MathNet 2008-part-02 — 165 problems (EN–VI) Question 6

Let $p$ be a prime number and $f(x)$ be a polynomial with rational coefficients and is of degree $p$. Furthermore (i) $f(m)$ is an integer for any integer $m$ and (ii) $p$ divides $f(p+1) - f(1)$. Find all such polynomials.