let a,b,c be positive real numbers such that ab+bc+ac=3 prove the inquality ((a(b^2 +c^2 ))/(a^2 +bc))+((b(c^2 +a^2 ))/(b^2 +ac))+((c(b^2 +a^2 ))/(c^2 +ab))≥3


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Question Number 98983 by M±th+et+s last updated on 17/Jun/20

$l e t a, b, c b e p o s i t i v e r e a l n u m b e r s s u c h$ $t h a t a b + b c + a c = 3$ $p r o v e t h e i n q u a l i t y$ $\frac{a (b^{2} + c^{2})}{a^{2} + b c} + \frac{b (c^{2} + a^{2})}{b^{2} + a c} + \frac{c (b^{2} + a^{2})}{c^{2} + a b} ⩾ 3$
Commented by MJS last updated on 17/Jun/20

$due to symmetry extremes at a = b = c \Rightarrow$ $a b + b c + a c = 3 \Leftrightarrow 3 a^{2} = 3 \Rightarrow a = \pm 1$ $but a > 0 \Rightarrow a = b = c = 1$ $the inequation with a = b = c turns into$ $3 a ⩾ 3 \Rightarrow true for a = 1$ $now test if this is \min or \max by putting$ $a = . 999; b = 1.001 \Rightarrow c = 1.0000005$ $\Rightarrow lhs > 3$ $\Rightarrow proven$ $I know you want a different kind of proof$ $but this is the easiest path$
Commented by M±th+et+s last updated on 17/Jun/20

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