# A Hard Way And A Hard Way

Algebra Level 5

$\large \frac{9a^{2}+9ab}{c^{2}+bc}+\frac{12b^{2}+12bc}{a^{2}+ac}+\frac{16c^{2}+16ac}{b^{2}+ab}$

Let $$a,b,c$$ be positive real numbers. The ratio of $$c:a$$ such that the above expression is minimized can be expressed as a fraction in the form

$\large \frac{m+\sqrt{n}}{p}$

where $$m,n,p$$ are integers and $$n$$ is not divisible by the square of any prime.

Find $$m+n+p$$.

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