\section*{Problem 4 - \(330914=331014\)}

From the area of a rectangle with side lengths \(a, b\), the areas of four quarter-circles are to be subtracted. These four quarter-circles should all have the same radius \(r\), with the condition that segments of the rectangle's sides still remain (see figure).

a) Determine all real numbers \(x\) for which there exist lengths \(a, b, r\) of the specified type such that exactly \(x\) percent of the rectangle's area is cut off!

b) Determine all ratio values \(k=\frac{b}{a} \geq 1\) for which it is possible

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1135.jpg?height=326&width=220&top_left_y=385&top_left_x=1523)
to choose a radius \(r\) of the specified type such that exactly half of the rectangle's area is cut off!
}

The condition to be satisfied by \(a, b, r\) states: If, without loss of generality, the labels are chosen such that \(00\) for \(b=a\) and \(r=\sqrt{\frac{x \cdot a b}{100 \pi}}\) is, on the one hand,

\[
00\) with \(\frac{b}{a}=k\) and \(r=\sqrt{\frac{a b}{2 \pi}}\) is, on the one hand,

\[
r=\sqrt{\frac{a \cdot a k}{2 \pi}}<\sqrt{\frac{a^{2}}{2 \pi} \cdot \frac{\pi}{2}}=\frac{a}{2}
\]

so that \(a, b, r\) are lengths of the assumed type; on the other hand, (2) holds, so exactly half of the rectangle's area is cut off.

With I. and II., it is proven: The sought numbers \(k\) are exactly all real numbers \(x\) with \(1 \leq k<\frac{\pi}{2}\) \((\approx 1.5708)\)