2003 AMC 8 Exam Problems
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Used with permission of the Mathematical Association of America.
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1.
Jamie counted the number of edges of a cube, Jimmy counted the numbers of corners, and Judy counted the number of faces. They then added the three numbers. What was the resulting sum?
Answer: E
Solution(s):
A cube has edges, corners, and faces. Adding these together yields
Thus, E is the correct answer.
2.
Which of the following numbers has the smallest prime factor?
Answer: C
Solution(s):
Note that the smallest prime number is This means that any even number would be our answer.
Thus, C is the correct answer.
3.
A burger at Ricky C's weighs grams, of which grams are filler. What percent of the burger is not filler?
Answer: D
Solution(s):
We get that grams are not filler. The percentage is therefore
Thus, D is the correct answer.
4.
A group of children riding on bicycles and tricycles rode past Billy Bob's house. Billy Bob counted children and wheels. How many tricycles were there?
Answer: C
Solution(s):
Let be the number of bicycles and be the number of tricycles. Then we can set up the following system of equations: Multiplying the first equation by and subtracting from the second equation, we get
Thus, C is the correct answer.
5.
If of a number is what is of the same number?
Answer: B
Solution(s):
Let the be the number. Then From this we get that
Thus, B is the correct answer.
6.
Given the areas of the three squares in the figure, what is the area of the interior triangle?
Answer: B
Solution(s):
We get that the side lengths of the squares are respectively. Note that these lengths form a Pythagorean triple.
Therefore, the interior triangle is right. Its area is
Thus, B is the correct answer.
7.
Blake and Jenny each took four -point tests. Blake averaged on the four tests. Jenny scored points higher than Blake on the first test, points lower than him on the second test, and points higher on both the third and fourth tests. What is the difference between Jenny's average and Blake's average on these four tests?
Answer: A
Solution(s):
The total point difference between the two is The average of this difference is
Thus, A is the correct answer.
8.
Who gets the fewest cookies from one batch of cookie dough?
Answer: A
Solution(s):
Note that the person who has the largest cookie would make the fewest cookies.
Art's cookie has an area of
Roger's is Paul's is and Trisha's is
Thus, A is the correct answer.
9.
Art's cookies sell for each. To earn the same amount from a single batch, how much should one of Roger's cookies cost?
Answer: C
Solution(s):
From Problem we know that Art's cookie has an area of Since there are cookies in a batch, each batch has of dough.
Since Roger's cookie has an area of Roger can make cookies.
Art would make from his cookies, so Roger would need to charge per cookie.
Thus, C is the correct answer.
10.
How many cookies will be in one batch of Trisha's cookies?
Answer: E
Solution(s):
From Problem we know that a batch has of cookie dough. We also know that Trisha's cookies have an area of
Therefore, Trisha can make cookies per batch.
Thus, E is the correct answer.
11.
Business is a little slow at Lou's Fine Shoes, so Lou decides to have a sale. On Friday, Lou increases all of Thursday's prices by Over the weekend, Lou advertises the sale: "Ten percent off the listed price. Sale starts Monday." How much does a pair of shoes cost on Monday that cost $40 on Thursday?
$36
$39.60
$40
$40.40
$44
Answer: B
Solution(s):
On Friday, the shoes would cost 40 \cdot 1.1 = 44 $. On Monday, they would cost 44 \cdot 0.9 = 39.6 $.
Thus, B is the correct answer.
12.
When a fair six-sided dice is tossed on a table top, the bottom face cannot be seen. What is the probability that the product of the faces that can be seen is divisible by
Answer: E
Solution(s):
Note that if is not the bottom case, then the product is automatically divisible by since is a factor.
If is on the bottom, then and are seen, also ensuring that the product is divisible by
Therefore, every possible scenario allows for the product to be divisible by
Thus, E is the correct answer.
13.
Fourteen white cubes are put together to form the figure on the right. The complete surface of the figure, including the bottom, is painted purple. The figure is then separated into individual cubes. How many of the individual cubes have exactly four purple faces?
Answer: B
Solution(s):
The cubes on top have exposed faces, so they don't work. The corners in the bottom row have exposed sides, so they don't work either.
Every other cube has exposed sides, so those work. There are cubes that work.
Thus, B is the correct answer.
14.
In this addition problem, each letter stands for a different digit. If and the letter represents an even number, what is the only possible value for
Answer: D
Solution(s):
Since both 's are we get that is either or Since is even, we get that
Then, we get that We also know that doesn't carry over, since otherwise would be
Therefore, is less than and cannot be or If then which is also not allowed.
This makes
Thus, D is the correct answer.
15.
