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PAMO Problem 5: Inequality involving fractions with strange condition

Source: 2018 Pan-African Mathematics Olympiad

July 3, 2018
inequalitiesalgebraAMGM

Problem Statement

Let aa, bb, cc and dd be non-zero pairwise different real numbers such that ab+bc+cd+da=4 and ac=bd. \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} = 4 \text{ and } ac = bd.
Show that ac+bd+ca+db≤−12 \frac{a}{c} + \frac{b}{d} + \frac{c}{a} + \frac{d}{b} \leq -12 and that −12-12 is the maximum.
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