2008 AMC amc10b 真题 答案 详解

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英文真题

2008 AMC 10B Problems
Problem 1
A basketball player made 5 baskets during a game. Each basket was worth either 2 or 3 points. How many different numbers could represent the total points scored by the player?
(A) 2 (B) 3 (C) 4 (D) 5 (E) 6

Problem 2
A 4 × 4 block of calendar dates is shown. The order of the numbers in the second row is to be reversed. Then the order of the numbers in the fourth row is to be reversed. Finally, the numbers on each diagonal are to be added. What will be the positive difference between the two diagonal sums?
[ \begin{array}{cccc} 1 & 2 & 3 & 4 \ 8 & 9 & 10 & 11 \ 15 & 16 & 17 & 18 \ 22 & 23 & 24 & 25 \ \end{array} ]
(A) 2 (B) 4 (C) 6 (D) 8 (E) 10

Problem 3
Assume that ( x ) is a positive real number. Which is equivalent to ( \sqrt[3]{x \sqrt{x}} )?
(A) ( x^{1/6} ) (B) ( x^{1/4} ) (C) ( x^{3/8} ) (D) ( x^{1/2} ) (E) ( x )

Problem 4
A semipro baseball league has teams with 21 players each. League rules state that a player must be paid at least $15,000 and that the total of all players' salaries for each team cannot exceed $700,000. What is the maximum possible salary, in dollars, for a single player?
(A) 270,000 (B) 385,000 (C) 400,000 (D) 430,000 (E) 700,000

Problem 5
For real numbers ( a ) and ( b ), define ( a \$ b = (a - b)^2 ). What is ( (x - y)^2 \$(y - x)^2 )?
(A) 0 (B) ( x^2 + y^2 ) (C) ( 2x^2 ) (D) ( 2y^2 ) (E) ( 4xy )

Problem 6
Points B and C lie on AD. The length of AB is 4 times the length of BD, and the length of AC is 9 times the length of CD. The length of BC is what fraction of the length of AD?
(A) ( \frac{1}{36} ) (B) ( \frac{1}{13} ) (C) ( \frac{1}{10} ) (D) ( \frac{5}{36} ) (E) ( \frac{1}{5} )

Problem 7
An equilateral triangle of side length 10 is completely filled in by non-overlapping equilateral triangles of side length 1. How many small triangles are required?
(A) 10 (B) 25 (C) 100 (D) 250 (E) 1000

Problem 8
A class collects $50 to buy flowers for a classmate who is in the hospital. Roses cost $3 each, and carnations cost $2 each. No other flowers are to be used. How many different bouquets could be purchased for exactly $50?
(A) 1 (B) 7 (C) 9 (D) 16 (E) 17

真题文字详解(英文)

Problem 1 A basketball player made 5 baskets during a game. Each basket was worth either 2 or 3 points. How many different numbers could represent the total points scored by the player?
(A) 2 (B) 3 (C) 4 (D) 5 (E) 6

Solution The number of points could have been 10, 11, 12, 13, 14, or 15. This is because the minimum is 25=10 and the maximum is 35=15. The numbers between 10 and 15 are possible as well. Thus, the answer is (E) 6.

See also Problem 2 The following problem is from both the 2008 AMC 12B #2 and 2008 AMC 10B #2, so both problems redirect to this page. A 4 x 4 block of calendar dates is shown. First, the order of the numbers in the second and the fourth rows are reversed. Then, the numbers on each diagonal are added. What will be the positive difference between the two diagonal sums?

[ \begin{array}{|c|c|c|c|} \hline 1 & 2 & 3 & 4 \ \hline 8 & 9 & 10 & 11 \ \hline 15 & 16 & 17 & 18 \ \hline 22 & 23 & 24 & 25 \ \hline \end{array} ]

(A) 2 (B) 4 (C) 6 (D) 8 (E) 10

Solution 1 After reversing the numbers on the second and fourth rows, the block will look like this:

[ \begin{array}{|c|c|c|c|} \hline 1 & 2 & 3 & 4 \ \hline 11 & 10 & 9 & 8 \ \hline 15 & 16 & 17 & 18 \ \hline 25 & 24 & 23 & 22 \ \hline \end{array} ]

The positive difference between the two diagonal sums is then ((4 + 9 + 16 + 25) - (1 + 10 + 17 + 22) = 3 - 1 - 1 + 3 = (B) 4).

Solution 2 Notice that at baseline the diagonals sum to the same number (52). Therefore we need only compute the effect of the swap. The positive difference between 9 and 10 is 1 and the positive difference between 22 and 25 is 3. Adding gives (1 + 3 = (B) 4).

Problem 3 Assume that (x) is a positive real number. Which is equivalent to (\sqrt[3]{x\sqrt{x}})?
(A) (x^{1/6}) (B) (x^{1/4}) (C) (x^{3/8}) (D) (x^{1/2}) (E) (x)

Solution 1 (\sqrt[3]{x\sqrt{x}} = \sqrt[3]{\sqrt{x^3}} = \sqrt[6]{x^3} = x^{3/6} = x^{1/2} (D))

真题文字详解(中英双语)

Problem 1
A basketball player made 5 baskets during a game. Each basket was worth either 2 or 3 points. How many different numbers could represent the total points scored by the player?
一位篮球运动员在比赛中投中了5个篮。每个篮要么得2分，要么得3分。这位运动员得分总数可能有多少种不同的数字？
(A) 2 (B) 3 (C) 4 (D) 5 (E) 6

Solution
The number of points could have been 10, 11, 12, 13, 14, or 15. This is because the minimum is 25=10 and the maximum is 35=15. The numbers between 10 and 15 are possible as well. Thus, the answer is (E) 6.
得分可能是10、11、12、13、14或15。因为最小值是25=10，最大值是35=15。10到15之间的数字也是可能的。因此，答案是(E) 6。

See also
Problem 2
The following problem is from both the 2008 AMC 12B #2 and 2008 AMC 10B #2, so both problems redirect to this page.
以下问题既来自2008年AMC 12B第2题，也来自2008年AMC 10B第2题，因此这两个问题都重定向到这个页面。
A 4 × 4 block of calendar dates is shown. First, the order of the numbers in the second and the fourth rows are reversed. Then, the numbers on each diagonal are added. What will be the positive difference between the two diagonal sums?
展示了一个4×4块日历日期。首先，第二行和第四行中的数字顺序被反转。然后，对每个对角线上的数字求和。问两个对角线和的正差是多少？
(A) 2 (B) 4 (C) 6 (D) 8 (E) 10

Solution 1
After reversing the numbers on the second and fourth rows, the block will look like this:
将第二行和第四行的数字颠倒后，块将如下所示：
[ \begin{array}{cccc} 1 & 2 & 3 & 4 \ 11 & 10 & 9 & 8 \ 15 & 16 & 17 & 18 \ 25 & 24 & 23 & 22 \ \end{array} ]
The positive difference between the two diagonal sums is then
(4 + 9 + 16 + 25) - (1 + 10 + 17 + 22) = 3 - 1 - 1 + 3 = (B) 4
两个对角线和的正差是 (4 + 9 + 16 + 25) - (1 + 10 + 17 + 22) = 3 - 1 - 1 + 3 = (B) 4。

Solution 2