Problem

Source: Philippine Mathematical Olympiad 2017

Tags: inequalities, function



Find all positive real numbers \((a,b,c) \leq 1\) which satisfy \[ \huge \min \Bigg\{ \sqrt{\frac{ab+1}{abc}}\, \sqrt{\frac{bc+1}{abc}}, \sqrt{\frac{ac+1}{abc}} \Bigg \} = \sqrt{\frac{1-a}{a}} + \sqrt{\frac{1-b}{b}} + \sqrt{\frac{1-c}{c}}\]