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n Tans

Source: USAMO 1998

October 9, 2005
trigonometryfunctioninequalitiesinequalities proposedn-variable inequalityHi

Problem Statement

Let a0,a1,⋯ ,ana_0,a_1,\cdots ,a_n be numbers from the interval (0,π/2)(0,\pi/2) such that tan⁡(a0−π4)+tan⁡(a1−π4)+⋯+tan⁡(an−π4)≥n−1. \tan (a_0-\frac{\pi}{4})+ \tan (a_1-\frac{\pi}{4})+\cdots +\tan (a_n-\frac{\pi}{4})\geq n-1. Prove that tan⁡a0tan⁡a1⋯tan⁡an≥nn+1. \tan a_0\tan a_1 \cdots \tan a_n\geq n^{n+1}.
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