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Integer Polynomials!

Source: INAMO 2015 Shortlist A7

December 30, 2018
integer polynomialsalgebrapolynomial

Problem Statement

Suppose P(n)P(n) is a nonconstant polynomial where all of its coefficients are nonnegative integers such that ∑i=1nP(i)∣nP(n+1) \sum_{i=1}^n P(i) | nP(n+1) for every n∈Nn \in \mathbb{N}. Prove that there exists an integer k≥0k \ge 0 such that P(n)=(n+kn−1)P(1) P(n) = \binom{n+k}{n-1} P(1) for every n∈Nn \in \mathbb{N}.
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