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cyclic product of (1/a + 1/bc) >= 1728

Source: 26th annual VJIMC (2016), Category II, Problem 1

April 10, 2016
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Problem Statement

Let a,b,ca,b,c be positive real numbers such that a+b+c=1a + b + c = 1. Show that
(1a+1bc)(1b+1ca)(1c+1ab)≥1728\left(\frac{1}{a} + \frac{1}{bc}\right)\left(\frac{1}{b} + \frac{1}{ca}\right)\left(\frac{1}{c} + \frac{1}{ab}\right) \geq 1728
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