I came to ask this because I am really stuck at this problem. I have tried everything from arithmetic mean, geometric mean and harmonic mean. Also, I have tried playing with the variables and such, but it got me to nowhere.
If $a+b+c=1$; $a,b,c$ nonnegative, calculate the minimum of $$\frac{4+3abc}{ab+bc+ac}$$ All I've got so far is: $$\frac{3abc}{ab+bc+ac} \le \frac{1}{3}$$ But this is obviously on the wrong side of the inequality. Also, I think that $$\frac{1}{ab+bc+ac}\ge3$$ But I haven't been able to prove it.
Playing with the most possible and obvious values, one could think that the answer is 37/3, but the excercise is about proving it. Any help and little hints are greatly apprecieated.