Problem

Source: Nikolai Beluhov, USAMTS 2017

Tags: algebra, polynomial, USAMTS



A positive integer is called downhill if the digits in its decimal representation form a nonstrictly decreasing sequence from left to right. Suppose that a polynomial $P(x)$ with rational coefficients takes on an integer value for each downhill positive integer $x$. Is it necessarily true that $P(x)$ takes on an integer value for each integer $x$?