2017 IMO Shortlist N7

Say that an ordered pair $(x, y)$ of integers is an irreducible lattice point if $x$ and $y$ are relatively prime. For any finite set $S$ of irreducible lattice points, show that there is a homogenous polynomial in two variables, $f(x, y)$, with integer coefficients, of degree at least 1 , such that $f(x, y)=1$ for each $(x, y)$ in the set $S$. Note: A homogenous polynomial of degree $n$ is any nonzero polynomial of the form $$f(x, y)=a_{0} x^{n}+a_{1} x^{n-1} y+a_{2} x^{n-2} y^{2}+\cdots+a_{n-1} x y^{n-1}+a_{n} y^{n} .$$

如果 $x$ 和 $y$ 互质，则称有序整数对 $(x, y)$ 是不可约格点。对于任何不可约格点的有限集 $S$，证明两个变量 $f(x, y)$ 中存在齐次多项式，其整数系数的次数至少为 1 ，使得对于集合 $S$ 中的每个 $(x, y)$，$f(x, y)=1$。注意：$n$ 次齐次多项式是 $$f(x, y)=a_{0} x^{n}+a_{1} x^{n-1} y+a_{2} x^{n-2} y^{2}+\cdots+a_{n-1} x y^{n-1}+a_{n} y^{n} 形式的任何非零多项式。$$