Monday, September 2, 2019

contest math - Help with inequality problem



Given a , b , c0 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)2cyc(a+b)(a+c)34, where the last inequality it's 4cyc(a2+2ab)3cyc(a2+3ab) or cyc(a2ab)0 or cyc(2a22ab)0 or cyc(a2+b22ab)0 or cyc(ab)20.


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