Problem 1. Let $a$ be a strictly positive real number. Determine the minimum value of the expression

$$
\left(\int_{0}^{1} f(x) \mathrm{d} x\right)^{2}-(a+1) \int_{0}^{1} x^{2 a} f(x) \mathrm{d} x
$$

as $f$ ranges over the set of concave functions $f:[0,1] \rightarrow \mathbb{R}$, with $f(0)=1$.