IMO 1971 LL GDR18

Let a1, a2, . . . , an be positive numbers, mg = (a1a2 \cdot \cdot \cdot an)1/n

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IMO 1971 LL GDR18

Origin: GDR

Problem

Let a1, a2, . . . , an be positive numbers, mg = (a1a2 \cdot \cdot \cdot an)1/n their geometric mean, and ma = (a1 + a2 + \cdot \cdot \cdot + an)/n their arithmetic mean. Prove that (1 + mg)n \leq(1 + a1) \cdot \cdot \cdot (1 + an) \leq(1 + ma)n.