2010 AMC 8 真题与答案解析

Problem 1

At Euclid Middle School the mathematics teachers are Miss Germain, Mr. Newton, and Mrs. Young. There are students in Mrs. Germain's class, students in Mr. Newton's class, and students in Mrs. Young's class taking the AMC 8 this year. How many mathematics students at Euclid Middle School are taking the contest?

 

Problem 2

If for positive integers, then what is ?

 

Problem 3

The graph shows the price of five gallons of gasoline during the first ten months of the year. By what percent is the highest price more than the lowest price?

 

Problem 4

What is the sum of the mean, median, and mode of the numbers ?

 

Problem 5

Alice needs to replace a light bulb located centimeters below the ceiling in her kitchen. The ceiling is meters above the floor. Alice is meters tall and can reach centimeters above the top of her head. Standing on a stool, she can just reach the light bulb. What is the height of the stool, in centimeters?

 

Problem 6

Which of the following figures has the greatest number of lines of symmetry?

 

Problem 7

Using only pennies, nickels, dimes, and quarters, what is the smallest number of coins Freddie would need so he could pay any amount of money less than a dollar?

 

Problem 8

As Emily is riding her bicycle on a long straight road, she spots Emerson skating in the same direction mile in front of her. After she passes him, she can see him in her rear mirror until he is mile behind her. Emily rides at a constant rate of miles per hour, and Emerson skates at a constant rate of miles per hour. For how many minutes can Emily see Emerson?

 

Problem 9

Ryan got of the problems correct on a -problem test, on a -problem test, and on a -problem test. What percent of all the problems did Ryan answer correctly?

 

Problem 10

Six pepperoni circles will exactly fit across the diameter of a -inch pizza when placed. If a total of circles of pepperoni are placed on this pizza without overlap, what fraction of the pizza is covered by pepperoni?

 

Problem 11

The top of one tree is feet higher than the top of another tree. The heights of the two trees are in the ratio . In feet, how tall is the taller tree?

 

Problem 12

Of the balls in a large bag, are red and the rest are blue. How many of the red balls must be removed from the bag so that of the remaining balls are red?

 

Problem 13

The lengths of the sides of a triangle in inches are three consecutive integers. The length of the shortest side is of the perimeter. What is the length of the longest side?

 

Problem 14

What is the sum of the prime factors of ?

 

Problem 15

A jar contains five different colors of gumdrops: are blue, are brown, red, yellow, and the other gumdrops are green. If half of the blue gumdrops are replaced with brown gumdrops, how many gumdrops will be brown?

 

Problem 16

A square and a circle have the same area. What is the ratio of the side length of the square to the radius of the circle?

 

Problem 17

The diagram shows an octagon consisting of unit squares. The portion below is a unit square and a triangle with base . If bisects the area of the octagon, what is the ratio ?

 

Problem 18

A decorative window is made up of a rectangle with semicircles on either end. The ratio of to is , and is 30 inches. What is the ratio of the area of the rectangle to the combined areas of the semicircles?

 

Problem 19

The two circles pictured have the same center . Chord is tangent to the inner circle at , is , and chord has length . What is the area between the two circles?

 

Problem 20

In a room, of the people are wearing gloves, and of the people are wearing hats. What is the minimum number of people in the room wearing both a hat and a glove?

 

Problem 21

Hui is an avid reader. She bought a copy of the best seller Math is Beautiful. On the first day, Hui read of the pages plus more, and on the second day she read of the remaining pages plus pages. On the third day she read of the remaining pages plus pages. She then realized that there were only pages left to read, which she read the next day. How many pages are in this book?

 

Problem 22

The hundreds digit of a three-digit number is more than the units digit. The digits of the three-digit number are reversed, and the result is subtracted from the original three-digit number. What is the units digit of the result?

 

Problem 23

Semicircles and pass through the center . What is the ratio of the combined areas of the two semicircles to the area of circle ?

 

Problem 24

What is the correct ordering of the three numbers, , , and ?

 

Problem 25

Everyday at school, Jo climbs a flight of stairs. Jo can take the stairs , , or at a time. For example, Jo could climb , then , then . In how many ways can Jo climb the stairs?

 

 

