Putnam 1964 A2

Find all continuous positive functions $f(x)$ , for $0\leq x \leq 1$ , such that $$\int_{0}^{1} f(x) dx =1,$$ $$\int_{0}^{1} xf(x) dx =\alpha,$$ $$\int_{0}^{1} x^2 f(x) dx =\alpha^2,$$ where $\alpha$ is a given real number.