## Task 18/85

A regular n-gon is inscribed in a circle, and its sides are spanned by semicircular arcs. Which n-gon satisfies the condition that the sum of the areas of the crescent-shaped regions bounded by the semicircular arcs and the corresponding circular arcs of the circumcircle is equal to the area of the n-gon?
The sum of the 5 crescent-shaped areas is given as the sum of the $n$-gon area $A_{n}$ and $n$ semicircle areas with the radius being half the $n$-gon side $a$, reduced by the circumcircle area $r^{2} \pi$ of the $n$-gon (where $r$ is the circumradius):

$$
S=A_{n}+n \cdot \frac{1}{2} \cdot\left(\frac{a}{2}\right)^{2} \pi-r^{2} \pi
$$

Since $S=A_{n}$, it follows that $\frac{1}{8} n a^{2} \pi=r^{2} \pi$ and, given $a=2 r \sin \frac{\pi}{n}$ and $r^{2} \pi \neq 0$, we finally get $n \sin ^{2} \frac{\pi}{n}=2$.

The left-hand side of this equation forms a strictly decreasing sequence for $n \geq 3$ (which can be shown, for example, by taking the first derivative of the function $y=x \sin ^{2} \frac{\pi}{x}$). Therefore, it takes each value, specifically the value 2, at most once starting from $n=3$. Trying quickly yields $n=4$. The sought $n$-gon is thus the square.