USAMO 2019 Problem 3

Let $K$ be the set of all positive integers that do not contain the digit $7$ in their base- $10$ representation. Find all polynomials $f$ with nonnegative integer coefficients such that $f(n)\in K$ whenever $n\in K$ .

Proposed by Titu Andreescu, Cosmin Pohoata, and Vlad Matei