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a^2+b^2+c^2≥p(ab+bc), and the four variable variation

Source: Bulgaria 1974 P3

June 20, 2021
inequalities

Problem Statement

(a) Find all real numbers pp for which the inequality x12+x22+x32≥p(x1x2+x2x3)x_1^2+x_2^2+x_3^2\ge p(x_1x_2+x_2x_3) is true for all real numbers x1,x2,x3x_1,x_2,x_3. (b) Find all real numbers qq for which the inequality x12+x22+x32+x42≥q(x1x2+x2x3+x3x4)x_1^2+x_2^2+x_3^2+x_4^2\ge q(x_1x_2+x_2x_3+x_3x_4) is true for all real numbers x1,x2,x3,x4x_1,x_2,x_3,x_4.
I. Tonov
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