III. Tutorial problems. Symmetric and alternating functions. RMO and IITJEE math

Simplify: $(b^{-1}+c^{1})(b+c-a)+(c^{-1}+a^{-1})(c+a-b)+(a^{-1}+b^{-1})(a+b=c)$
Simplify: $\frac{(x-b)(x-c)}{(a-b)(a-c)} + \frac{(x-c)(x-a)}{(b-c)(b-a)} + \frac{(x-a)(x-b)}{(c-a)(c-b)}$
Simplify: $\frac{b^{2}+c^{2}-a^{2}}{(a-b)(a-c)} + \frac{c^{2}+a^{2}-b^{2}}{(b-c)(b-a)} + \frac{a^{2}+b^{2}-c^{2}}{(c-a)(c-b)}$
Simplify: $\frac{b-c}{1+bc} + \frac{c-a}{1+ca} + \frac{a-b}{1+ab}$
Simplify: $\frac{a(b-c)}{1+bc} + \frac{b(c-a)}{1+ca} + \frac{c(a-b)}{1+ab}$
Factorize: $(b-c)^{2}(b+c-2a)+(c-a)^{2}(c+a-2b)+(a-b)^{2}(a+b-2c)$ . Put $b-c=x, c-a=y, a-b=z$ and $b+c-2a=y-z$
Factorize: $8(a+b+c)^{2}-(b+c)^{2}-(c+a)^{2}-(a+b)^{2}$ . Put $b+c=x, c+a=y, a+b=z$ .
Factorize: $(a+b+c)^{2}-(b+c-a)^{2}-(c+a-b)^{2}+(a+b-c)^{2}$
Factorize: $(1-a^{2})(1-b^{2})(1-c^{2})+(a-bc)(b-ac)(c-ab)$
Express the following substitutions as the product of transpositions: (i) $\left(\begin{array}{cccccc}123456\\654321\end{array}\right)$ (ii) $\left(\begin{array}{cccccc}123456\\246135\end{array}\right)$ (iii) $\left(\begin{array}{cccccc}123456\\641235\end{array}\right)$