Difference between revisions of "1989 USAMO Problems/Problem 5"
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so <math>u<v</math>, as desired. <math>\blacksquare</math> | so <math>u<v</math>, as desired. <math>\blacksquare</math> | ||
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* [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=356639#356639 Discussion on AoPS/MathLinks] | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=356639#356639 Discussion on AoPS/MathLinks] | ||
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[[Category:Olympiad Algebra Problems]] | [[Category:Olympiad Algebra Problems]] |
Latest revision as of 19:16, 18 July 2016
Problem
Let and be real numbers such that Determine, with proof, which of the two numbers, or , is larger.
Solution
The answer is .
We define real functions and as follows: We wish to show that if , then .
We first note that when , , , and , so Similarly, .
We also note that if , then Similarly . It then follows that .
Now, for all , Since and are both strictly increasing functions over the nonnegative reals, it then follows that so , as desired.
See Also
1989 USAMO (Problems • Resources) | ||
Preceded by Problem 4 |
Followed by Last Question | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.