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英文真题

2006 AMC 12B Problems

Problem 1

What is $(-1)^{1} + (-1)^{2} + \cdots + (-1)^{2006}$? (A) -2006
(B) -1
(C) 0
(D) 1
(E) 2006

Problem 2

For real numbers (x) and (y), define (x \spadesuit y = (x+y)(x-y)). What is (3 \spadesuit (4 \spadesuit 5))? (A) -72
(B) -27
(C) -24
(D) 24
(E) 72

Problem 3

A football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of 34 points, and the Cougars won by a margin of 14 points. How many points did the Panthers score?
(A) 10
(B) 14
(C) 17
(D) 20
(E) 24

Problem 4

Mary is about to pay for five items at the grocery store. The prices of the items are \$7.99, \$4.99, \$2.99, \$1.99, and \$0.99. Mary will pay with a twenty-dollar bill. Which of the following is closest to the percentage of the \$20.00 that she will receive in change?
(A) 5
(B) 10
(C) 15
(D) 20
(E) 25

Problem 5

John is walking east at a speed of 3 miles per hour, while Bob is also walking east, but at a speed of 5 miles per hour. If Bob is now 1 mile west of John, how many minutes will it take for Bob to catch up to John?
(A) 30
(B) 50
(C) 60
(D) 90
(E) 120

Problem 6

Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water to make lemonade. There are 25 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar. Water contains no calories. How many calories are in 200 grams of her lemonade?
(A) 129
(B) 137
(C) 174
(D) 223
(E) 411

Problem 7

Mr. and Mrs. Lopez have two children. When they get into their family car, two people sit in the front, and the other two sit in the back. Either Mr. Lopez or Mrs. Lopez must sit in the driver's seat. How many seating arrangements are possible?
(A) 4
(B) 12
(C) 16
(D) 24
(E) 48

Problem 8

The lines (x=\frac{1}{4}y+a) and (y=\frac{1}{4}x+b) intersect at the point ((1,2)). What is (a+b)?
(A) 0
(B) (\frac{3}{4})
(C) 1
(D) 2
(E) (\frac{9}{4})

Problem 9

How many even three-digit integers have the property that their digits, read left to right, are in strictly increasing order?
(A) 21
(B) 34
(C) 51
(D) 72
(E) 150

真题文字详解(英文)

Problem 1 What is $(-1)^{1}+(-1)^{2}+\cdots+(-1)^{2006}$?
(A) -2006 (B) -1 (C) 0 (D) 1 (E) 2006

Solution
$(-1)^n = 1$ if n is even and -1 if n is odd. So we have
$-1 + 1 - 1 + 1 - \cdots - 1 + 1 = 0 + 0 + \cdots + 0 + 0 = 0 \Rightarrow (C)$

See also

Problem 2 For real numbers x and y, define $x\spadesuit y=(x+y)(x-y)$. What is $3\spadesuit(4\spadesuit5)$?
(A) -72 (B) -27 (C) -24 (D) 24 (E) 72

Solution
$4\spadesuit 5=-9$
$3\spadesuit -9=-72 \Rightarrow (A)$

See also

Problem 3 A football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of 34 points, and the Cougars won by a margin of 14 points. How many points did the Panthers score?
(A) 10 (B) 14 (C) 17 (D) 20 (E) 24

Solution
Solution 1
If the Cougars won by a margin of 14 points, then the Panthers' score would be half of $(34-14)$. That's $10 \Rightarrow (A)$.

Solution 2
Let the Panthers' score be $x$. The Cougars then scored $x+14$. Since the teams combined scored 34, we get $\rightarrow x=10$, and the answer is $(A)$.

See also

Problem 4 Mary is about to pay for five items at the grocery store. The prices of the items are $7.99, 4.99, 2.99, 1.99,$ and $0.99$. Mary will pay with a twenty-dollar bill. Which of the following is closest to the percentage of the $20.00 that she will receive in change?
(A) 5 (B) 10 (C) 15 (D) 20 (E) 25

真题文字详解(中英双语)

Problem 1 What is $(-1)^{1}+(-1)^{2}+\cdots+(-1)^{2006}$?
(A) -2006 (B) -1 (C) 0 (D) 1 (E) 2006

Solution
$(-1)^n = 1$ if n is even and -1 if n is odd. So we have
$-1 + 1 - 1 + 1 - \cdots - 1 + 1 = 0 + 0 + \cdots + 0 + 0 = 0 \Rightarrow (C)$

See also

Problem 2 For real numbers x and y, define $x\spadesuit y=(x+y)(x-y)$. What is $3\spadesuit(4\spadesuit5)$?
(A) -72 (B) -27 (C) -24 (D) 24 (E) 72

Solution
$4\spadesuit 5=-9$
$3\spadesuit -9=-72 \Rightarrow (A)$

See also

Problem 3 A football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of 34 points, and the Cougars won by a margin of 14 points. How many points did the Panthers score?
(A) 10 (B) 14 (C) 17 (D) 20 (E) 24

Solution
Solution 1
If the Cougars won by a margin of 14 points, then the Panthers' score would be half of $(34-14)$. That's 10 $\Rightarrow (A)$.

Solution 2
Let the Panthers' score be $x$. The Cougars then scored $x+14$. Since the teams combined scored 34, we get
$x+x+14=34$
$\rightarrow 2x+14=34$
$\rightarrow 2x=20$
$\rightarrow x=10$
and the answer is $(A)$.

See also

Problem 4 Mary is about to pay for five items at the grocery store. The prices of the items are 7.99, 4.99, 2.99, 1.99, and 0.99. Mary will pay with a twenty-dollar bill. Which of the following is closest to the percentage of the 20.00 that she will receive in change?