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p(1) + p(2) + p(3) + ... + p(n) = p(n)q(n) [CMO 2018 - P4]

Source: 2018 Canadian Mathematical Olympiad - P4

March 31, 2018
algebranumber theorypolynomial

Problem Statement

Find all polynomials p(x)p(x) with real coefficients that have the following property: there exists a polynomial q(x)q(x) with real coefficients such that p(1)+p(2)+p(3)+⋯+p(n)=p(n)q(n)p(1) + p(2) + p(3) +\dots + p(n) = p(n)q(n) for all positive integers nn.
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