Problem
ALG-B2-M02-P008 Sum normalization
#8
★★★★☆ Level 4 of 5
Let \(a,b,c>0\). Prove \[\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\ge3.\]
Hint 1. Multiply the three fractions.
Hint 2. Their product is \(1\).
By AM-GM, \[\frac{\frac{a}{b}+\frac{b}{c}+\frac{c}{a}}{3}\ge\sqrt[3]{\frac{a}{b}\cdot\frac{b}{c}\cdot\frac{c}{a}}=1.\] The result follows.
AM-GM module training problem. Method tags: am-gm, homogeneous.