Problem

Source: ELMO 2014 Shortlist N3, by Michael Kural

Tags: algebra, polynomial, induction, binomial theorem, number theory proposed, number theory



Let $t$ and $n$ be fixed integers each at least $2$. Find the largest positive integer $m$ for which there exists a polynomial $P$, of degree $n$ and with rational coefficients, such that the following property holds: exactly one of \[ \frac{P(k)}{t^k} \text{ and } \frac{P(k)}{t^{k+1}} \] is an integer for each $k = 0,1, ..., m$. Proposed by Michael Kural