Putnam 1983 A3

Let $p$ be an odd prime and let $$F(n)=1+2n+3n^2+\ldots+(p-1)n^{p-2}.$$ Prove that if $a$ and $b$ are distinct integers in $\{0,1,2,\ldots,p-1\}$ then $F(a)$ and $F(b)$ are not congruent modulo $p$ .