Problem
ALG-B2-M07-P009 Fractions with neighboring sums
#9
★★★☆☆ 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. This is a homogeneous inequality of degree \(1\); apply Cauchy.
By Cauchy, \[\sum\frac{a^2}{a+b}\ge\frac{(a+b+c)^2}{(a+b)+(b+c)+(c+a)}=\frac{(a+b+c)^2}{2(a+b+c)}=\frac{a+b+c}{2}.\]
A good example where sum normalization is allowed, but Cauchy is faster.