Prove that $a+b+c\le \frac {a^3}{bc} + \frac {b^3}{ac} + \frac {c^3}{ab}$ One of my friend had just given me an inequality to solve which is stated below. 
Consider the three positive reals $a, b, c$ then prove that 

$$a+b+c\le \frac {a^3}{bc} + \frac {b^3}{ac} + \frac {c^3}{ab}$$

I have solved this inequality very easily using Muirhead. But my friend has no idea what Muirhead inequality is. So I want to know whether there is any other method to solve this problem except for Muirhead's inequality.

$$\frac{a^3}{bc}+\frac{b^3}{ca}+\frac{c^3}{ab}\ge\sqrt{\frac{a^3b^3}{abc^2}}+\sqrt{\frac{b^3c^3}{bca^2}}+\sqrt{\frac{c^3a^3}{cab^2}}=\frac{ab}{c}+\frac{bc}{a}+\frac{ca}{b}\ge\sqrt{\frac{ab^2c}{ac}}+\sqrt{\frac{bc^2a}{ba}}+\sqrt{\frac{ca^2b}{cb}}=a+b+c$$
Above, we have twice used the known inequality $x^2+y^2+z^2\ge xy+yz+xz$, or (equivalently)$x+y+z\ge\sqrt{xy}+\sqrt{yz}+\sqrt{xz}$.
