ELMO SL 2014 N3

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