An integral condition

Question

Let $f$ be a non-decreasing and continuous function on $[0,1]$, such that $\int_0^1f(x)dx=2\int_0^1xf(x)dx$. Given that $f(1)=10.5$. Find the value of $f(0)+f(0.5)$.

Answer 1

Tricky exercise. We have: $$ 0 = \int_{0}^{1}(1-2x)\,f(x)\,dx = \int_{0}^{1/2}(1-2x)\left(f(x)-f(1-x) \right)\,dx $$ but since $(1-2x)>0$ over $I=\left(0,\frac{1}{2}\right)$ while $f(x)\leq f(1-x)$, the only way in which the above integral can be zero is with a constant $f$, hence from $f(1)=10.5$ we get $f(0)+f(1/2)=21$.