MathDB

N6

Part of 2019 IMO Shortlist

Problems(1)

is it true c shortlist has 9 problems?

Source: ISL 2019 N6

9/22/2020
Let H={⌊i2⌋:i∈Z>0}={1,2,4,5,7,… }H = \{ \lfloor i\sqrt{2}\rfloor : i \in \mathbb Z_{>0}\} = \{1,2,4,5,7,\dots \} and let nn be a positive integer. Prove that there exists a constant CC such that, if A⊆{1,2,…,n}A\subseteq \{1,2,\dots, n\} satisfies ∣A∣≥Cn|A| \ge C\sqrt{n}, then there exist a,b∈Aa,b\in A such that a−b∈Ha-b\in H. (Here Z>0\mathbb Z_{>0} is the set of positive integers, and ⌊z⌋\lfloor z\rfloor denotes the greatest integer less than or equal to zz.)
number theoryIMO ShortlistIMO Shortlist 2019floor function