IMO 1966 LL GDR2

Given n positive real numbers a1, a2, . . . , an such that a1a2 \cdot \cdot \cdot an

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IMO 1966 LL GDR2

Origin: GDR

Problem

Given n positive real numbers a1, a2, . . . , an such that a1a2 \cdot \cdot \cdot an = 1, prove that (1 + a1)(1 + a2) \cdot \cdot \cdot (1 + an) \geq2n.