2009 AIME II Problems

2009 AIME II (Answer Key) | AoPS Contest Collections • PDF

Instructions This is a 15-question, 3-hour examination. All answers are integers ranging from to , inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers. No aids other than scratch paper, graph paper, ruler, compass, and protractor are permitted. In particular, calculators and computers are not permitted.

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Problem 1

Before starting to paint, Bill had ounces of blue paint, ounces of red paint, and ounces of white paint. Bill painted four equally sized stripes on a wall, making a blue stripe, a red stripe, a white stripe, and a pink stirpe. Pink is a mixture of red and white, not necessarily in equal amounts. When Bill finished, he had equal amounts of blue, red, and white paint left. Find the total number of ounces of paint Bill had left.

Solution

Problem 2

Suppose that , , and are positive real numbers such that , , and . Find

Solution

Problem 3

In rectangle , . Let be the midpoint of . Given that line and line are perpendicular, find the greatest integer less than .

Solution

Problem 4

A group of children held a grape-eating contest. When the contest was over, the winner had eaten grapes, and the child in -th place had eaten grapes. The total number of grapes eaten in the contest was . Find the smallest possible value of .

Solution

Problem 5

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Problem 6

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Problem 7

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Problem 8

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Problem 9

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Problem 10

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Problem 11

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Problem 12

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Problem 13

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Problem 14

Solution

Problem 15

Solution

See also

• American Invitational Mathematics Examination
• AIME Problems and Solutions
• Mathematics competition resources