Let us have an $f(x)$ Integer-valued polynomial, which gains the value $1$ in $4$ different places. Prove, that in that case, it can't gain the value $-1$ on integer places.
I tried with $f(x)-1=0$, which is true $4$ times, so the polynom's degree must be at least 4. But how do I go on? Derivating seemed a good idea, but I don't know what do after, any ideas?
What you want to do is factor $f(x)-1$, which you know to have at least four distict linear factors. This then lets you write $f(x)$ in the form $(x-n_1)(x-n_2)(x-n_3)(x-n_4)g(x)+1$, then to show that $f(m) \ne -1$ for any integer $m$, you need to show that the product $(m-n_1)(m-n_2)(m-n_3)(m-n_4)g(m)$ cannot possibly take on the value $-2$. Here you will have to figure our how to take advantage of the fact that the $n_i$ are distinct.