Problem
ALG-B1-M07-P022 Half of the sum
#22
★★★★☆ Level 4 of 5
Let \(a,b,c>0\). Prove \[\frac{a^2}{a+b}+\frac{b^2}{b+c}+\frac{c^2}{c+a}\ge\frac{a+b+c}{2}.\]
Apply Cauchy and carefully add the denominators.
By Cauchy, the left-hand side is at least \(\frac{(a+b+c)^2}{(a+b)+(b+c)+(c+a)}=\frac{(a+b+c)^2}{2(a+b+c)}=\frac{a+b+c}{2}\).
Equality can occur when \(a=b=c\), which is useful to check separately.