Problem

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

Tags: inequalities, induction, rearrangement inequality, inequalities proposed



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