MathDB
2008 ToT Spring Senior A P3 sum x_i^2/n > sum y_i^2/(n-1)

Source:

March 7, 2020
algebrapolynomialinequalitiesSumroots

Problem Statement

A polynomial xn+a1xn−1+a2xn−2+...+an−2x2+an−1x+anx^n + a_1x^{n-1} + a_2x^{n-2} +... + a_{n-2}x^2 + a_{n-1}x + a_n has nn distinct real roots x1,x2,...,xnx_1, x_2,...,x_n, where n>1n > 1. The polynomial nxn−1+(n−1)a1xn−2+(n−2)a2xn−3+...+2an−2x+an−1nx^{n-1}+ (n - 1)a_1x^{n-2} + (n - 2)a_2x^{n-3} + ...+ 2a_{n-2}x + a_{n-1} has roots y1,y2,...,yn1y_1, y_2,..., y_{n_1}. Prove that x12+x22+...+xn2n>y12+y22+...+yn−12n−1\frac{x^2_1+ x^2_2+ ...+ x^2_n}{n}>\frac{y^2_1 + y^2_2 + ...+ y^2_{n-1}}{n - 1}
Back to ProblemsView on AoPS