1981 IMO Problems/Problem 3
Determine the maximum value of , where and are integers satisfying and .
We first observe that since , and are relatively prime. In addition, we note that , since if we had , then would be the sum of two negative integers and therefore less than . We now observe
i.e., is a solution iff. is also a solution. Therefore, for a solution , we can perform the Euclidean algorithm to reduce it eventually to a solution . It is easy to verify that if is a positive integer, it must be either 2 or 1. Thus by trivial induction, all the positive integer solutions are of the form , where the are the Fibonacci numbers. Simple calculation reveals and to be the greatest Fibonacci numbers less than , giving as the maximal value.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
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