英文真题

2008 AMC 12B 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 $4 \times 4$ 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?

1  2  3  4  
8  9  10 11  
15 16 17 18  
22 23 24 25  

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

Problem 3

A semipro baseball league has teams with 21 players each. League rules state that a player must be paid at least $15,000$ dollars, and that the total of all players' salaries for each team cannot exceed $700,000$ dollars. 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 4

On circle O, points C and D are on the same side of diameter AB, ∠AOC = 30°, and ∠DOB = 45°. What is the ratio of the area of the smaller sector COD to the area of the circle?
(A) ( \frac{2}{9} )
(B) ( \frac{1}{4} )
(C) ( \frac{5}{18} )
(D) ( \frac{7}{24} )
(E) ( \frac{3}{10} )

Problem 5

A class collects $50$ dollars to buy flowers for a classmate who is in the hospital. Roses cost $3$ dollars each, and carnations cost $2$ dollars each. No other flowers are to be used. How many different bouquets could be purchased for exactly $50$ dollars?
(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 1 If the basketball player makes ( x ) three-point shots and ( 5 - x ) two-point shots, he scores ( 3x + 2(5 - x) = 10 + x ) points. Clearly every value of ( x ) yields a different number of total points. Since he can make any number of three-point shots between 0 and 5 inclusive, the number of different point totals is ( 6 \Rightarrow E ).

Solution 2 Stars and bars can also be utilized to solve this problem. Since we need to decide what number of 2's and 3's are scored, and there are a total of 5 shots. It can be written like such: | Solving this, we get ( 6 \Rightarrow E ).

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 \times 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}{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

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

[ \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 = \boxed{(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 = \boxed{(B) 4} ).

Problem 3 A semipro baseball league has teams with 21 players each. League rules state that a player must be paid at least $15,000 dollars, and that the total of all players' salaries for each team cannot exceed $700,000 dollars. What is the maximum possible salary, in dollars, for a single player?

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进了5个篮。每个篮要么值2分，要么值3分。这位运动员得分的总数可能有多少种不同的数值？ (A) 2 (B) 3 (C) 4 (D) 5 (E) 6 Solution 1 If the basketball player makes x three-point shots and 5 - x two-point shots, he scores 3x + 2(5 - x) = 10 + x points. Clearly every value of x yields a different number of total points. Since he can make any number of three-point shots between 0 and 5 inclusive, the number of different point totals is 6 => E. 如果这位篮球运动员投进了x个三分球和5-x个两分球，他得到了3x+2(5-x)=10+x分。显然，每个x的值都会得到不同的总分数。由于他可以投进介于0和5之间的任意数量的三分球，因此不同的得分总数有6 => E种。 Solution 2 Stars and bars can also be utilized to solve this problem. Since we need to decide what number of 2's and 3's are scored, and there are a total of 5 shots. It can be written like such: ||. Solving this, we get 6 => E. 星与条也可以用来解决这个问题。因为我们需要决定投进了多少个两分球和三分球，总共有5次投篮。可以写成如下形式：___||__。解出后，我们得到6 => E。 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. 下面的問題來自2008年AMC 12B #2和2008年AMC 10B #2，因此这两个问题都指向这一页。 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? 显示了一个日期块。首先，第二行和第四行的数字顺序被反转。然后，对每个对角线上的数字求和。那么，两个对角线之和的正差是多少？ (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: 反转第二行和第四行上的数字后，块看起来像这样：