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a^2 + b^2 + c^2 + a^2b^2 + b^2c^2 + c^2a^2 \geq 15/16 if 8abc=1

Source: JBMO Shortlist 2014 A2

April 24, 2019

algebrainequalitiespositive realthree variable inequality

Problem Statement

Let

a, b, c

be positive real numbers such that

abc = \dfrac {1} {8}

. Prove the inequality:

a ^ 2 + b ^ 2 + c ^ 2 + a ^ 2b ^ 2 + b ^ 2c ^ 2 + c ^ 2a ^ 2 \geq \dfrac {15} {16}

When the equality holds?