For Problem 10, rearrange the inequality to

$$a^3b^3 + b^3c^3 + c^3a^3 - 3a^2b^2c^2\geq a^3+b^3+c^3-3abc,$$

and factor both sides.  The fact that $a\geq 1$, $b\geq1$ and $c\geq1$ is important.

Problem 12 is a typical proof.  Remember $x$, $y$ and $z$ are symmetric, and you may assume $x\geq y\geq z$ without loss of generality.