MathDB

4

Part of 2018 South East Mathematical Olympiad

Problems(2)

Does there exists set of positive integers?

Source: 2018 China Southeast MO Grade 10 P4

7/30/2018
Does there exist a set A⊆N∗A\subseteq\mathbb{N}^* such that for any positive integer nn, A∩{n,2n,3n,...,15n}A\cap\{n,2n,3n,...,15n\} contains exactly one element? Please prove your conclusion.
number theory
Harder than Grade 10 P4

Source: 2018 China Southeast MO Grade 11 P4

7/30/2018
Does there exist a set A⊆N∗A\subseteq\mathbb{N}^* such that for any positive integer nn, A∩{n,2n,3n,...,15n}A\cap\{n,2n,3n,...,15n\} contains exactly one element and there exists infinitely many positive integer mm such that {m,m+2018}⊂A\{m,m+2018\}\subset A? Please prove your conclusion.
number theory