Problem
ALG-B2-M12-P007 Set 2. Roots of pairs
#7
★★★★★ Level 5 of 5
Let \(a,b,c\ge0\), \(a+b+c=3\). Prove \[\sqrt{a^2+ab+b^2}+\sqrt{b^2+bc+c^2}+\sqrt{c^2+ca+a^2}\ge3\sqrt{3}.\]
Hint. Prove \(\sqrt{x^2+xy+y^2}\ge\frac{\sqrt{3}}2(x+y)\).
Since \(x^2+xy+y^2-\frac34(x+y)^2=\frac14(x-y)^2\ge0\), \(\sqrt{x^2+xy+y^2}\ge\frac{\sqrt{3}}2(x+y)\). Summing gives \(3\sqrt{3}\).
Mock set 2: the problem is intended for independent method selection without an explicit cue in the statement.