The (very nice) final problem of [IMO 2017](http://imo-official.org/year_info.aspx?year=2017) asked contestants to show:



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> If $S$ is a finite set of lattice points $(x,y)$ with $\gcd(x,y)=1$, then there is a nonconstant homogeneous polyonmial $f \in \mathbb Z[x,y]$ such that $f(x,y) = 1$ for all $(x,y) \in S$.
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It's claimed [in this forum post](https://artofproblemsolving.com/community/c6h1480686p8648257) that the above IMO problem is a special case of [Lemma 7.3 of arXiv:16040.01704](https://arxiv.org/pdf/1604.01704.pdf). The former post phrases the lemma as follows:



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> If $X$ is a finite scheme over $\operatorname{Spec} \mathbb Z$ then $\operatorname{Pic}(X)$ is finite.
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Being unknowledgable as I am, I do not see how to deduce the IMO problem from the lemma. Can someone make the connection explicit?



IMO 2017/6 via arithmetic geometry
The set $S$ gives rise to a subscheme (which let's also denote by $S$) of $\mathbb{P}^1\_{\mathbb{Z}},$ because relatively a prime pair $(x,y)$ corresponds to a section of $\mathbb{P}^1\_{\mathbb{Z}}\rightarrow\operatorname{Spec}\mathbb{Z}$, and we take the union of these divisors in $\mathbb{P}^1\_\mathbb{Z}$.


Now, we have a map $\mathcal{O}(d)\_{\mathbb{P}^1}\rightarrow\mathcal{O}(d)\_S,$ and it will suffice to show that for some $d$ some element in the image of the induced map on global sections is nowhere zero. (If one composes this map with the map $\mathcal{O}(d)\_S\rightarrow\mathcal{O}(d)\_{\operatorname{Spec}\mathbb{Z}}$ corresponding to a section $(x,y)$, one gets at the level of global sections the evaluation map $\mathbb{Z}[x,y]\_d\rightarrow\mathbb{Z}$ on degree $d$ homogeneous polynomials.)


For this, it suffices that one can find $d$ so that this map is surjective on global sections and so that $\mathcal{O}(d)\_S$ is trivial. The map will be surjective on global sections for large enough $d$ by ampleness of $\mathcal{O}(1)$, and the triviality of a power of $\mathcal{O}(1)\_S$ follows from the finiteness of $\operatorname{Pic}(S)$.


It's maybe slightly inaccurate to say just that it's a case in Lemma 7.3 in the above paper; rather, this exact argument is ran in the proof of Corollary 1.3 (which immediately follows the aforementioned lemma) to prove a much more general result.


