elementary number theory - Prove that there is no polynomial with integer coefficients such that $p(a)=b,,p(b)=c,,p(c)=a$ for distinct integers $a,b,c$

Our teacher gave us this question but I am very stuck. I drew graphs to see why it cant be true but I didnt find anything. I see that if $p$ existed then: $$p(\cdots p(a))=a,\;p(\cdots p(b))=b,\;p(\cdots p(c))=c$$

(With $3n$ $p$'s). But I don't know where the contradiction is... Maybe we can work on the divisibility of the polynomials coefficients or something?

Answer

Hint $\ $ Writing $\rm\,px\,$ for $\rm\,p(x)\,$ and applying the Factor Theorem we have

$$\rm\,\ a\!-\!b\mid pa\!-\!pb = b\!-\!c\mid pb\!-\!pc=c\!-\!a\mid pc\!-\!pa = a\!-\!b\ $$

therefore we deduce $\rm\,\ a\!-\!b\mid b\!-\!c\mid c\!-\!a\mid a\!-\!b\ $ hence $\rm\ \ldots$