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p(x) has at most n-1 different rational roots

Source: 2023 China South East Mathematical Olympiad Grade 10 P8 CSMO

April 6, 2024
algebrapolynomialgeometrygeometric transformation

Problem Statement

Let p(x)p(x) be an nn-degree (n≥2)(n \ge 2) polynomial with integer coefficients. If there are infinitely many positive integers mm, such that p(m)p(m) at most n−1n -1 different prime factors ff, prove that p(x)p(x) has at most n−1n-1 different rational roots .
[color=#f00]a help in translation is welcome
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