2006 AMC 12A Problems

Problem 1

Sandwiches at Joe's Fast Food cost $3$ each and sodas cost $2$ each. How many dollars will it cost to purchase $5$ sandwiches and $8$ sodas?

Solution

Problem 2

Define $x\otimes y=x^3-y$ . What is $h\otimes (h\otimes h)$ ?

Solution

Problem 3

The ratio of Mary's age to Alice's age is $3:5$ . Alice is $30$ years old. How old is Mary?

Solution

Problem 4

A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display?

Solution

Problem 5

Doug and Dave shared a pizza with $8$ equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half the pizza. The cost of a plain pizza was $8$ , and there was an additional cost of $2$ for putting anchovies on one half. Dave ate all the slices of anchovy pizza and one plain slice. Doug ate the remainder. Each paid for what he had eaten. How many more dollars did Dave pay than Doug?

Solution

Problem 6

(Missing Diagram)

The $8\times 18$ rectangle $ABCD$ is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is $y$ ?

Solution

Problem 7

Mary is $20%$ (Error compiling LaTeX. Unknown error_msg) older than Sally, and Sally is $40%$ (Error compiling LaTeX. Unknown error_msg) younger than Danielle. The sum of their ages is $23.2$ years. How old will Mary be on her next birthday?

Solution

Problem 8

How many sets of two or more consecutive positive integers have a sum of $15$ ?

Solution

Problem 9

Oscar buys $13$ pencils and $3$ erasers for $1.00$ . A pencil costs more than an eraser, and both items cost a whole number of cents. What is the total cost, in cents, of one pencil and one eraser?

Solution

Problem 10

For how many real values of $x$ is $\sqrt{120-\sqrt{x}}$ an integer?

Solution

Problem 11

Which of the following describes the graph of the equation $(x+y)^2=x^2+y^2$ ?

Solution

Problem 12

(Missing Diagram)

A number of linked rings, each 1 cm thick, are hanging on a peg. The top ring has an outisde diameter of 20 cm. The outside diameter of each of the outer rings is 1 cm less than that of the ring above it. The bottom ring has an outside diameter of 3 cm. What is the distance, in cm, from the top of the top ring to the bottom of the bottom ring?

Solution

Problem 13

(Missing Diagram)

The vertices of a $3-4-5$ right triangle are the centers of three mutually externally tangent circles, as shown. What is the sum of the areas of the three circles?

Solution

Problem 14

Two farmers agree that pigs are worth $300$ and that goats are worth $210$ . When one farmer owes the other money, he pays the debt in pigs or goats, with "change" received in the form of goats or pigs as necessary. (For example, a $390$ debt could be paid with two pigs, with one goat received in change.) What is the amount of the smallest positive debt that can be resolved in this way?