USAMO 1982 Problem 2

Show that if $m, n$ are positive integers such that $\left(x^{m+n}+y^{m+n}+z^{m+n}\right) /(m+n)=\left(x^{m}+y^{m}+z^{m}\right) / m\left(x^{n}+y^{n}+\right.$ $\left.\mathrm{z}^{\mathrm{n}}\right) / \mathrm{n}$ for all real $\mathrm{x}, \mathrm{y}, \mathrm{z}$ with sum 0 , then $\{\mathrm{m}, \mathrm{n}\}=\{2,3\}$ or $\{2,5\}$.