Problem
ALG-B2-M03-P010 Homogeneous fractional sum
#10
★★★★★ Level 5 of 5
Prove for \(a,b,c>0\): \[\frac{a^2}{a^2+ab+b^2}+\frac{b^2}{b^2+bc+c^2}+\frac{c^2}{c^2+ca+a^2}\ge\frac12.\]
Hint 1. Apply Engel form to numerators \(a,b,c\).
Hint 2. Compare the denominator sum with \(2(a+b+c)^2\).
By Cauchy, the left side is at least \[\frac{(a+b+c)^2}{2(a^2+b^2+c^2)+ab+bc+ca}.\] Since \(2(a^2+b^2+c^2)+ab+bc+ca\le2(a+b+c)^2\), we get the lower bound \(\frac12\).
Cauchy-Schwarz module training problem. Method tags: cauchy, homogeneous.