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2010 AMC 8
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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?
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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
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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
Ermenson 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. Ermenson
skates at a constant rate of miles
per hour. For how many minutes can Emily see Ermenson?
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?
http://www.artofproblemsolving.com/Forum/code.php?hash=fe5dbbcea5ce7e2988b8c69bcfdfde8904aabc1f&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=b1d5781111d84f7b3fe45a0852e59758cd7a87e5&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=1574bddb75c78a6fd2251d61e2993b5146201319&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=a6a849b746fe12c9e546ad9e4eae9dff8e651318&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=f83a383c0fa81f295d057f8f5ed0ba4610947817&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=2d8dd32633c14d11f45a9a459e4e53273570144d&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=9be642acfe79327d490affd899d711cbc34d26b9&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=7f2306429106e15a9ec530f6c14681ed1c864f87&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=f83a383c0fa81f295d057f8f5ed0ba4610947817&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=ca9e8166ed0546a29f36c58b8d152a87e87c0a69&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=a6a849b746fe12c9e546ad9e4eae9dff8e651318&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=da4b9237bacccdf19c0760cab7aec4a8359010b0&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=1574bddb75c78a6fd2251d61e2993b5146201319&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=19e735269c83ffeb39e7283653734dd9f9256549&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=4d134bc072212ace2df385dae143139da74ec0ef&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=7b52009b64fd0a2a49e6d8a939753077792b0554&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=c1dfd96eea8cc2b62785275bca38ac261256e278&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=37c73be71ba78d98ff01ff091608cadd89fd01e2&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=b1d5781111d84f7b3fe45a0852e59758cd7a87e5&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=ab9ace0bc2e3f369aa05403ab24c5008e1ffa894&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=af3e133428b9e25c55bc59fe534248e6a0c0f17b&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=cc15779c1914283f35b677b2a30e9f2caa43d744&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=f6e1126cedebf23e1463aee73f9df08783640400&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=7f2306429106e15a9ec530f6c14681ed1c864f87&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=d8ffccd325347ec48b7723c94acd5a0cd35ab26c&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=fe5dbbcea5ce7e2988b8c69bcfdfde8904aabc1f&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=7b52009b64fd0a2a49e6d8a939753077792b0554&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=ee2caa07b76d9053386202d6a9c27c949c65c58f&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=ee2caa07b76d9053386202d6a9c27c949c65c58f&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=568675b6b9c747c037ebb926fc0ee89dcbd708c9&sid=c9aa8f19565158a0703262bf2dcdc2d8
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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 ?
http://www.artofproblemsolving.com/Forum/code.php?hash=48c1216cae3fab8b397d5e7b56286e51f8fd9e22&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=cb4a6abf141bc67d1d7feee8596b5e248ecce124&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=5786196c254133b2f6fcd96afca560a69802dcb0&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=4cc6fe824c9e3fa0daf27f6cd0bc9f618c6f5c8f&type=2&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=41bdd60809433155fb52e93ed7feb9c968dc74ce&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=11816ef9a80f3b8271708949264a8f583f185489&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=ac3478d69a3c81fa62e60f5c3696165a4e5e6ac4&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=11816ef9a80f3b8271708949264a8f583f185489&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=b1d5781111d84f7b3fe45a0852e59758cd7a87e5&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=c7027fc0537e5168d232f65d57e22cd4f61fc43d&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=4ce7afcf48678fc8da1772a18795abb7b619e1dd&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=22d200f8670dbdb3e253a90eee5098477c95c23d&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=5786196c254133b2f6fcd96afca560a69802dcb0&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=cb4a6abf141bc67d1d7feee8596b5e248ecce124&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=a0f8ddcf04419a1581c3c83320c929008ff73471&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=48c1216cae3fab8b397d5e7b56286e51f8fd9e22&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=ac3478d69a3c81fa62e60f5c3696165a4e5e6ac4&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=208b22c126ad4dfab52eda44364ef7f33e83c2f0&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=e22cd461c068aea5dff1c3462214880d76b3e39c&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=adec537942454243a09be0c2ed2d7df8f8b69613&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=48c1216cae3fab8b397d5e7b56286e51f8fd9e22&sid=c9aa8f19565158a0703262bf2dcdc2d8
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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?
