IMC 1995 Problem 2

Let $f$ be a continuous function on $[0,1]$ such that for every $x\in [0,1]$ , we have $\int_{x}^{1}f(t)dt \geq\frac{1-x^{2}}{2}$ . Show that $\int_{0}^{1}f(t)^{2}dt \geq \frac{1}{3}$ .