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Non homogeneous inequality of degree n-2

Source: Indian IMOTC 2013, Practice Test 2, Problem 1

July 30, 2013
inequalitiesinductionrearrangement inequalityinequalities proposed

Problem Statement

Let a,b,ca, b, c be positive real numbers such that a+b+c=1a + b + c = 1. If nn is a positive integer then prove that (3a)n(b+1)(c+1)+(3b)n(c+1)(a+1)+(3c)n(a+1)(b+1)≥2716 . \frac{(3a)^n}{(b + 1)(c + 1)} + \frac{(3b)^n}{(c + 1)(a + 1)} + \frac{(3c)^n}{(a + 1)(b + 1)} \ge \frac{27}{16} \,.
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