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Maximize 4-variable ratio

Source: Austrian-Polish 1985, Problem 7

July 5, 2015
inequalitiesmaximizationratiofour variables

Problem Statement

Find an upper bound for the ratio x1x2+2x2x3+x3x4x12+x22+x32+x42\frac{x_1x_2+2x_2x_3+x_3x_4}{x_1^2+x_2^2+x_3^2+x_4^2} over all quadruples of real numbers (x1,x2,x3,x4)≠(0,0,0,0)(x_1,x_2,x_3,x_4)\neq (0,0,0,0).
Note. The smaller the bound, the better the solution.
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