INMO 2019 Problem 6

Let $f$ be function defined from the set $\{(x, y): x, y$ reals, $x y \neq 0\}$ in to the set of all positive real numbers such that

(i) $f(x y, z)=f(x, z) f(y, z)$, for all $x, y \neq 0$;

(ii) $f(x, y z)=f(x, y) f(x, z)$, for all $x, y \neq 0$;

(iii) $f(x, 1-x)=1$, for all $x \neq 0,1$.

Prove that

(a) $f(x, x)=f(x,-x)=1$, for all $x \neq 0$;

(b) $f(x, y) f(y, x)=1$, for all $x, y \neq 0$.