Prove $\frac{a}{b} + \frac{b}{c}+\frac{c}{a} \geq \frac{c+a}{c+b} + \frac{a+b}{a+c} + \frac{b+c}{b+a}$ Prove that $\frac{a}{b} + \frac{b}{c}+\frac{c}{a} \geq \frac{c+a}{c+b} + \frac{a+b}{a+c} + \frac{b+c}{b+a}$ with a,b,c > 0 

Show ${a\over b} +{b\over c} +{c\over a}  \ge {c+a\over c+b}  + {a+b\over a+c}    +{b+c\over b+a}$
But this is the same as (1) a/b+b/c+c/a >= (2) avg(c,a)/avg(c,b)+avg(a,b)/avg(a,c)+avg(b,c)/avg(b,a)  . Thus it is sufficient to show a (mean preserving) spread of the inputs increases avg{a/b,b/c,c/a} . This is easily verified. 
