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: $$C_k=x_k+y_{k}i\text{and}M=\{C_1,\dots ,C_n\}\subset \mathbb{Z}[i].$$ Let $p$ be an arbitrary prime divisor of $d$. Let us show that there is a rotation of the $R$ plane such that $RM\subset p\mathbb{Z}[i]$, that is, $R(\frac{1}{p}M)\subset \mathbb{Z}[i]$. 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 $\pi \in \mathbb{Z}[i]$ 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 $N(\pi /\pi )=1$: $$R_{\pi }:x+yi\to \frac{\bar{\pi }}{\pi }(x+yi),$$ 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 $R_{\pi }M\subset p\mathbb{Z}[i]$ as required. Let us now consider a decomposition of the number $d=p_1{p}_2\cdots p_m$ into prime factors (not necessarily different). According to what has been proven, there is a rotation $R_1$ such that $R_1(\frac{1}{p_1}M)\subset \mathbb{Z}[i]$. Similarly, there is a rotation $R_2$ such that $R_2(R_1(\frac{1}{p_1{p}_2}M))\subset \mathbb{Z}[i]$, and so on, there is a rotation $R_m$ such that $$R_m\left(\dots R_2\left(R_1\left(\frac{1}{p_1{p}_2\cdots p_m}M\right)\right)\right)\subset \mathbb{Z}[i].$$ Since $\frac{1}{p_1{p}_2\cdots p_m}M=\{A_1,\dots ,A_n\}$, the composition of rotations $R_m\circ \cdots \circ R_2\circ R_1$ sends the set $A_1,\dots ,A_n$ to the set of integer points $B_1,\dots ,B_n$ with the required properties.