2006 AMC 12B Problems

Problem 1

What is $( - 1)^1 + ( - 1)^2 + \cdots + ( - 1)^{2006}$ ?

$\text {(A) } - 2006 \qquad \text {(B) } - 1 \qquad \text {(C) } 0 \qquad \text {(D) } 1 \qquad \text {(E) } 2006$

Problem 2

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

Solution

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? \[ \text {(A) } 10 \qquad \text {(B) } 14 \qquad \text {(C) } 17 \qquad \text {(D) } 20 \qquad \text {(E) } 24 \]

Solution

Problem 4

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

Solution

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? \[ \text {(A) } 30 \qquad \text {(B) } 50 \qquad \text {(C) } 60 \qquad \text {(D) } 90 \qquad \text {(E) } 120 \]

Solution

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. \[ \text {(A) } 129 \qquad \text {(B) } 137 \qquad \text {(C) } 174 \qquad \text {(D) } 223 \qquad \text {(E) } 411 \]

Solution

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? \[ \text {(A) } 4 \qquad \text {(B) } 12 \qquad \text {(C) } 16 \qquad \text {(D) } 24 \qquad \text {(E) } 48 \]

Solution

Problem 8

The lines $x = \frac 14y + a$ and $y = \frac 14x + b$ intersect at the point $(1,2)$ . What is $a + b$ ? \[ \text {(A) } 0 \qquad \text {(B) } \frac 34 \qquad \text {(C) } 1 \qquad \text {(D) } 2 \qquad \text {(E) } \frac 94 \]