Problem
ALG-B2-M12-P006 Set 2. Denominators with squares
#6
★★★★★ Level 5 of 5
Prove for \(a,b,c>0\): \[\sum_{\mathrm{cyc}}\frac{a^2}{b^2+c^2+a(b+c)}\ge\frac12.\]
Hint. Find the sum of denominators and apply Cauchy.
The sum of denominators is \(2(a^2+b^2+c^2)+2(ab+bc+ca)=2(a+b+c)^2\). Thus by Cauchy the left side is at least \((a+b+c)^2/(2(a+b+c)^2)=1/2\).
Mock set 2: the problem is intended for independent method selection without an explicit cue in the statement.