Difference between revisions of "2010 AIME I Problems/Problem 3"
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== Problem == | == Problem == | ||
Suppose that <math>y = \frac34x</math> and <math>x^y = y^x</math>. The quantity <math>x + y</math> can be expressed as a rational number <math>\frac {r}{s}</math>, where <math>r</math> and <math>s</math> are relatively prime positive integers. Find <math>r + s</math>. | Suppose that <math>y = \frac34x</math> and <math>x^y = y^x</math>. The quantity <math>x + y</math> can be expressed as a rational number <math>\frac {r}{s}</math>, where <math>r</math> and <math>s</math> are relatively prime positive integers. Find <math>r + s</math>. |
Revision as of 16:02, 9 August 2018
Contents
[hide]Problem
Suppose that and . The quantity can be expressed as a rational number , where and are relatively prime positive integers. Find .
Solution
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 2
Taking the logarithm base of both sides, we arrive with:
Where the last two simplifications were made since . Then,
Then, , and thus:
See Also
2010 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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