2010 AMC 8

Problem 1

At Euclid High School, the mathematics teachers are Mrs. Germain, Mr. Newton, and Mrs. Young. There are students in Mrs. Germain's class, 8 in Mr. Newton, and in Mrs. Young's class are taking the AMC this year. How many mathematics students at Euclid High 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, medium, and mode of the numbers, ?

Problem 5

Alice needs to replace a light bulb located centimeters below the ceiling of her kitchen. The ceiling is meters above the floor. Alice is meters tall and can reach centimeters above 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 has the greatest number of line 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 one dollar?

Problem 8

As Emily is riding her bike 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. 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 on a -problem test, on a -problem test, and on a -problem test. What percent of all problems did Ryan answer correctly?

Problem 10

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 with pepperoni?

Problem 11

The top of one tree is feet higher than the top of another tree. The height of the trees are at a ratio of . 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 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 shorter 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 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 ?

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);
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Problem 18

A decorative window is made up of a rectangle with semicircles at either end. The ratio of to is . And is 30 inches. What is the ratio of the area of the rectangle to the combined area of the semicircle.

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, she read of the pages plus more, and on the second day she read of the remaining pages plus more. On the third day she read of the remaining pages plus more. She then realizes she has pages left, which she finishes 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 of circle . 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, , $5^1^2$ (Error compiling LaTeX. Unknown error_msg), and $2^2^4$ (Error compiling LaTeX. Unknown error_msg)?

$\textbf{(A)}\ 2^2^4<10^8<5^1^2$ (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)

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?