Problem
ALG-B2-M07-P008 Squares of ratios
#8
★★★☆☆ Level 3 of 5
Prove for \(a,b,c>0\): \[\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}\ge3.\]
Hint. Use AM-GM again; the product of the terms is \(1\).
By AM-GM, \[\sum\frac{a^2}{b^2}\ge3\sqrt[3]{\frac{a^2}{b^2}\cdot\frac{b^2}{c^2}\cdot\frac{c^2}{a^2}}=3.\]
Reinforces degree \(0\) and products of ratios.