MathNet 2016-part-02 — 184 problems (EN–VI) Question 43

Determine all positive integers $n$ for which there exist pairwise distinct positive real numbers $a_1, a_2, \dots, a_n$ satisfying $\{a_i + \frac{(-1)^i}{a_i} \mid 1 \le i \le n\} = \{a_i \mid 1 \le i \le n\}$.