$f(x)$ is a function continuous in $[0,1]$, differentiable in $(0,1)$ and satisfies $$f(1)= 3\int_0^{\frac{1}{3}}e^{1-x^2}f(x) \, dx.$$ Prove that there is some $\xi$ such that $f'(\xi)=2\xi f(\xi)$.
I tried to make $g(x)=e^{1-x^2}f(x)$, then we can have $g'(x)=e^{1-x^2}(f'(x)-2xf(x))$. If we can apply Rolle's Theorem on $g(x)$ then problem solved, but I do know how to use the condition on $f(1)$.
Apply the first mean value theorem for integrals to $g$. This will tell you that there is some $c \in (0,1/3)$ such that $g(c)=f(1)$. Then proceed as you were already planning to.