Harder X1 inequalities (1 Viewer)

[leekeenyan]

Prove that \frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4
\\and \frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq 3

Or just tell me what steps I take so that I actually understand it. Thanks.

(I know this is one of the easier proofs but meh)

 

[Trebla]

Just to make it easier to read, it says prove that:



and



Both can be proved from the arithmetic mean and geometric mean inequality:

For the second one, one can derive the result:



Replace a with a/b, b with b/c and c with c/a, which leaves



The first one can be approached in a similar way...

 

[seanieg89]

If you do not want to prove AM-GM for n=3 as a separate step, the second inequality follows from simplifying: