IOM 2019 Problem 6

Let $p$ be a prime and let $f(x)$ be a polynomial of degree $d$ with integer coefficients. Assume that the numbers $f(1),f(2),\dots,f(p)$ leave exactly $k$ distinct remainders when divided by $p$ , and $1<k<p$ . Prove that $$\frac{p-1}{d}\leq k-1\leq (p-1)\left(1-\frac1d \right) .$$

Dániel Domán, Gauls Károlyi, and Emil Kiss