Prove $Sigma_{cyc}(frac{a}{b-c}-3)^4ge193$
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The inequality is expected original question of this MSE question. The exact statement is "If $a$ , $b$ and $c$ are positive real numbers and none of them are equal pairwise, prove the following inequality." $$Sigma_{cyc}left(frac{a}{b-c}-3right)^4ge193$$ Full expanding gives 12-degree polynomial with about 90 terms. It starts with $Sigma_{cyc}(a^{12}-16a^{11}b+8a^{11}c)$ and it does not look good for Muirhead or Schur. Also I tried substitution of $frac{a}{b-c}=x$ , $frac{b}{c-a}=y$ and $frac{c}{a-b}=z$ . Then by $uvw$ , it suffices to show when $x=y$ (See answer to linked question for details). That is, $frac{a}{b-c}=frac{b}{c-a}$ or $c=frac{a^2+b^2}{a+b}$ , therefore either $a<c<b$ or $b<c<a$ . Given the constraints, it is clear that $x>0$ and substitutin...