A figure is constructed from unit cubes. Each cube shares at least one face with another cube. What is the minimum number of cubes needed to build a figure with the front and side views shown?
Answer: B
Solution(s):
We need at least cubes to form the front view. Then we also need another cube to ensure that the side views are correct.
Thus, B is the correct answer.
16.
Ali, Bonnie, Carlo and Dianna are going to drive together to a nearby theme park. The car they are using has four seats: one driver's seat, one front passenger seat and two back passenger seat. Bonnie and Carlo are the only ones who know how to drive the car. How many possible seating arrangements are there?
Answer: D
Solution(s):
There are options for who sits in the driver's seat. There are options for the other front seat, and options for the first passenger seat.
The last person has to sit in the last seat, for a total of possible seating arrangements.
Thus, D is the correct answer.
17.
The six children listed below are from two families of three siblings each. Each child has blue or brown eyes and black or blond hair. Children from the same family have at least one of these characteristics in common. Which two children are Jim's siblings?
Nadeen and Austin
Benjamin and Sue
Benjamin and Austin
Nadeen and Tevyn
Austin and Sue
Answer: E
Solution(s):
Note that Nadeen, Austin, and Sue are the only individuals who share a characteristic with Jim. We need to find which of the are completely different from the others.
Austin and Sue both have blue eyes, which makes Nadeen the odd one out. Therefore, Austin and Sue are Jim's siblings.
Thus, E is the correct answer.
18.
Each of the twenty dots on the graph below represents one of Sarah's classmates. Classmates who are friends are connected with a line segment. For her birthday party, Sarah is inviting only the following: all of her friends and all of those classmates who are friends with at least one of her friends. How many classmates will not be invited to Sarah's party?
Answer: D
Solution(s):
Note that Sarah invites only the dots that are most line segments away from Sarah.
There are people who are completely disconnected from Sarah, and there are people who are lines away from Sarah.
Sarah will not invite any of these people.
Thus, D is the correct answer.
19.
How many integers between and have all three of the numbers and as factors?
Answer: C
Solution(s):
If a number has these three numbers as factors, then their least common multiple must also divide
These numbers have the following prime factorizations:
From these values, we get that the least common multiple is
Therefore, the multiples of between and are and
Thus, C is the correct answer.
20.
What is the measure of the acute angle formed by the hands of the clock at a.m.?
Answer: D
Solution(s):
At the hour hand will be a of the way between and
Each hour represents This means the hour hand will be past
Note that at minutes, the minute hand will be at This means the degree formed is
Thus, D is the correct answer.
21.
The area of trapezoid is The altitude is cm, is cm, and is cm. What is in centimeters?
Answer: B
Solution(s):
Drop perpendiculars from and to and let them hit at and
Then, using the Pythagorean theorem, we get that
Then the area of can be expressed as Note that: Substituting, we get that
Thus, B is the correct answer.
22.
The following figures are composed of squares and circles. Which figure has a shaded region with largest area?
only
only
only
both and
all are equal
Answer: C
Solution(s):
The shaded area of is
The shaded area of is
Note that the diagonal of the square in is equal to the diameter of the circle.
Therefore, the side length is This makes the area of the shaded region
Thus, C is the correct answer.
23.
In the pattern below, the cat moves clockwise through the four squares and the mouse moves counterclockwise through the eight exterior segments of the four squares.
If the pattern is continued, where would the cat and mouse be after the th move?
Answer: A
Solution(s):
We can find the positions of the mouse and cat individually. Note that the mouse's position repeats every moves and the cat's every moves.
has a remainder of when divided by which means the cat is in the same position as after the rd move, which is the bottom right square.
has a remainder of when divided by which means the cat is in the same position as after the th move, which is the bottom left segment.
Thus, A is the correct answer.
24.
A ship travels from point to point along a semicircular path, centered at Island Then it travels along a straight path from to Which of these graphs best shows the ship’s distance from Island as it moves along its course?
Answer: B
Solution(s):
Note that any point along a circle is the same distance from its center. This means that travelling from to the distance from remains constant.
We also know that as we move from to the midpoint of we get closer to
Similarly, as we move from the midpoint towards we get farther from This can be represented by a downwards facing semicircle.
Thus, B is the correct answer.
25.
In the figure, the area of square is The four smaller squares have sides cm long, either parallel to or coinciding with the sides of the large square. In and when is folded over side point coincides with the center of square What is the area of in square centimeters?
Answer: C
Solution(s):
We get that the side lengths of are cm, since
We also know that the distance from to is since it is the sum of the side lengths of unit squares.
Finally, the distance from to is the same as the distance from to which is
Now, we can find which is
Therefore, the area of is
Thus, C is the correct answer.