2010 AMC 8 Problems

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?

Solution

Problem 2

If for positive integers, then what is ?

Solution

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?

Solution

Problem 4

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

Solution

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?

Solution

Problem 6

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

Solution

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?

Solution

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?

Solution

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?

Solution

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?

Solution

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?

Solution

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?

Solution

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?

Solution

Problem 14

What is the sum of the prime factors of ?

Solution

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?

Solution

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?

Solution

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 ?

import graph; size(300); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen xdxdff = rgb(0.49,0.49,1);
draw((0,0)--(6,0),linewidth(1.2pt)); draw((0,0)--(0,1),linewidth(1.2pt)); draw((0,1)--(1,1),linewidth(1.2pt)); draw((1,1)--(1,2),linewidth(1.2pt)); draw((1,2)--(5,2),linewidth(1.2pt)); draw((5,2)--(5,1),linewidth(1.2pt)); draw((5,1)--(6,1),linewidth(1.2pt)); draw((6,1)--(6,0),linewidth(1.2pt)); draw((1,1)--(5,1),linewidth(1.2pt)+linetype("2pt 2pt")); draw((1,1)--(1,0),linewidth(1.2pt)+linetype("2pt 2pt")); draw((2,2)--(2,0),linewidth(1.2pt)+linetype("2pt 2pt")); draw((3,2)--(3,0),linewidth(1.2pt)+linetype("2pt 2pt")); draw((4,2)--(4,0),linewidth(1.2pt)+linetype("2pt 2pt")); draw((5,1)--(5,0),linewidth(1.2pt)+linetype("2pt 2pt")); draw((0,0)--(5,1.5),linewidth(1.2pt));
dot((0,0),ds); label("$P$", (-0.23,-0.26),NE*lsf); dot((0,1),ds); dot((1,1),ds); dot((1,2),ds); dot((5,2),ds); label("$X$", (5.14,2.02),NE*lsf); dot((5,1),ds); label("$Y$", (5.12,1.14),NE*lsf); dot((6,1),ds); dot((6,0),ds); dot((1,0),ds); dot((2,0),ds); dot((3,0),ds); dot((4,0),ds); dot((5,0),ds); dot((2,2),ds); dot((3,2),ds); dot((4,2),ds); dot((5,1.5),ds); label("$Q$", (5.14,1.51),NE*lsf); clip((-4.19,-5.52)--(-4.19,6.5)--(10.08,6.5)--(10.08,-5.52)--cycle);
 (Error making remote request. Unknown error_msg)

Solution

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.

Solution

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?

Solution

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?

Solution

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?

Solution

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?

Solution

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 ?

Solution

Problem 24

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

$\textbf{(A)}\ 2^2^4<10^8<5^1^2$ (Error compiling LaTeX. Unknown error_msg) $\textbf{(B)}\ 2^2^4<5^1^2<10^8$ (Error compiling LaTeX. Unknown error_msg) $\textbf{(C)}\ 5^1^2<2^2^4<10^8$ (Error compiling LaTeX. Unknown error_msg) $\textbf{(D)}\ 10^8<5^1^2<2^2^4$ (Error compiling LaTeX. Unknown error_msg) $\textbf{(E)}\ 10^8<2^2^4<5^1^2$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 25

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

Solution

See Also

• AMC 8
• AMC 8 Problems and Solutions
• Mathematics competition resources

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.