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Centroamerican 2018, problem 5

Source: 2018 Centroamerican and Caribbean Math Olympiad

June 25, 2018
algebraPolynomialsCentroamericanpolynomial

Problem Statement

Let nn be a positive integer, 1<n<20181<n<2018. For each i=1,2,…,ni=1, 2, \ldots ,n we define the polynomial Si(x)=x2−2018x+liS_i(x)=x^2-2018x+l_i, where l1,l2,…,lnl_1, l_2, \ldots, l_n are distinct positive integers. If the polynomial S1(x)+S2(x)+⋯+Sn(x)S_1(x)+S_2(x)+\cdots+S_n(x) has at least an integer root, prove that at least one of the lil_i is greater or equal than 20182018.
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