Let a,b,c ≥ 0 and a^2 +b^2 +c^2 = 3. Prove that (1) Σ_(cyc) a^3 +Σ_(cyc) (a+b)^3 ≤ 27 (2) a^3 +b^3 +(b+c)^3 +(c+a)^3 ≥ (1/2)[c^3 +(a+b)^3 ] (3) For a≥b≥c≥0, a^3 +b^3 +(b+c)^3 +(c+a)^3 ≤ 2[c^3 +(a+b)^3 ]


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Question Number 145240 by loveineq last updated on 03/Jul/21

$Let a, b, c ⩾ 0 and a^{2} + b^{2} + c^{2} = 3 . Prove that$ $(1) \sum_{c y c} a^{3} + \sum_{c y c} {(a + b)}^{3} ⩽ 27$ $(2) a^{3} + b^{3} + {(b + c)}^{3} + {(c + a)}^{3} ⩾ \frac{1}{2} [c^{3} + {(a + b)}^{3}]$ $(3) For a ⩾ b ⩾ c ⩾ 0,$ $a^{3} + b^{3} + {(b + c)}^{3} + {(c + a)}^{3} ⩽ 2 [c^{3} + {(a + b)}^{3}]$
	Answered by mitica last updated on 03/Jul/21

	Commented by loveineq last updated on 03/Jul/21

	$t h a n k s y o u . I f i n d y o u!!!!!$
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