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TOT 2006 Fall - Senior A-Level p7 sum x_i^2< (sum x_i)/n, sum x_i<(sum x_i^3)/2

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February 25, 2020
inequalitiesalgebraSum

Problem Statement

Positive numbers x1,...,xkx_1,..., x_k satisfy the following inequalities: x12+...+xk2<x1+...+xk2  and  x1+...+xk<x13+...+xk32x_1^2+...+ x_k^2 <\frac{x_1+...+x_k}{2} \ \ and \ \ x_1+...+x_k < \frac{x_1^3+...+ x_k^3}{2} a) Show that k>50k > 50, (3) b) Give an example of such numbers for some value of kk (3) c) Find minimum kk, for which such an example exists. (3)
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