2011 AIME I Problems/Problem 9
Problem
Suppose is in the interval and . Find .
Solution 1
We can rewrite the given expression as Square both sides and divide by to get Rewrite as Testing values using the rational root theorem gives as a root, does fall in the first quadrant so it satisfies the interval. There are now two ways to finish this problem.
First way: Since , we have Using the Pythagorean Identity gives us . Then we use the definition of to compute our final answer. .
Second way: Multiplying our old equation by gives So, .
Solution 2
Like Solution 1, we can rewrite the given expression as Divide both sides by . Square both sides. Substitute the identity . Let . Then . Since , we can easily see that is a solution. Thus, the answer is .
Video Solution
~IceMatrix
See also
2011 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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