Problem 5, 2023

A disc \(K\) of radius \(R\) is cut into three circular sectors in such a way that the areas of the two smaller sectors add up to the area of the largest sector, while the difference of the areas of the two smaller sectors equals one third of the area of the largest sector.

In each of the three sectors the largest possible circle is inscribed; call these circles \(K_{1}\), \(K_{2}\) and \(K_{3}\). Express, in terms of \(R\), the area of the region

\[ K \setminus \left(K_{1} \cup K_{2} \cup K_{3}\right) . \]