Subgroups of integers with density guarantees on intervals

[al-mahed]

Given a positive whole number n, $$\exists$$ N with the following property: if A is a subgroup of {1,2,...,N} with at least N/2 elements, then there is a positive whole number m<= N - n such that

|A $$\cap$${m+1, m+2,..., m+k}|>=k/2

$$\forall$$ k = 1, 2, …, n.

 

[CRGreathouse]

Just look at the top half and the bottom half.

 

[al-mahed]

Hi, I'll be glad if you put your solution here. I already saw a proof, but I don't know if it's correct.

 

[al-mahed]

this is an olympic problem, by the way