So this is a problem I found on a qualifying exam and it has been nagging at me for some a while. It seems overly easy.
Assume that $f$ is a nonnegative function on $[0,1]$ and \begin{align} \int_{[0,1]}f\,dx=\int_{[0,1]}f^2\,dx=\int_{[0,1]}f^3\,dx. \end{align} Show that $f(1-f)=0$ almost everywhere.
I assume that the difficulty may lay in the part "almost everywhere", but I don't think it's the problem. Any ideas?