We are given four positive real numbers $a,b,c,d$. I would like to prove that $$\frac ab+\frac bc+\frac cd+\frac da\ge4$$ To start solving this I assumed $$a\ge b\ge c\ge d$$ Therefore $$\frac ab,\frac bc,\frac cd\ge 1,\ \frac da\le 1$$ and the last one is causing the problem.
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$\begingroup$ What are you asking? $\endgroup$– mosheOct 6, 2016 at 14:02
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$\begingroup$ to prove the above. $\endgroup$– Lucifer -Oct 6, 2016 at 14:03
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3$\begingroup$ Use AM GM inequality $\endgroup$– lab bhattacharjeeOct 6, 2016 at 14:03
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$\begingroup$ Is $a\geq b\geq c\geq d$ part of the questions definitions? $\endgroup$– mosheOct 6, 2016 at 14:04
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$\begingroup$ no. i assumed it $\endgroup$– Lucifer -Oct 6, 2016 at 14:05
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1 Answer
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As lab bhattacharjee commented, You can use the AM-GM inequality. By applying it here, we can obtain:
$$ \begin{align*} \frac{\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a}}{4} & \ge \sqrt[4]{\frac{abcd}{bcda}}\\ & = \sqrt[4]{1}\\ & = 1 \end{align*} $$
Therefore:
$$ \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} \ge 4 $$
