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Understand the problem

If $P(x)$ is a polynomial with integer coefficients and $a$, $b$, $c$, three distinct integers, then show that it is impossible to have $P(a)=b$, $P(b)=c$, $P(c)=a$.

Algebra
Easy
Suggested Book
An Excursion in Mathematics

Do you really need a hint? Try it first!

Note that, for any two integers $x$ and $y$ we have $x-y|P(x)-P(y)$.
Also, if $x|y$ then $|x|\le |y|$. Use this along with hint 1 to get $|a-b|=|b-c|=|c-a|$.
Prove that, hint 2 gives $a-b=b-c=c-a$.
Hint 3 gives $2a=b+c, 2b=c+a,2c=a+b$. This is impossible unless $a=b=c$. As this contradicts the hypothesis in the question, such a $P$ cannot exist.

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