Let a≥b≥c≥0 , c^3 +(a+b)^3 ≠0 and a^2 +b^2 +c^2 = 3. Prove that (1/2) ≤ ((a^3 +(b+c)^3 +b^3 +(c+a)^3 )/(c^3 +(a+b)^3 )) ≤ 2 Determine when equality holds.


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Question Number 145198 by loveineq last updated on 03/Jul/21

$Let a ⩾ b ⩾ c ⩾ 0, c^{3} + {(a + b)}^{3} \neq 0 and a^{2} + b^{2} + c^{2} = 3 . Prove that$ $\frac{1}{2} ⩽ \frac{a^{3} + {(b + c)}^{3} + b^{3} + {(c + a)}^{3}}{c^{3} + {(a + b)}^{3}} ⩽ 2$ $Determine when equality holds .$
	Answered by mitica last updated on 03/Jul/21

	$u^{3} + v^{3} = (u + v) (u^{2} - u v + v^{2});$ $\Leftrightarrow \frac{1}{2} ⩽ \frac{6 + a c + b c - 2 a b}{3 + 2 a b - a c - b c} ⩽ 2 ∙$ $y = 2 a b - a c - b c \Rightarrow y = 2 a \cdot b - (a + b) \cdot c$ $2 a ⩾ a + b; b ⩾ c \Rightarrow y ⩾ 0$ $y ⩽ a^{2} + b^{2} - c (a + b) = 3 - c^{2} - c (a + b) = 3 - c (a + b + c) ⩽ 3$ $∙ \Leftrightarrow \frac{1}{2} ⩽ \frac{6 - y}{3 + y} ⩽ 2 \Leftrightarrow 0 ⩽ y ⩽ 3$
	Commented by loveineq last updated on 03/Jul/21

	$t h a n k s$
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