Problem
ALG-B2-M08-P015 Squares of cyclic ratios
#15
★★★★★ Level 5 of 5
Prove for \(a,b,c>0\): \[\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}\ge\frac{a}{c}+\frac{b}{a}+\frac{c}{b}.\]
Hint. Set \(x=a/b\), \(y=b/c\), \(z=c/a\), so \(xyz=1\).
After substitution, we need \(x^2+y^2+z^2\ge xy+yz+zx\). This is true because \(\frac12((x-y)^2+(y-z)^2+(z-x)^2)\ge0\).
Teaching goal: recognize the appropriate substitution and check the domain.