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Infinite sequence of polynomials

Source: 2024 USEMO/4

October 27, 2024
algebrapolynomialUSEMO 2024

Problem Statement

Find all sequences a1a_1, a2a_2, …\dots of nonnegative integers such that for all positive integers nn, the polynomial 1+xa1+xa2+⋯+xan1+x^{a_1}+x^{a_2}+\dots+x^{a_n} has at least one integer root. (Here x0=1x^0=1.)
Kornpholkrit Weraarchakul
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