Finding Subgroups of Size N/2 in {1, 2,..., N} with Property m<=N-n

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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.
 
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Just look at the top half and the bottom half.
 
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.
 
this is an olympic problem, by the way
 

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