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max of a/c+c/a+b/d+d/b. if a/b + b/c + c/d + d/a=4, ac = bd,

Source: 2012 Romania JBMO TST 3.1

June 2, 2020

algebramaxinequalitiessystem of equations

Problem Statement

Let

a, b, c, d

be distinct non-zero real numbers satisfying the following two conditions:

ac = bd

and

\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}= 4

. Determine the largest possible value of the expression

\frac{a}{c}+\frac{c}{a}+\frac{b}{d}+\frac{d}{b}