Putnam 1941 A6

If the $x$ -coordinate $\overline{x}$ of the center of mass of the area lying between the $x$ -axis and the curve $y=f(x)$ with $f(x)>0$ , and between the lines $x=0$ and $x=a$ is given by $$\overline{x}=g(a),$$ show that $$f(x)=A\cdot \frac{g'(x)}{(x-g(x))^{2}} \cdot e^{\int \frac{1}{t-g(t)} dt},$$ where $A$ is a positive constant.