SIMON-MARAIS 2022 A3

Let $0 < a < 1$ be a fixed real number. Show that there are at least two values of $x$ in the interval $(0,1)$ such that

$$\int_0^x \left( \sin \left( \frac{\pi \sin \frac{\pi t}{2}}{2} \right) + \frac{2}{\pi} \arcsin \left( \frac{2}{\pi} \arcsin t \right) - 2t \right) \mathrm{d}t = \frac{1}{2} \left( a - \frac{2}{\pi} \arcsin \left( \frac{2}{\pi} \arcsin a \right) \right) \left( \sin \left( \frac{\pi \sin \frac{\pi a}{2}}{2} \right) - a \right).$$