National problem · Geometry

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

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

The shaded part of the disc is the region whose area is asked for. The three sectors are drawn at arbitrary sizes - determining their actual angles is part of the problem.