1. [4 points] Around a flower, in the same plane as it, a bumblebee and a bee are flying along two circles. The speed of the bee is one and a half times the speed of the bumblebee. In the specified plane, a rectangular coordinate system is introduced, in which the flower (the common center of the circles) is at the point $O(0 ; 0)$. The bee moves clockwise, while the bumblebee moves counterclockwise. At the initial moment, the bee and the bumblebee are at points $M_{0}\left(-\frac{1}{2} ; \frac{\sqrt{3}}{2}\right)$ and $N_{0}(2 ; 0)$, respectively. Determine the coordinates of all positions of the bumblebee where the distance between it and the bee will be the shortest.
Answer: $(\sqrt{3} ; 1),(-1 ; \sqrt{3}),(-\sqrt{3} ;-1),(1 ;-\sqrt{3})$.

Solution. Let the points where the bee and the bumblebee are located be $M(\alpha)$ and $N(\beta)$, respectively, where $\alpha$ and $\beta$ are the angles that the radius vectors of points $M$ and $N$ form with the positive direction of the x-axis. Note that the angle between $\overrightarrow{A M_{0}}$ and $\overrightarrow{A N_{0}}$ is $\frac{2 \pi}{3}$, and at this point $\alpha_{0}=\frac{2 \pi}{3}, \beta_{0}=0$ - are the angles corresponding to the initial positions of the insects.

The distance between the bumblebee and the bee will be the smallest when the angle between the vectors $\overrightarrow{A M}$ and $\overrightarrow{A N}$ is zero. Since $\left|A M_{0}\right|=1$ and $\left|A N_{0}\right|=2$ are the radii of the circles, and $\left|A N_{0}\right|=2\left|A M_{0}\right|$, the angular velocity of the bee is three times the angular velocity of the bumblebee.

Let at the moment of the vectors $\overrightarrow{A M}$ and $\overrightarrow{A N}$ aligning, the bumblebee has moved by an angle $\omega$. Then $\alpha_{0}-3 \omega+2 \pi k=\beta_{0}+\omega$, where $k \in \mathbb{Z}$. Therefore, $\omega=\frac{\alpha_{0}-\beta_{0}}{4}+\frac{\pi k}{2}=\frac{\pi}{6}+\frac{\pi k}{2}$, where $k=0,1,2 \ldots$ There will be four different points (for $k=0,1,2,3$). For $k=0$, we get $\beta_{1}=\beta_{0}+\frac{\pi}{6}=\frac{\pi}{6}$. The coordinates of the bumblebee's position are found using the formulas $x_{N}=2 \cos \beta_{1}=\sqrt{3}, y_{N}=2 \sin \beta_{1}=1$.

The other points are obtained by rotating point $N_{1}$ around the origin by angles $\frac{\pi}{2}, \pi, \frac{3 \pi}{2}$ and have, respectively, coordinates $(-1 ; \sqrt{3}),(-\sqrt{3} ;-1),(1 ;-\sqrt{3})$.