If is a polynomial with integer coefficients and , , , three distinct integers, then show that it is impossible to have , , .

Indian National Mathematical Olympiad 1986

Algebra

Easy

An Excursion in Mathematics

Do you really need a hint? Try it first!

Note that, for any two integers and we have .

Also, if then . Use this along with hint 1 to get .

Prove that, hint 2 gives .

Hint 3 gives . This is impossible unless . As this contradicts the hypothesis in the question, such a cannot exist.

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