2022 SSMO Speed Round Problems/Problem 4
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