Problem 2. (D. Mukhin) Mitya bought a round cake with a diameter of 36 centimeters and 13 thin candles for a birthday. Mitya doesn't like it when the candles are too close to each other, so he wants to place them at least 10 centimeters apart. Will all the candles fit on the cake?
Answer. They will fit.

Solution. Place the candles as shown in the figure. We will prove that the segment $O A$ is less than the radius, that is, the farthest points from the center lie inside the circle. Indeed, $A B C$ is an equilateral triangle with a side length of 20 cm. Then its height $A O=10 \sqrt{3}$ cm $<18$ cm, since $(10 \sqrt{3})^{2}=300<324=$ $18^{2}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_89d73d034ee0bef24fcbg-1.jpg?height=531&width=562&top_left_y=1905&top_left_x=747)

Comment. 1. In fact, $2 \cdot O A=20 \sqrt{3}$ cm is even less than 35 cm, so this construction will fit on a cake with a diameter of 35 cm.