Problem

Source: INAMO Shortlist 2015 A2

Tags: polynomial, functional equation, Natural Number, algebra



Suppose $a$ real number so that there is a non-constant polynomial $P (x)$ such that $\frac{P(x+1)-P(x)}{P(x+\pi)}= \frac{a}{x+\pi}$ for each real number $x$, with $x+\pi \ne 0$ and $P(x+\pi)\ne 0$. Show that $a$ is a natural number.