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The big inequality with condition abcd = 1

Source: International Zhautykov Olympiad 2013 - D1 - P3

January 17, 2013
inequalitiesinequalities unsolved

Problem Statement

Let a,b,ca, b, c, and dd be positive real numbers such that abcd=1abcd = 1. Prove that (a−1)(c+1)1+bc+c+(b−1)(d+1)1+cd+d+(c−1)(a+1)1+da+a+(d−1)(b+1)1+ab+b≥0.\frac{(a-1)(c+1)}{1+bc+c} + \frac{(b-1)(d+1)}{1+cd+d} + \frac{(c-1)(a+1)}{1+da+a} + \frac{(d-1)(b+1)}{1+ab+b} \geq 0. Proposed by Orif Ibrogimov, Uzbekistan.
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