Question 57
A positive integer $n$ is *downhill* if its decimal representation $\overline{a_k a_{k-1} \dots a_0}$ satisfies $a_k \ge a_{k-1} \ge \dots \ge a_0$. A real-coefficient polynomial $P$ is *integer-valued* if $P(n)$ is an integer for all integer $n$, and *downhill-integer-valued* if $P(n)$ is an integer for all downhill positive integers $n$. Is it true that every downhill-integer-valued polynomial is also integer-valued?