# Triple Inequalities

Algebra Level 5

Given positive real numbers $$(x,y,z)$$ such that $$xy + yz + xz = 5$$. The minimum value of $\dfrac{x^2}{y^3} + \dfrac{y^2}{z^3} + \dfrac{z^2}{x^3} + x + y + z$

is $$\frac{ a \sqrt{b} } { c}$$, where $$a, b,$$ and $$c$$ are positive integers, $$\gcd(a,c) = 1$$ , and $$b$$ is square free. What is the value of $$a + b + c$$?