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Polynomial divides another one for all $n$ in $N$

Source: RMO (Mumbai Region) 2015 P3

December 6, 2015

algebrapolynomialalgebra proposedInteger PolynomialDivisibility

Problem Statement

Let

P(x)

be a polynomial whose coefficients are positive integers. If

P(n)

divides

P(P(n)-2015)

for every natural number

n

, prove that

P(-2015)=0

.
One additional condition must be given that

P

is non-constant, which even though is understood.