Balti Tee 2021 · Ülesanne 4

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 $(\sqrt{2}/2, \sqrt{2}/2)$, 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. (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 $z_n = \exp(ix_n)$ with $x_n \in ]0, 2\pi[$. At this time, Mario is located at $z_i^3$ and Daisy at $(z_i^3 + z_j)/2$.

According to our assumption we have $(z_1^3 + z_1)/2 = (z_2^3 + z_2)/2$ or, equivalently, $(z_1 - z_2)(z_1^2 + z_1z_2 + z_2^2 + 1) = 0$. We have $z_1 \neq z_2$, so that we must have $z_1^2 + z_1z_2 + z_2^2 = -1$. We proceed with an observation of the structure of $\Gamma$ as a set of complex numbers. Suppose that $z \in \Gamma \setminus \{S\}$. Then $z+1+z^{-1} \in \Gamma$ if and only if $z \in \{i, -i, -i, i\}$. 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 $z+1+\bar{z} = -\bar{z_1} \cdot \bar{z_2} \in \Gamma$. Therefore, one of the following cases must occur. (i) We have $z = \pm i$, that is, $z_1 = \pm iz_2$. Without loss of generality we may assume $z_2 = iz_1$. It follows that $-1 = z_1^2 + z_1z_2 + z_2^2 = iz_1^2$, so that $z_1 = \exp(i\pi/4)$ or $z_1 = \exp(5i\pi/4)$. In the former case $(z_1, z_2) = (\exp(i\pi/4), \exp(3i\pi/4))$, which matches case (1) above. In the latter case $(z_1, z_2) = (\exp(5i\pi/4), \exp(7i\pi/4))$, which matches case (2) above. (ii) We have $z = -1$, that is, $z_2 = -z_1$. It follows that $-1 = z_1^2 + z_1z_2 + z_2^2 = z_1^2$, so that $z_1 = i$ or $z_1 = -i$. This matches case (3) above.