1. [4 points] Around a bird feeder, in the same plane as it, a tit and a bullfinch are flying along two circles at the same speed. In the plane, a rectangular coordinate system is introduced, in which the feeder (the common center of the circles) is located at point $O(0 ; 0)$. The tit is moving clockwise, while the bullfinch is moving counterclockwise. At the initial moment, the tit and the bullfinch are at points $M_{0}(\sqrt{3} ; 3)$ and $N_{0}(6 ;-2 \sqrt{3})$ respectively. Determine the coordinates of all positions of the bullfinch where the distance between the birds will be the shortest.
Answer: $(4 \sqrt{3} ; 0),(-2 \sqrt{3} ; 6),(-2 \sqrt{3} ;-6)$.

Solution. Let the points where the blue tit and the snow bunting 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 $\frac{\pi}{2}$, and at this point $\alpha_{0}=\frac{\pi}{3}$, $\beta_{0}=-\frac{\pi}{6}$ - the angles corresponding to the initial positions of the birds.

The distance between the blue tit and the snow bunting will be the smallest when the angle between the vectors $\overrightarrow{A M}$ and $\overrightarrow{A N}$ is zero. Since $\left|A M_{0}\right|=2 \sqrt{3}$ and $\left|A N_{0}\right|=4 \sqrt{3}$ are the radii of the circles, and $\left|A N_{0}\right|=2\left|A M_{0}\right|$, the angular velocity of the blue tit is twice the angular velocity of the snow bunting.

Let at the moment when the directions of the vectors $\overrightarrow{A M}$ and $\overrightarrow{A N}$ coincide, the snow bunting has moved by an angle $\omega$. Then $\alpha_{0}-2 \omega=\beta_{0}+\omega+2 \pi n$, where $n \in \mathbb{Z}$. Therefore, $\omega=\frac{\alpha_{0}-\beta_{0}}{3}-\frac{2 \pi n}{3}=\frac{\pi}{6}-\frac{2 \pi n}{3}$, where $n=0,-1,-2 \ldots$

There will be three different points (for $n=0,-1,-2$). For $n=0$, we get $\beta_{1}=\beta_{0}+\frac{\pi}{6}=0$. The coordinates of the snow bunting's position are found using the formulas $x_{N}=4 \sqrt{3} \cos \beta_{1}=4 \sqrt{3}, y_{N}=4 \sqrt{3} \sin \beta_{1}=0$.

The other points are obtained by rotating point $N_{1}$ around the origin by angles $\frac{2 \pi}{3}, \frac{4 \pi}{3}$ and have, respectively, coordinates $(-2 \sqrt{3} ; 6)$ and $(-2 \sqrt{3} ;-6)$.