Let P(x) be a non-constatnt polynomial whose coefficients are positive integers.

Question

Let P(x) be a non-constatnt polynomial whose coefficients are positive integers.

Let P(x) be a non-constatnt polynomial whose coefficients are positive integers. If P(n) divides P(P(n) - 2015) for every natural number n, prove that P(- 2015) = 0.

Monjur Alahi

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2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Note that P(n) - 2015 - (-2015) = P(n) divides P(P(n) - 2015) - P(-2015) for every positive integer n. But P(n) divides P(P(n) - 2015) for every positive integer n. Therefore P(n) divides P(-2015) for every positive integer n. Hence P(-2015) = 0.

Note

In the original version of the problem the word 'non-constant' was missing. The falsity of the statement was brought to the attention of the examiners by a contestant who mentioned it in a remark at the end of a perfect solution to the problem assuming that the polynomial is non-constant. Many students assumed that the polynomial is non-constant and completed the solution. They deserved full credit for doing so.