http://www.artofproblemsolving.com/Forum/code.php?hash=913d6161cdc874d85f778046b535c978be395ad3&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=1574bddb75c78a6fd2251d61e2993b5146201319&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=913d6161cdc874d85f778046b535c978be395ad3&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=b1d5781111d84f7b3fe45a0852e59758cd7a87e5&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=b1fb3bec6fdb22e19a94fe4c6c4481ccba2ee9f0&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=ae4f281df5a5d0ff3cad6371f76d5c29b6d953ec&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=913d6161cdc874d85f778046b535c978be395ad3&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=32096c2e0eff33d844ee6d675407ace18289357d&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=25ffe68159d73e07266aac9339c7d442a9912634&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=f052a0905e778d48071a2af01eade4db05acf108&type=2&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=06d945942aa26a61be18c3e22bf19bbca8dd2b5d&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=3996b6ea950b55eb675068b6eb119dc2d2d09055&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=06d945942aa26a61be18c3e22bf19bbca8dd2b5d&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=6d95c1847219c633950f8f1ceca9761315abfc19&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=aabebe5af23c70874c8a036b51a7a479f00bd1fd&sid=c9aa8f19565158a0703262bf2dcdc2d8
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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
[i]Art of Problem Solving Hand Book[/i]. 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
How many whole numbers between do not contain the digit ?
Problem 23
http://www.artofproblemsolving.com/Forum/code.php?hash=f76a73f93f04c429d0f7eb86428ad2c38c2ff57f&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=cdd57ac962db2915610097cd593fa2be9dfaac65&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=356a192b7913b04c54574d18c28d46e6395428ab&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=f76a73f93f04c429d0f7eb86428ad2c38c2ff57f&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=8f58d07d79688697e1713992c4e55aaae2bf8b26&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=511a418e72591eb7e33f703f04c3fa16df6c90bd&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=9e6a55b6b4563e652a23be9d623ca5055c356940&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=803ab79310ab4c905d897e5b8b61272124906f8e&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=f1abd670358e036c31296e66b3b66c382ac00812&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=f9d4061486beb3393b2008158466b69b3fa998ce&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=7b52009b64fd0a2a49e6d8a939753077792b0554&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=c0aab0c11f992f78a63bdc1caf2da7c6b632a96e&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=33e3c5eaf987288d7f2a7e59e30e9aeeb771152e&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=cc80df1f23a810ec9e1b7b0863bf8d648ccf9930&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=cb347a0ce208d753df48aa9fbc31e2712ad0d091&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=1b96f768b5c681ae90290f38cf981138011e44c7&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=dcb80bb612893c04ba3cd00435c5a9852a51e519&type=2&sid=c9aa8f19565158a0703262bf2dcdc2d8
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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, , , and ?
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?
http://www.artofproblemsolving.com/Forum/code.php?hash=c1dfd96eea8cc2b62785275bca38ac261256e278&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=431344e5f1152ac5a2ae985d565c41f1a533c42a&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=da4b9237bacccdf19c0760cab7aec4a8359010b0&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=356a192b7913b04c54574d18c28d46e6395428ab&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=77de68daecd823babbb58edb1c8e14d7106e83bb&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=77de68daecd823babbb58edb1c8e14d7106e83bb&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=2219fecb861ae82ba2706e49a065c8d155e9f2a6&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=c1dfd96eea8cc2b62785275bca38ac261256e278&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=118cc960a2afa9101f6689fc9411947d91d58ae0&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=66579c6293f790c2b6a7807ac5cb674e57138d6f&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=61e34f8c67bf605d50ee06e507d3f4e249b028d0&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=5ab80478a0b7a220a2b5ff865996b7dd76561d6e&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=06aafc594d503ea2f02e18c7050e92948340e3c2&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=2b17e2a37bf1df419464d696c105247badb47789&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=83f6e5e001792910983be523f7b3b14305a16a74&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=6b90b8c5a30864f48be95d61056de0c151a164d6&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=5c932fbe61cb823632b958e937e76013207e000c&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=1537d151583c437eeaa50a417da8b9e5f3b7b655&type=2&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=08a914cde05039694ef0194d9ee79ff9a79dde33&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=08a914cde05039694ef0194d9ee79ff9a79dde33&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=88d24b96933a8cd4418c0e1ec6ae5086009c84b6&sid=c9aa8f19565158a0703262bf2dcdc2d8http://www.artofproblemsolving.com/Forum/code.php?hash=b18a2dd85fb63753fa3084863b338be0f8f2d618&sid=c9aa8f19565158a0703262bf2dcdc2d8
