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The non-existence of a polynomial

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$.

Source of the problem
Indian National Mathematical Olympiad 1986
Topic
Algebra
Difficulty Level
Easy
Suggested Book
An Excursion in Mathematics

Start with hints

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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