TR-MO 2014 Problem 5

Find all natural numbers $n$ for which there exist non-zero and distinct real numbers $a_1, a_2, \ldots, a_n$ satisfying $$\left\{a_i+\dfrac{(-1)^i}{a_i} \Big | 1 \leq i \leq n\right\} = \{a_i \mid 1 \leq i \leq n\}.$$