Let a,b,c be postive real numbers prove that ((a/(b+c))+(b/(a+c)))((b/(a+c))+(c/(a+b)))((c/(a+b))+(b/(a+c)))≥1


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Question Number 203891 by York12 last updated on 01/Feb/24

$Let a, b, c be postive real numbers prove that$ $(\frac{a}{b + c} + \frac{b}{a + c}) (\frac{b}{a + c} + \frac{c}{a + b}) (\frac{c}{a + b} + \frac{b}{a + c}) ⩾ 1$
	Answered by sniper237 last updated on 01/Feb/24

	$T h e l a s t f a c t o r s h o u l d b e (\frac{c}{a + b} + \frac{a}{b + c})$ $I f s o, L e t n a m e d P t h a t p r o d u c t$ $D i v i d e e a c h f a c t o r b y c, a, b r e s p$ $P = (\frac{a / c}{1 + b / c} + \frac{b / c}{1 + a / c}) . . . .$ $P ⩾ (\frac{a + b}{c}) (\frac{b + c}{a}) (\frac{c + a}{b}) ⩾ \frac{2 \sqrt{a b} 2 \sqrt{b c} 2 \sqrt{c a}}{a b c}$ $T h e n P ⩾ 8$
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