1. Vasya and Petya drew a five-pointed star each. All the angles at the vertices of Petya's star are acute, while Vasya's star has an obtuse angle. Each of them claims that the sum of the angles at the vertices of his star is greater. Who is right?
![](https://cdn.mathpix.com/cropped/2024_05_06_ad0ddfff017bcb029f08g-1.jpg?height=328&width=758&top_left_y=1595&top_left_x=654)
Answer: nobody.

Solution. Let's denote the angles of the star as shown in the figure

![](https://cdn.mathpix.com/cropped/2024_05_06_ad0ddfff017bcb029f08g-1.jpg?height=423&width=391&top_left_y=2056&top_left_x=841)

\[
\begin{aligned}
& \angle 1+\angle 3=180^{\circ}-\angle 6=\angle 9 \\
& \angle 2+\angle 4=180^{\circ}-\angle 7=\angle 8
\end{aligned}
\]

Then

\[
\angle 1+\angle 3+\angle 2+\angle 4+\angle 5=\angle 9+\angle 8+\angle 5=180^{\circ} .
\]

These calculations are valid for both stars, the sum of the angles at the vertices of each five-pointed star is \(180^{\circ}\).

Comment. An answer without justification - 0 points. Proven that the sum of the acute angles of a regular five-pointed star is \(180^{\circ}\) - 2 points. Proven that the sum of the acute angles of an acute-angled five-pointed star is \(180^{\circ}\) - 5 points.