Thursday, 31 August 2017

contest math - 3 variable symmetric inequality



Show that for positive reals a,b,c,
a2b+c+b2c+a+c2a+b3a3+3b3+3c32a2+2b2+2c2



What I did was WLOG a+b+c=1 (since the inequality is homogenous)
Then I substituted into the LHS to get cyca21a3a3+3b3+3c32a2+2b2+2c2. Now I'm not sure if I should clear the denominators and expand and try to use Muirhead+Schur? (Clearing the denominators seems quite tedious)?




Any ideas are appreciated.


Answer



your inequalitiy is equivalent to
(a32abc+b3+c3)(2a3a2ba2cab2ac2+2b3b2cbc2+2c3)0


since a3+b3+c33abc>2abc
is the first factor positive,
and
2(a3+b3+c3)ab(a+b)+ac(a+c)+bc(b+c)

since a3+b3=(a+b)(a2+b2ab)ab(a+b) etc is the second factor also positive.


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