1. Given that these are the only math teachers at Euclid Middle School and we are told how many from each class are taking the AMC 8, we simply add the three numbers to find the total.
2. Substitute and into the expression for to get:Thus, answer choice is correct.
3. The highest price was in Month 1, which was $17. The lowest price was in Month 3, which was $10. 17 is of 10, and is more than 10. Therefore, the answer is
4. Putting the numbers in numerical order we get the list The mode is The median is The average is The sum of all three is
5. Convert everything to the same unit. Since the answer is in centimeters, change meters to centimeters by moving the decimal place two places to the right.The ceiling is centimeters above the floor. The combined height of Alice and the light bulb when she reaches for it is centimeters. That means the stool's height needs to be
6. An equilateral triangle has lines of symmetry. A non-square rhombus has lines of symmetry. A non-square rectangle has lines of symmetry. An isosceles trapezoid has line of symmetry. A square has lines of symmetry.Therefore, the answer is .
7. You need dimes, nickel, and pennies for the first cents. From cents to cents, you only need to add quarter. From cents to cents, you also only need to add quarter. The same for cents to cents. Notice that we didn't put instead of . We are left with quarters, dimes, nickel, and pennies. Thus, the correct answer is .
8. Because they are both moving in the same direction, Emily is skating relative to Emerson mph. Now we can look at it as if Emerson is not moving at all[on his skateboard] and Emily is riding at mph. It takes herto skate the mile to reach him, and then the same amount of time to be mile ahead of him. This totals to
9. Ryan answered problems correct on the first test, on the second, and on the third. This amounts to a total of problems correct. The total number of problems is Therefore, the percentage is
10. The pepperoni circles' diameter is 2, since . From that we see that the area of the circles of pepperoni is . The large pizza's area is . Therefore, the ratio is
11. Let the height of the taller tree be and let the height of the smaller tree be . Since the ratio of the smaller tree to the larger tree is , we have . Solving for gives us
12. Since 80 percent of the 500 balls are red, there are 400 red balls. Therefore, there must be 100 blue balls. For the 100 blue balls to be 25% or of the bag, there must be 400 balls in the bag so 100 red balls must be removed. The answer is .
13. Since the length of the shortest side is a whole number and is equal to of the perimeter, it follows that the perimeter must be a multiple of . Adding the two previous integers to each answer choice, we see that . Thus, answer choice is correct.
14. First, we must find the prime factorization of . We add the factors up to get
15. We do to find the percent of gumdrops that are green. We find that of the gumdrops are green. That means there are gumdrops. If we replace half of the blue gumdrops with brown gumdrops, then of the jar's gumdrops are brown.
16. Let the side length of the square be , and let the radius of the circle be . Thus we have . Dividing each side by , we get . Since , we have
17. We see that half the area of the octagon is . We see that the triangle area is . That means that .Meaning,
18. We can set a proportion:We substitute with 30 and solve for AD.We calculate the combined area of semicircle by putting together semicircle and to get a circle with radius . Thus, the area is . The area of the rectangle is . We calculate the ratio:
19. Since is isosceles, bisects . Thus . From the Pythagorean Theorem, . Thus the area between the two circles is Note: The length is necessary information, as this tells us the radius of the larger circle. The area of the annulus is .
20. Let be the number of people wearing both a hat and a glove. Since the number of people wearing a hat or a glove must be whole numbers, the number of people in the room must be a multiple of . Since we are trying to find the minimum , we must use the smallest possible value for the number of people in the room. Similarly, we can assume that there are no people present who are wearing neither of the two items since this would unnecessarily increase the number of people in the room. Thus, we can say that there are people in the room, all of which are wearing at least a hat or a glove.It follows that there are people wearing gloves and people wearing hats. Then by applying the Principle of Inclusion Exclusion (PIE), the total number of people in the room wearing either a hat or a glove or both is . Since we know that this equals , it follows that , which implies that . Thus, is the correct answer.
21. Let be the number of pages in the book. After the first day, Hui had pages left to read. After the second, she had left. After the third, she had left. This is equivalent to
22. Let the hundreds, tens, and units digits of the original three-digit number be , , and , respectively. We are given that . The original three-digit number is equal to . The hundreds, tens, and units digits of the reversed three-digit number are , , and , respectively. This number is equal to. Subtracting this expression from the expression for the original number, we get . Thus, the units digit in the final result is
23. By the Pythagorean Theorem, the radius of the larger circle turns out to be . Therefore, the area of the larger circle is . Using the coordinate plane given, we find that the radius of each of the two semicircles to be 1. So, the area of the two semicircles is . Finally, the ratio of the combined areas of the two semicircles to the area of circle is .
24. Use brute force. , , and . Therefore, is the answer.
25. We will systematically consider all of the possibilities. A valid climb can be thought of as a sequence of some or all of the numbers , , and , in which the sum of the sequence adds to . Since there is only one way to create a sequence which contains all , all , or all , there are three possible sequences which only contain one number. If we attempt to create sequences which contain one and the rest , the sequence will contain one and four . We can place the in either the first, second, third, fourth, or fifth position, giving a total of five possibilities. If we attempt to create sequences which contain one and the rest , the sequence will contain one and three . We can place the in either the first, second, third, or fourth position, giving a total of four possibilities. For sequences which contain exactly two and the rest , the sequence will contain two and two . The two could be next to each other, separated by one in between, or separated by two in between. We can place the two next to each other in three ways, separated by one in two ways, and separated by two in only one way. This gives us a total of six ways to create a sequence which contains two and two . Note that we cannot have a sequence of only and since the sum will either be or greater than . We now only need to consider the case where we use all three numbers in the sequence. Since all three numbers add to , the number of permutations of the three numbers is . Adding up the number of sequences above, we get: . Thus, answer choice is correct.