Problem

Source: 2019 Saudi Arabia IMO TST II p2

Tags: algebra, polynomial, divisible



Let non-constant polynomial $f(x)$ with real coefficients is given with the following property: for any positive integer $n$ and $k$, the value of expression $$\frac{f(n + 1)f(n + 2)... f(n + k)}{ f(1)f(2) ... f(k)} \in Z$$Prove that $f(x)$ is divisible by $x$