For a, b, c, d ∈ R a+b+c+d=0 ab, ac, ad, bc, bd, cd ≠0 Prove the inequality: ((ab)/((a+b)^2 ))+((ac)/((a+c)^2 ))+((ad)/((a+d)^2 ))+((bc)/((b+c)^2 ))+((bd)/((b+d)^2 ))+((cd)/((c+d)^2 ))≤−(3/2) When is equality reached?


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Question Number 182043 by depressiveshrek last updated on 03/Dec/22

$F o r a, b, c, d \in R$ $a + b + c + d = 0$ $a b, a c, a d, b c, b d, c d \neq 0$ $P r o v e t h e i n e q u a l i t y :$ $\frac{a b}{{(a + b)}^{2}} + \frac{a c}{{(a + c)}^{2}} + \frac{a d}{{(a + d)}^{2}} + \frac{b c}{{(b + c)}^{2}} + \frac{b d}{{(b + d)}^{2}} + \frac{c d}{{(c + d)}^{2}} \leq - \frac{3}{2}$ $W h e n i s e q u a l i t y r e a c h e d ?$
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