2010 AIME I Problems/Problem 3
Contents
Problem
Suppose that and . The quantity can be expressed as a rational number , where and are relatively prime positive integers. Find .
Solution 1
Substitute into and solve.
Solution 2
We solve in general using instead of . Substituting , we have:
Dividing by , we get .
Taking the th root, , or .
In the case , , , , yielding an answer of .
Solution 3
Taking the logarithm base of both sides, we arrive with:
Where the last two simplifications were made since . Then,
Then, , and thus:
Solution 4 (another version of Solution 3)
Taking the logarithm base of both sides, we arrive with: Now we proceed by the logarithm rule . The equation becomes: Then find as in solution 3, and we get .
See Also
2010 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
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All AIME Problems and Solutions |
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