Let a , b , c be real positive numbers & abc=1 prove that ((ab)/(a^5 +b^5 +ab))+((bc)/(b^5 +c^5 +bc))+((ac)/(a^5 +c^5 +ac))≤1


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Question Number 194297 by York12 last updated on 02/Jul/23

$L e t a, b, c b e r e a l p o s i t i v e n u m b e r s &$ $a b c = 1$ $p r o v e t h a t$ $\frac{a b}{a^{5} + b^{5} + a b} + \frac{b c}{b^{5} + c^{5} + b c} + \frac{a c}{a^{5} + c^{5} + a c} ⩽ 1$
	Answered by deleteduser1 last updated on 02/Jul/23

	$a^{5} + b^{5} + a b ⩾ a^{3} b^{2} + b^{3} a^{2} + a b = a^{2} b^{2} (a + b) + a b$ $\Rightarrow Σ \frac{a b}{a^{5} + b^{5} + a b} ⩽ Σ \frac{1}{a b (a + b + c)} = Σ \frac{c}{a + b + c} = 1$
	Commented by York12 last updated on 02/Jul/23

	$s i r w h a t i s t h e m o t i v a t i o n t o u s e t h e f i r s t i n e q u a l i t y$
	Commented by deleteduser1 last updated on 02/Jul/23
	$Rearrangement inequality Let {a_i },{b_i }_(1≤i≤n) be increasing sequences or both decreasing. Then Σ_(i=1) ^n a_i b_i ≥a_1 b_i_p +a_2 b_i_p_2 +....+a_n b_i_p_n where i_p_j is the jth term in any permution of {n}$
	$R e a r r a n g e m e n t i n e q u a l i t y$ $L e t {a_{i}}, {b_{i}}_{1 ⩽ i ⩽ n} b e i n c r e a s i n g s e q u e n c e s o r b o t h$ $d e c r e a s i n g .$ $T h e n \underset{i = 1}{\sum^{n}} a_{i} b_{i} ⩾ a_{1} b_{i_{p}} + a_{2} b_{i_{p_{2}}} + . . . . + a_{n} b_{i_{p_{n}}}$ $w h e r e i_{p_{j}} i s t h e j t h t e r m i n a n y p e r m u t i o n o f {n}$
	Commented by York12 last updated on 02/Jul/23

	$T h a n k s s o m u c h s i r, I r e a l l y a p p r e c i a t e$ $i t$
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