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Polynomial with many different integer roots

Source: 239-School Open Olympiad 2022, Junior League P1

April 19, 2023
algebragamepolynomial

Problem Statement

Egor and Igor take turns (Igor starts) replacing the coefficients of the polynomial a99x99+⋯+a1x+a0a_{99}x^{99} + \cdots + a_1x + a_0with non-zero integers. Egor wants the polynomial to have as many different integer roots as possible. What is the largest number of roots he can always achieve?
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