The following problem is from Larson's problem solving through problems:
If $a,b$ and $c$ are the roots of the equation $x^3-x^2-x-1=0$, show that $$ \frac{a^{1000}-b^{1000}}{a-b}+ \frac{b^{1000}-c^{1000}}{b-c}+ \frac{c^{1000}-a^{1000}}{c-a} $$ is an integer.
My way is to observe that $$f(x,y,z)=\frac{x^{1000}-y^{1000}}{x-y}+\frac{y^{1000}-z^{1000}}{y-z}+\frac{z^{1000}-x^{1000}}{z-x}$$ is a symmetric polynomial and hence the result follows immediately from the fundamental theorem of symmetric polynomials together with Viete's theorem.
But I would like to ask if there is a more elementary solution to this problem because of the origin of this problem, thanks!!