Let \(a, b, c \) be non-negative real numbers. Prove that

\[ \frac{ ab}{a+4b+4c} + \frac{bc}{b+4c+4a} + \frac{ ac}{c+4a+4b} \leq \frac{ a+b+c}{9} \]

Hi! :) I saw this difficult inequality problem on the Studying Math FB Page but haven't been able to solve it for a very long time! (No copyright intended!) Any help would be appreciated. Thanks! :)

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TopNewestI have some solutions :) – Dinesh Chavan · 2 years, 11 months ago

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– Bogdan Simeonov · 2 years, 11 months ago

Could you please give it out?I tried homogenizing, linearization.The only other ways are expanding and then using some inequality like Muirhead or Holder.Log in to reply

Maybe since the inequality it homogeneous creating a condition like \(a+b+c\) might help? My other thought is try using Jensen's inequality because it's normally really useful with cyclic inequalities like this, and the \(9\) looks Jensen-esque. You could always just multiply everything up and AM-GM/ Murihead's? – Daniel Remo · 3 years, 1 month ago

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– Happy Melodies · 3 years, 1 month ago

Sure! I will try it out :) Thanks!Log in to reply