IMO 1969 Problem 2

Let $f(x)=\cos(a_1+x)+{1\over2}\cos(a_2+x)+{1\over4}\cos(a_3+x)+\ldots+{1\over2^{n-1}}\cos(a_n+x)$ , where $a_i$ are real constants and $x$ is a real variable. If $f(x_1)=f(x_2)=0$ , prove that $x_1-x_2$ is a multiple of $\pi$ .