Difference between revisions of "1974 IMO Problems/Problem 5"
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Latest revision as of 15:00, 29 January 2021
Problem 5
Determine all possible values of where are arbitrary positive numbers.
Solution
Note that We will now prove that can reach any range in between and .
Choose any positive number . For some variables such that and , let , , and . Plugging this back into the original fraction, we get The above equation can be further simplified to Note that is a continuous function and that is a strictly increasing function. We can now decrease and to make tend arbitrarily close to . We see , meaning can be brought arbitrarily close to . Now, set and for some positive real numbers . Then Notice that if we treat the numerator and denominator each as a quadratic in , we will get , where has a degree lower than . This means taking , which means can be brought arbitrarily close to . Therefore, we are done. ~Imajinary
See Also
1974 IMO (Problems) • Resources | ||
Preceded by Problem 4 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 6 |
All IMO Problems and Solutions |