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min length of EC and the polynomials for which this is attained wanted

Source: 2020 Balkan MO shortlist A4

September 14, 2021
algebrapolynomial

Problem Statement

Let P(x)=x3+ax2+bx+1P(x) = x^3 + ax^2 + bx + 1 be a polynomial with real coefficients and three real roots ρ1\rho_1, ρ2\rho_2, ρ3\rho_3 such that ∣ρ1∣<∣ρ2∣<∣ρ3∣|\rho_1| < |\rho_2| < |\rho_3|. Let AA be the point where the graph of P(x)P(x) intersects yy′yy' and the point B(ρ1,0)B(\rho_1, 0), C(ρ2,0)C(\rho_2, 0), D(ρ3,0)D(\rho_3, 0). If the circumcircle of △ABD\vartriangle ABD intersects yy′yy' for a second time at EE, find the minimum value of the length of the segment ECEC and the polynomials for which this is attained.
Brazitikos Silouanos, Greece
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