Let $\Gamma$ be a circle in the plane and $S$ be a point on $\Gamma$. Mario and Luigi drive around the circle $\Gamma$ with their go-karts. They both start at $S$ at the same time. They both drive for exactly 6 minutes at constant speed counterclockwise around the track. During these 6 minutes, Luigi makes exactly one lap around $\Gamma$ while Mario, who is three times as fast, makes three laps.

While Mario and Luigi drive their go-karts, Princess Daisy positions herself such that she is always exactly in the middle of the chord between them. When she reaches a point she has already visited, she marks it with a banana.

How many points in the plane, apart from $S$, are marked with a banana by the end of the race?
. Without loss of generality, we assume that $\Gamma$ is the unit circle and $S=(1,0)$. Three points are marked with bananas:

(i) After 45 seconds, Luigi has passed through an arc with a subtended angle of $45^{\circ}$ and is at the point $\left(\sqrt{2} / 2, \sqrt{2} / 2\right)$, whereas Mario has passed through an arc with a subtended angle of $135^{\circ}$ and is at the point $(-\sqrt{2} / 2, \sqrt{2} / 2)$. Therefore Daisy is at the point $(0, \sqrt{2} / 2)$ after 45 seconds. After 135 seconds, Mario and Luigi's positions are exactly the other way round, so the princess is again at the point $(0, \sqrt{2} / 2)$ and puts a banana there.

## Baltic Way

Reykjavík, November 11th - 15th

Solutions

(ii) Similarly, after 225 seconds and after 315 seconds, Princess Daisy is at the point $(0,-\sqrt{2} / 2)$ and puts a banana there.

(iii) After 90 seconds, Luigi is at $(0,1)$ and Mario at $(0,-1)$, so that Daisy is at the origin of the plane. After 270 seconds, Mario and Luigi's positions are exactly the other way round, hence Princess Daisy drops a banana at the point $(0,0)$.

We claim that no other point in the plane, apart from these three points and $S$, is marked with a banana. Let $t_{1}$ and $t_{2}$ be two different times when Daisy is at the same place. For $n \in\{1,2\}$ we write Luigi's position at time $t_{n}$ as a complex number $z_{n}=\exp \left(i x_{n}\right)$ with $x_{n} \in\right] 0,2 \pi[$. At this time, Mario is located at $z_{i}^{3}$ and Daisy at $\left(z_{i}^{3}+z_{i}\right) / 2$. According to our assumption we have $\left(z_{1}^{3}+z_{1}\right) / 2=\left(z_{2}^{3}+z_{2}\right) / 2$ or, equivalently, $\left(z_{1}-z_{2}\right)\left(z_{1}^{2}+z_{1} z_{2}+z_{2}^{2}+1\right)=0$. We have $z_{1} \neq z_{2}$, so that we must have $z_{1}^{2}+z_{1} z_{2}+z_{2}^{2}=-1$.

We proceed with an observation of the structure of $\Gamma$ as a set of complex numbers. Suppose that $z \in \Gamma \backslash\{S\}$. Then $z+1+z^{-1} \in \Gamma$ if and only if $z \in\{i,-1,-i\}$. For a proof of the observation note that $z+1+z^{-1}=z+1+\bar{z}$ is a real number for every $z \in \mathbb{C}$ with norm $|z|=1$. So it lies on the unit circle if and only if it is equal to 1, in which case the real part of $z$ is equal to 0, or it is equal to -1, in which case the real part of $z$ is equal to -1. We apply the observation to the number $z=z_{1} / z_{2}$, which satisfies the premise since $z+1+\bar{z}=-\overline{z_{1}} \cdot \overline{z_{2}} \in \Gamma$. Therefore, one of the following cases must occur.

(i) We have $z= \pm i$, that is, $z_{1}= \pm i z_{2}$. Without loss of generality we may assume $z_{2}=i z_{1}$. It follows that $-1=z_{1}^{2}+z_{1} z_{2}+z_{2}^{2}=i z_{1}^{2}$, so that $z_{1}=\exp (i \pi / 4)$ or $z_{1}=\exp (5 i \pi / 4)$. In the former case $\left(z_{1}, z_{2}\right)=(\exp (i \pi / 4), \exp (3 i \pi / 4))$, which matches case (1) above. In the latter case $\left(z_{1}, z_{2}\right)=(\exp (5 i \pi / 4), \exp (7 i \pi / 4))$, which matches case (2) above.

(ii) We have $z=-1$, that is, $z_{2}=-z_{1}$. It follows that $-1=z_{1}^{2}+z_{1} z_{2}+z_{2}^{2}=z_{1}^{2}$, so that $z_{1}=i$ or $z_{1}=-i$. This matches case (3) above.

![](https://cdn.mathpix.com/cropped/2024_04_17_e27e300a283338a2e031g-04.jpg?height=348&width=320&top_left_y=1865&top_left_x=865)

Depiction of the path Daisy takes

## Baltic Way

Reykjavík, November 11th - 15th

Solutions