Question 21
Determine all continuous functions $f:[0,1]\longrightarrow[0,\infty)$ satisfying
$$
\left(\int_0^1 f(x)\,dx\right)\left(\int_0^1 f^2(x)\,dx\right)\cdots\left(\int_0^1 f^{2020}(x)\,dx\right)
=\left(\int_0^1 f^{2021}(x)\,dx\right)^{1010}.
$$
$$
\left(\int_0^1 f(x)\,dx\right)\left(\int_0^1 f^2(x)\,dx\right)\cdots\left(\int_0^1 f^{2020}(x)\,dx\right)
=\left(\int_0^1 f^{2021}(x)\,dx\right)^{1010}.
$$