Solving simultaneous equation and inequation subject to constraints.

Let a, b, c be positive real number satisfying:

$\displaystyle \frac{1}{a+b+1}+\frac{1}{b+c+1} +\frac{1}{c+a+1}\geq1$

Prove that $\displaystyle a+b+c=ab+bc+ca$

Find all real numbers k and m and satisfying:

$\displaystyle k(x^3+y^3+z^3)+mxyz\geq(x+y+z)^3$

with all $\displaystyle x,y,z$non-negative.

Solve this equations:

$\displaystyle \sqrt[2]{y^2-8x+9}-\sqrt[3]{xy-6x+12}\leq1$and $\displaystyle \sqrt{2(x-y)^2+10x-6y+12}-\sqrt{y}=\sqrt{x+2}$

Thank you very much!