IMO Shortlist 2018 problem A2

Find all integers $n \geq 3$ for which there exist real numbers $a_1, a_2, \dots a_{n + 2}$ satisfying $a_{n + 1} = a_1$ , $a_{n + 2} = a_2$ and $a_ia_{i + 1} + 1 = a_{i + 2},$ for $i = 1, 2, \dots, n$ .

Izvor: https://www.imo-official.org/problems/IMO2018SL.pdf

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IMO Shortlist 2018 problem A2