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SMO 2015 open q4

Source: SMO 2015 open

March 31, 2018
number theoryFibonacci sequence

Problem Statement

Let f0,f1,...f_0, f_1,... be the Fibonacci sequence: f0=f1=1,fn=fn−1+fn−2f_0 = f_1 = 1, f_n = f_{n-1} + f_{n-2} if n≥2n \geq 2. Determine all possible positive integers nn so that there is a positive integer aa such that fn≤a≤fn+1f_n \leq a \leq f_{n+1} and that a(1f1+1f1f2+⋯+1f1f2...fn)a( \frac{1}{f_1}+\frac{1}{f_1f_2}+\cdots+\frac{1}{f_1f_2...f_n} ) is an integer.
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