IMO 1974 LL VIE48

Let a be a number different from zero. For all integers n define

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IMO 1974 LL VIE48

Origin: VIE

Problem

Let a be a number different from zero. For all integers n define Sn = an + a−n. Prove that if for some integer k both Sk and Sk+1 are integers, then for each integer n the number Sn is an integer.