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Found problems: 2

2024 Brazil Cono Sur TST, 3
Find all positive integers $m$ that have some multiple of the form $x^2+5y^2+2024$, with $x$ and $y$ integers.
quadratic residue chinese remainder theorem number theory hensel s lemma divisibility bijection pigeonhole principle quadratic residues

2022 Korea -Final Round, P5
Find all positive integers $m$ such that there exists integers $x$ and $y$ that satisfies $$m \mid x^2+11y^2+2022.$$
quadratic residue chinese remainder theorem number theory hensel s lemma divisibility Cauchy-Schwarz inequality Chinese remainder theorem mathematical induction modular arithmetic quadratic residues