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Max and min value with constraint

Source: 2018 Latvia BW TST P1

June 4, 2022
algebrainequalitiesalgebra unsolved

Problem Statement

Let p1,p2,...,pnp_1,p_2,...,p_n be n≥2n\ge 2 fixed positive real numbers. Let x1,x2,...,xnx_1,x_2,...,x_n be nonnegative real numbers such that x1p1+x2p2+...+xnpn=1.x_1p_1+x_2p_2+...+x_np_n=1. Determine the (a) maximal; (b) minimal possible value of x12+x22+...+xn2x_1^2+x_2^2+...+x_n^2.
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