5. Find the area of the shaded figure described by a semicircle rotated about one of its ends by an angle $\alpha=20^{\circ} .(15$ points $)$
Solution. Let the area of the semicircle be \( S_{0}=\frac{\pi R^{2}}{2} \), the area of the lune \( A B_{1} C \) be \( x \), the lune \( C B_{1} B \) be \( y \), and the sector \( A C B \) be \( a \). Then, if the area of the segment \( A C \) is \( b \), we have: \( x + b = S_{0} = b + a \Rightarrow x = a \).

The entire shaded area is \( x + y = a + y \), but \( a + y \) is the area of the circular sector \( A B B_{1} \) with radius \( 2R \). Thus, the area of \( A B B_{1} = \frac{1}{2}(2R)^{2} \frac{\pi}{9} = \frac{2 \pi R^{2}}{9} \).

Answer. \( \frac{2 \pi R^{2}}{9} \).

Solutions to the final round of the school students' Olympiad "The Rock of Science" in the Natural Sciences profile for the subject of Mathematics in the 2020/2021 academic year

## MATHEMATICS

## VARIANT 1