Problem

Source: 2012 Romania JBMO TST 3.1

Tags: algebra, max, inequalities, system of equations



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