Problem
ALG-B2-M11-P003 Fractions with a reverse cycle
#3
★★★☆☆ Level 3 of 5
Prove for \(a,b,c>0\): \[\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\ge a+b+c.\]
Hint. Apply Cauchy to the three fractions.
By Cauchy, \[\sum\frac{a^2}{b}\ge\frac{(a+b+c)^2}{a+b+c}=a+b+c.\]
Teaching goal: the student should first recognise the type of estimate, then choose the tool.