Let a,b and c be three positive real numbers . Prove that a+b+c ≤ ((a^2 +bc)/(b+c))+((b^2 +ca)/(c+a))+((c^2 +ab)/(a+b))


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Question Number 131465 by bramlexs22 last updated on 05/Feb/21

$L e t a, b a n d c b e t h r e e p o s i t i v e r e a l n u m b e r s$ $. P r o v e t h a t a + b + c ⩽ \frac{a^{2} + b c}{b + c} + \frac{b^{2} + c a}{c + a} + \frac{c^{2} + a b}{a + b}$
	Answered by EDWIN88 last updated on 05/Feb/21

	$A c c o r d i n g t o t h e i d e n t i t y \frac{a^{2} + b c}{b + c} - a = \frac{(a - b) (a - c)}{b + c}$ $w e c a n c h a n g e o u r i n e q u a l i t y i n t o t h e f o r m$ $x (a - b) (a - c) + y (b - a) (b - c) + z (c - a) (c - b) ⩾ 0$ $i n w h i c h x = \frac{1}{b + c}, y = \frac{1}{c + a}, z = \frac{1}{a + b}$ $W L O G, a s s u m e a ⩾ b ⩾ c t h e n x ⩽ y ⩽ z$
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