KMO 2008 Problem 7

Prove that the only function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying the following is $f(x)=x$ . (i) $\forall x \not= 0$ , $f(x) = x^2f(\frac{1}{x})$ . (ii) $\forall x, y$ , $f(x+y) = f(x)+f(y)$ . (iii) $f(1)=1$ .