Selection Tests of the Belarusian Team to the IMO

Let $d$ be the least common multiple of the denominators of all coordinates of the points $A_1, \dots, A_n$. After homothety with center at the origin $O$ and coefficient $d$, these points will go to points $C_1, \dots, C_n$, respectively, with integer coordinates. Let us switch to complex numbers and work with Gaussian integers. Let us introduce the following notation:
$$C_k = x_k + y_k i \quad \text{and} \quad M = \{C_1, \dots, C_n\} \subset \mathbb{Z}*.$$
Let $p$ be an arbitrary prime divisor of $d$. Let us show that there is a rotation of the $R$ plane such that $RM \subset p\mathbb{Z}*$, that is, $R(\frac{1}{p}M) \subset \mathbb{Z}*$.
We have $p^2 \mid N(z)$ and $p^2 \mid N(z-u)$ for any $z, u \in M$, where $N(a+bi) = a^2 + b^2$ means norm. If $p \mid z$ for any $z \in M$, then everything is proven. Suppose that there exists $v \in M$ such that $p \nmid v$. Then it can be shown that there exists $\pi \in \mathbb{Z}*$ such that $N(\pi) = p$ and $\pi^2 \mid v$. Then for any $u \in M$:
- either $p \nmid u$, whence $\pi^2 \mid u$,
- or $p \mid u$ and $p \nmid v-u$, from which $\pi^2 \mid v-u$ and again $\pi^2 \mid u$.
Apply the following plane transformation, which is a rotation, since $N(\pi/\pi) = 1$:
$$R_{\pi}: x + yi \rightarrow \frac{\bar{\pi}}{\pi}(x + yi),$$
Each number from $M$ can be represented as $\pi^2(a + bi)$, which is sent by the rotation $R_{\pi}$ to the number $\bar{\pi}\pi(a + bi) = p(a + bi)$. Thus, we get $R_{\pi}M \subset p\mathbb{Z}*$ as required.
Let us now consider a decomposition of the number $d = p_1p_2 \cdots p_m$ into prime factors (not necessarily different). According to what has been proven, there is a rotation $R_1$ such that $R_1(\frac{1}{p_1}M) \subset \mathbb{Z}*$. Similarly, there is a rotation $R_2$ such that $R_2(R_1(\frac{1}{p_1p_2}M)) \subset \mathbb{Z}*$, and so on, there is a rotation $R_m$ such that
$$R_m \left( \dots R_2 \left( R_1 \left( \frac{1}{p_1 p_2 \cdots p_m} M \right) \right) \right) \subset \mathbb{Z}*.$$
Since $\frac{1}{p_1p_2\cdots p_m}M = \{A_1, \dots, A_n\}$, the composition of rotations $R_m \circ \dots \circ R_2 \circ R_1$ sends the set $A_1, \dots, A_n$ to the set of integer points $B_1, \dots, B_n$ with the required properties.