Given a , b , c≥0 show that a2(a+b)(a+c)+b2(b+a)(b+c)+c2(c+a)(c+b)≥34.
I tried using Titu's lemma on it, resulting in
a2(a+b)(a+c)+b2(b+a)(b+c)+c2(c+a)(c+b)≥(a+b+c)2a2+b2+c2+3(ab+bc+ca)
And I am stuck here.
Answer
By C-S ∑cyca2(a+b)(a+c)≥(a+b+c)2∑cyc(a+b)(a+c)≥34, where the last inequality it's 4∑cyc(a2+2ab)≥3∑cyc(a2+3ab) or ∑cyc(a2−ab)≥0 or ∑cyc(2a2−2ab)≥0 or ∑cyc(a2+b2−2ab)≥0 or ∑cyc(a−b)2≥0.
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