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Polynomial with 2023 roots.

Source: BdMO 2023 Higher Secondary National P10

February 12, 2023
algebrapolynomialinequalities

Problem Statement

Let all possible 20232023-degree real polynomials: P(x)=x2023+a1x2022+a2x2021+⋯+a2022x+a2023P(x)=x^{2023}+a_1x^{2022}+a_2x^{2021}+\cdots+a_{2022}x+a_{2023}, where P(0)+P(1)=0P(0)+P(1)=0, and the polynomial has 2023 real roots r1,r2,⋯r2023r_1, r_2,\cdots r_{2023} [not necessarily distinct] so that 0≤r1,r2,⋯r2023≤10\leq r_1,r_2,\cdots r_{2023}\leq1. What is the maximum value of r1⋅r2⋯r2023?r_1 \cdot r_2 \cdots r_{2023}?
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