Putnam 2012 A3

Let $f: [-1, 1] \to \mathbb{R}$ be a continuous function such that

(i) $f(x) = \frac{2-x^2}{2} f \left( \frac{x^2}{2-x^2} \right)$ for every $x$ in $[-1, 1]$,

(ii) $f(0) = 1$, and

(iii) $\lim_{x \to 1^-} \frac{f(x)}{\sqrt{1-x}}$ exists and is finite.

Prove that $f$ is unique, and express $f(x)$ in closed form.