Saudi Arabia booklet 2024

Let $R$ be the radius of the circle ($O$) and let $M,K$ be the midpoints of $IO,AI$ respectively. Then, according to the property of the midline in triangle $AIO$, we have $MK=\frac{AO}{2}$ and $MK\parallel AO$. Thus, we immediately have $MK\perp SP$ and $MK=\frac{R}{2}$. Similarly for the other sides.



Therefore, the point $M$ is equidistant from the sides of the quadrilateral $PQRS$ and is also inside the quadrilateral (since $I,O$ lie inside it), so $M$ is the incenter of the quadrilateral $PQRS$. $\square$