KöMaL 2026 A940

Let $p$ be a prime number and let $k$ be an integer with $1\le k\le p$. Suppose that $f(x)$ is a polynomial with integer coefficients such that its value at every integer point gives a remainder of $0$ or $1$ when divided by $p^k$, and both cases occur. Show that the degree of $f$ is at least $k(p-1)$.