MathDB
P_n (x) = x^{2n} + a_1x^{2n - 2} + a_2x^{2n - 4} +... + a_{n -1}x^2 + a_n

Source: 2020 Dutch IMO TST 2.1

November 22, 2020
algebrapolynomialinequalities

Problem Statement

Given are real numbers a1,a2,...,a2020a_1, a_2,..., a_{2020}, not necessarily different. For every n≥2020n \ge 2020, define an+1a_{n + 1} as the smallest real zero of the polynomial Pn(x)=x2n+a1x2n−2+a2x2n−4+...+an−1x2+anP_n (x) = x^{2n} + a_1x^{2n - 2} + a_2x^{2n - 4} +... + a_{n -1}x^2 + a_n, if it exists. Assume that an+1a_{n + 1} exists for all n≥2020n \ge 2020. Prove that an+1≤ana_{n + 1} \le a_n for all n≥2021n \ge 2021.
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