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Find all n such that two sets are identical

Source: Turkey National Olympiad 2014 P5

November 17, 2014
integrationalgebra proposedalgebra

Problem Statement

Find all natural numbers nn for which there exist non-zero and distinct real numbers a1,a2,…,ana_1, a_2, \ldots, a_n satisfying {ai+(−1)iai ∣ 1≤i≤n}={ai∣1≤i≤n}. \left\{a_i+\dfrac{(-1)^i}{a_i} \, \Big | \, 1 \leq i \leq n\right\} = \{a_i \mid 1 \leq i \leq n\}.
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