<s>[INST] Then find the value of ... Let $a, \: b, \: c$ be three variables which can take any value (Real or Complex). Given that $ab + bc + ca = \frac{1}{2}$; $a + b + c = 2$; $abc = 4$. Then find the value of $$\frac{1}{ab + c - 1} + \frac{1}{bc + a - 1} + \frac{1}{ac + b - 1}$$
I always get stuck in such problems. Please give a detailed solution and also give me an approach that I can use to solve other such kind of problems.
 [/INST] $$\sum_{cyc}\frac{1}{ab+c-1}=\frac{\sum\limits_{cyc}(ab+c-1)(ac+b-1)}{\prod\limits_{cyc}(ab+c-1)}=$$
$$=\frac{\sum\limits_{cyc}(a^2bc+a^2b+a^2c-ab-2a+1)}{a^2b^2c^2+abc-1+\sum\limits_{cyc}(a^3bc+a^2b^2-a^2bc-a^2b-a^2c+a)}=$$
$$=\frac{2\cdot4+2\cdot\frac{1}{2}-3\cdot4-\frac{1}{2}-2\cdot2+3}{4^2+4-1+4\cdot\left(2^2-2\cdot\frac{1}{2}\right)+\left(\frac{1}{2}\right)^2-2\cdot4\cdot2-2\cdot\frac{1}{2}+3\cdot4+2}=-\frac{18}{113}.$$
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