Difference between revisions of "2009 AIME II Problems"
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== Problem 1 == | == Problem 1 == | ||
| + | Before starting to paint, Bill had <math>130</math> ounces of blue paint, <math>164</math> ounces of red paint, and <math>188</math> 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. | ||
[[2009 AIME II Problems/Problem 1|Solution]] | [[2009 AIME II Problems/Problem 1|Solution]] | ||
== Problem 2 == | == Problem 2 == | ||
| + | Suppose that <math>a</math>, <math>b</math>, and <math>c</math> are positive real numbers such that <math>a^{\log_3 7} = 27</math>, <math>b^{\log_7 11} = 49</math>, and <math>c^{\log_{11}25} = \sqrt{11}</math>. Find | ||
| + | <cmath> a^{(\log_3 7)^2} + b^{(\log_7 11)^2} + c^{(\log_{11} 25)^2}. </cmath> | ||
[[2009 AIME II Problems/Problem 2|Solution]] | [[2009 AIME II Problems/Problem 2|Solution]] | ||
== Problem 3 == | == Problem 3 == | ||
| + | In rectangle <math>ABCD</math>, <math>AB=100</math>. Let <math>E</math> be the midpoint of <math>\overline{AD}</math>. Given that line <math>AC</math> and line <math>BE</math> are perpendicular, find the greatest integer less than <math>AD</math>. | ||
[[2009 AIME II Problems/Problem 3|Solution]] | [[2009 AIME II Problems/Problem 3|Solution]] | ||
== Problem 4 == | == Problem 4 == | ||
| + | A group of children held a grape-eating contest. When the contest was over, the winner had eaten <math>n</math> grapes, and the child in <math>k</math>-th place had eaten <math>n+2-2k</math> grapes. The total number of grapes eaten in the contest was <math>2009</math>. Find the smallest possible value of <math>n</math>. | ||
[[2009 AIME II Problems/Problem 4|Solution]] | [[2009 AIME II Problems/Problem 4|Solution]] | ||
Revision as of 12:16, 6 April 2009
| 2009 AIME II (Answer Key) | AoPS Contest Collections • PDF | ||
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Instructions
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| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
to 
, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.Contents
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.
Problem 2
Suppose that 
, 
, and 
are positive real numbers such that 
, 
, and 
. Find
![\[a^{(\log_3 7)^2} + b^{(\log_7 11)^2} + c^{(\log_{11} 25)^2}.\]](http://latex.artofproblemsolving.com/6/8/0/680c5fe965e912d7a3cbf4cf823fe104d86d3124.png)
Problem 3
In rectangle 
, 
. Let 
be the midpoint of 
. Given that line 
and line 
are perpendicular, find the greatest integer less than 
.
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 .
