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A polynomial with rational roots

Source: 2018 Balkan MO Shortlist N4

May 22, 2019
polynomialnumber theory

Problem Statement

Let P(x)=adxd+⋯+a1x+a0P(x)=a_d x^d+\dots+a_1 x+a_0 be a non-constant polynomial with non-negative integer coefficients having dd rational roots.Prove that lcm(P(m),P(m+1),…,P(n))≥m(nm)\text{lcm} \left(P(m),P(m+1),\dots,P(n) \right)\geq m \dbinom{n}{m} for all n>mn>m
(Navid Safaei, Iran)
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