Problem
ALG-B2-M03-P013 Product and sum
#13
★★★★★ Level 5 of 5
Let \(a,b,c>0\) and \(abc=1\). Prove \[\frac{a^2}{a^2+ab+b^2}+\frac{b^2}{b^2+bc+c^2}+\frac{c^2}{c^2+ca+a^2}\ge\frac12.\]
Hint 1. The condition \(abc=1\) is not the main point here.
Hint 2. Use the same denominator sum as in the homogeneous problem.
By Cauchy, the left side is at least \[\frac{(a+b+c)^2}{2(a^2+b^2+c^2)+ab+bc+ca}.\] This denominator is at most \(2(a+b+c)^2\), so the left side is at least \(\frac12\). The condition \(abc=1\) shows that the problem may look multiplicative, but the method is homogeneous.
Cauchy-Schwarz module training problem. Method tags: cauchy, normalisation.