Find $f(x)$ and $g(x)$ or some relations between the two functions knowing that $$ \frac{f(x)}{\int_0^{x_1} f(x)dx} = \frac{g(x)}{\int_0^{x_1} g(x)dx}, \quad \forall x \in (0, x_1), \ \forall x_1 \in (0, x_{\max}). $$
One solution is $g(x)=a f(x)$ where $a$ is a constant.
I would be intrested in other solutions less evident.