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Logarithm

Source: 1993 National High School Mathematics League, Exam One, Problem 11

February 27, 2020
logarithms

Problem Statement

Four real numbers x0>x1>x2>x3>0x_0>x_1>x_2>x_3>0, if log⁡x0x11993+log⁡x1x21993+log⁡x2x31993≥k⋅log⁡x0x31993\log_{\frac{x_0}{x_1}}1993+\log_{\frac{x_1}{x_2}}1993+\log_{\frac{x_2}{x_3}}1993\geq k\cdot\log_{\frac{x_0}{x_3}}1993 for all x0,x1,x2,x3x_0,x_1,x_2,x_3, then the maximum value of kk is________.
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