Let $f,g$ be Riemann integrable on $[0,1]$ such that $\int_0^1 f=\int_0^1 g=1$. Show that there exists $0\leq a<b\leq 1$ such that $\int_a^b f=\int_a^b g=\frac{1}{2}$.
If there is only one function $f$, it is easy. What about two functions?
from Hot Weekly Questions - Mathematics Stack Exchange
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