Nemotron-IMO-Bench 2026 N16

Let $p2$ be an odd prime. For $i\in \{0,1,..,p-1\}$ call $f=a_0+a_1X+...+a_nX^n\in \mathbf{Z}[X]$ $i$-remainder if $$\sum_{j>0, p-1\mid j} a_j\equiv i\pmod p.$$ Prove that $f, f^2,..., f^{p-2}$ are $0$-remainder and $f^{p-1}$ is $1$-remainder if and only if $f(0), f(1),..., f(p-1)$ give pairwise distinct remainders modulo $p$.