Problem
ALG-B2-M12-P014 Set 4. Two factors
#14
★★★★★ Level 5 of 5
Prove for \(a,b,c>0\): \[\sum_{\mathrm{cyc}}\frac{a^2}{(a+b)(a+c)}\ge\frac34.\]
Hint. After Cauchy, it remains to use \(a^2+b^2+c^2\ge ab+bc+ca\).
By Cauchy, the left side is at least \(\frac{(a+b+c)^2}{a^2+b^2+c^2+3ab+3bc+3ca}\). Comparing this fraction with \(3/4\) is equivalent to \(a^2+b^2+c^2\ge ab+bc+ca\).
Mock set 4: the problem is intended for independent method selection without an explicit cue in the statement.