Difference between revisions of "2023 SSMO Speed Round Problems/Problem 5"
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Revision as of 14:17, 3 July 2023
Problem
Let and for all be the Fibonacci numbers. If distinct positive integers satisfies , find the minimum possible value of
Solution
Suppose that are taken such is minimal and
Then, there are no consecutive and such .
Generalize to any . Let be maximal such We claim that .
If is a Fibonacci number the result follows.
Note that and follow inductively.
Thus, if then contradiction.
Then, be must maximal so we can reduce greedily to get and the answer is