USAMO 1991 Problem 4

$\mathrm{a}$ and $\mathrm{b}$ are positive integers and $\mathrm{c}=\left(\mathrm{a}^{\mathrm{a}+1}+\mathrm{b}^{\mathrm{b}+1}\right) /\left(\mathrm{a}^{\mathrm{a}}+\mathrm{b}^{\mathrm{b}}\right)$. By considering $\left(\mathrm{x}^{\mathrm{n}}-\mathrm{n}^{\mathrm{n}}\right) /(\mathrm{x}-\mathrm{n})$ or otherwise, show that $c^{a}+c^{b} \geq a^{a}+b^{b}$.