Olympiad problems

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Found problems: 26

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2022 Baltic Way, 7

The writer Arthur has $n \ge1$ co-authors who write books with him. Each book has a list of authors including Arthur himself. No two books have the same set of authors. At a party with all his co-author, each co-author writes on a note how many books they remember having written with Arthur. Inspecting the numbers on the notes, they discover that the numbers written down are the first $n$ Fibonacci numbers (defined by $F_1 = F_2 = 1$ and $F_{k+2}= F_{k+1} + F_k$). For which $n$ is it possible that none of the co-authors had a lapse of memory?

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