Problem
ALG-B2-M11-P007 Sum equal to one
#7
★★★★☆ Level 4 of 5
Let \(a,b,c>0\), \(a+b+c=1\). Prove \[\frac{a^2}{a+b}+\frac{b^2}{b+c}+\frac{c^2}{c+a}\ge\frac12.\]
Hint. This is Problem 2 after normalization.
By Cauchy, \[\sum\frac{a^2}{a+b}\ge\frac{(a+b+c)^2}{2(a+b+c)}=\frac12.\]
Teaching goal: the student should first recognise the type of estimate, then choose the tool.