每天Jenny投中的罚球数都是下一次练习的一半。因此在第四天她投中了0个罚球，在第三天投中了1个，在第二天投中了2个，在第一天投中了3个。

Because there are five days, or four transformations between days (day 1 → day 2 → day 3 → day 4 → day 5), she makes (48 \cdot \frac{1}{2^4} = (A) 3)

因为有五天，或者是四天之间的转换（第一天0个，第二天1个，第三天2个，第四天3个，第五天），她投中了4个。

Problem 2

The following problem is from both the 2004 AMC 12B #2 and 2004 AMC 10B #5, so both problems redirect to this page.

下面的問題來自2004年AMC 12B第2題和2004年AMC 10B第5题，因此这两个问题都重定向到这个页面。

Problem 2

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?

在表达式(c \cdot a^{b} - d)中，(a)、(b)、(c)和(d)的值分别是(0)、(1)、(2)和(3)，尽管不一定按这个顺序。求结果的最大可能值是多少？

(A) 5 (B) 6 (C) 8 (D) 9 (E) 10

Solution

If (a = 0) or (c = 0), the expression evaluates to (-d < 0).

如果(a = 0)或(c = 0)，表达式求值为(-d < 0)。

If (b = 0), the expression evaluates to (c - d \leq 2).

如果(b = 0)，表达式求值为(c - d \leq 2)。

Case (d = 0) remains. In that case, we want to maximize (c \cdot a^{b}) where ({a,b,c}={1,2,3}). Trying out the six possibilities we get that the greatest is ((a,b,c)=(3,2,1)), where (c \cdot a^{b}=1 \cdot 3^{2}=(D) 9).

情况(d = 0)仍然存在。在这种情况下，我们希望最大化(c \cdot a^{b})，其中({a,b,c}={1,2,3})。尝试六种可能性后，我们发现最大值是((a,b,c)=(3,2,1))，其中(c \cdot a^{b}=1 \cdot 3^{2}=(D) 9)。

Problem 3