Problem
ALG-B2-M11-P002 Half of the sum
#2
★★★☆☆ Level 3 of 5
Prove for \(a,b,c>0\): \[\frac{a^2}{a+b}+\frac{b^2}{b+c}+\frac{c^2}{c+a}\ge\frac{a+b+c}{2}.\]
Hint. The sum of denominators is \(2(a+b+c)\).
By Cauchy, \[\sum\frac{a^2}{a+b}\ge\frac{(a+b+c)^2}{2(a+b+c)}=\frac{a+b+c}{2}.\]
Teaching goal: the student should first recognise the type of estimate, then choose the tool.