Problem
ALG-B1-M07-P013 A cyclic fraction
#13
★★★☆☆ Level 3 of 5
Let \(a,b,c>0\). Prove that \(\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\ge a+b+c\).
Apply Cauchy in Engel form to the three fractions.
By Cauchy, \(\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\ge\frac{(a+b+c)^2}{a+b+c}=a+b+c\). Equality holds when \(a=b=c\).
The cyclic form should not hide the single sum of denominators.