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, 2015

Number theory Difficulty 6.0 National Olympiad Prove it Hungary

Let $k$ be a nonnegative integer. Prove that there are only finitely many positive integers $n$ for which there exist two disjoint sets $A$ and $B$ satisfying $A \cup B = \{1; 2; \ldots; n\}$ and $\text{| a}$ A} {a} - $\prod\limits _{b \in}$ B} $\left. {b}\right|=k\$.$
Proposed by: Balázs Maga, Budapest
(5 pont)

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