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Let $a,b,c$ be 3 distinct positive real numbers such that abc = 1. Prove that $$\frac{a^3}{\left(a-b\right)\left(a-c\right)}\ +\frac{b^3}{\left(b-c\right)\left(b-a\right)}\ +\ \frac{c^3}{\left(c-a\right)\left(c-b\right)}\ \geq 3$$

I tried AM-GM in many different ways, but it doesn't work since one of the terms on the LHS inevitably becomes negative. Any help is greatly appreciated.

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  • $\begingroup$ Try $a=\frac{1}{x},b=\frac{1}{y},c=\frac{1}{z}$ so $xyz=1$ $\endgroup$ Oct 16, 2016 at 20:04

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By AM-GM $$\sum\limits_{cyc}\frac{a^3}{(a-b)(a-c)}=-\sum\limits_{cyc}\frac{a^3}{(a-b)(c-a)}=-\sum\limits_{cyc}\frac{a^3(b-c)}{\prod\limits_{cyc}(a-b)}=a+b+c\geq3$$

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  • $\begingroup$ Could you explain how you got $- \sum\limits_{cyc}\frac{a^3(b-c)}{\prod\limits_{cyc}(a-b)}=a+b+c$ $\endgroup$
    – Airdish
    Oct 17, 2016 at 10:48
  • $\begingroup$ @Airdish Because $-\sum\limits_{cyc}a^3(b-c)=(a+b+c)(a-b)(b-c)(c-a)$ $\endgroup$ Oct 17, 2016 at 11:03
  • $\begingroup$ Yes that is what I'd like elaborated on. Maybe how it is factorized $\endgroup$
    – Airdish
    Oct 17, 2016 at 11:17
  • $\begingroup$ @Airdish $-\sum\limits_{cyc}(a^3c-a^3b)=a^3c-a^3b+b^3a-c^3a+c^3b-b^3c=(b-c)(-a^3+a(b^2+bc+c^2)-bc(b+c))=...$ $\endgroup$ Oct 17, 2016 at 11:47
  • $\begingroup$ @Airdish $-a^3+ab^2+abc+ac^2-b^2c-bc^2=(a-b)(-a^2-ab+bc+c^2)=(a-b)(c-a)(a+b+c)$ $\endgroup$ Oct 17, 2016 at 12:07

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