3. A regular pentagon and a regular icosagon are inscribed in the same circle. Which is greater: the sum of the squares of the lengths of all sides of the pentagon or the sum of the squares of the lengths of all sides of the icosagon?
Answer: the sum of the squares of the side lengths of a regular pentagon is greater.

Solution. Note that in a regular 20-sided polygon, the vertices taken every other one form a regular 10-sided polygon, and the vertices of this 10-sided polygon, taken every other one, form a regular 5-sided polygon. Therefore, it is sufficient to compare two quantities: $4 a_{20}^{2}$ and $a_{5}^{2}$, where $a_{20}$ and $a_{5}$ are the side lengths of the regular 20-sided and 5-sided polygons, respectively. Consider the corresponding fragment and introduce the vertex notations as shown in Fig. 11.3a, b. We can reason in different ways from here.

First method. We use the fact that for $n>4$, the angle of any regular $n$-sided polygon is obtuse. Consider the triangle $A_{1} A_{2} A_{3}$ with the obtuse angle $A_{2}$ (see Fig. 11.3a). By the corollary of the cosine theorem, $2 a_{20}^{2}\frac{8}{3}\left(x^{2}+y^{2}\right)>4 a_{20}^{2}$, from which the answer follows.

Third method. We use the formula for calculating the side lengths of regular $n$-sided polygons inscribed in a circle of radius $R: a_{n}=2 R \sin \frac{180^{\circ}}{n}$. Then $\frac{a_{5}^{2}}{4 a_{20}^{2}}=$ $\left(\frac{a_{5}}{2 a_{20}}\right)^{2}=\left(\frac{\sin 36^{\circ}}{2 \sin 9^{\circ}}\right)^{2}=\left(\frac{4 \cos 18^{\circ} \cos 9^{\circ} \sin 9^{\circ}}{2 \sin 9^{\circ}}\right)^{2}=$

![](https://cdn.mathpix.com/cropped/2024_05_06_97c3a8739e7290d11eb7g-2.jpg?height=317&width=766&top_left_y=270&top_left_x=1156)
$\left(2 \cos 18^{\circ} \cos 9^{\circ}\right)^{2}>2.25>1$, since $\cos 9^{\circ}>\cos 18^{\circ}>\cos 30^{\circ}=\frac{\sqrt{3}}{2}$. Therefore, $a_{5}^{2}>4 a_{20}^{2}$.

## Grading Criteria.

“+" - a complete and well-reasoned solution is provided

“士” - a generally correct reasoning is provided, containing minor gaps or inaccuracies

“-” - only the answer is provided

“-" - the problem is not solved or solved incorrectly