USAMO 2019 — Days 1 and 2 Question 3

Let $K$ be the positive integers whose decimal representations contain no digit $7$. Determine all polynomials $f$ with nonnegative coefficients such that $f(x)\in K$ for every $x\in K$.

Proposed by Titu Andreescu, Vlad Matei, Cosmin Pohoata. Source: Evan Chen’s third-party solution notes.