Suppose a polynomial with rational coefficients takes an integer value for all non-decreasing integers. In a nondecreasing integer, the digits never decrease from left to right. Examples include 122, 123, 224, and 237. Must it be true that the polynomial takes an integer value for all integer inputs?
My idea was to use induction on the degree. Basically it goes something like this: For a degree $n$ polynomial $P(x)$ define $d(x)=P(x+1)-P(x)$. But the induction sadly fails, since we can't conclude that $d(x)$ is an integer on all non-decreasing integers (we can get close). Any ideas?