IrMO 1993

Let $f(x)=x^n+a_{n-1} x^{n-1}+...+a_0$ $(n \ge 1)$ be a polynomial with real coefficients such that $|f(0)|=f(1)$ and each root $\alpha$ of $f$ is real and lies in the interval $[0,1]$ . Prove that the product of the roots does not exceed $\frac{1}{2^n}$ .