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

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In summary, the significance of finding subgroups of size N/2 with the property m<=N-n is that it has various applications in mathematics, computer science, and other fields. This problem involves the values N, m, and n, where N represents the total number of elements in the set and m and n are specific values on opposite sides of the midpoint N/2. There is no one specific method for solving this problem, but some common approaches include algebraic techniques, graph theory, and combinatorial methods. Real-life examples where this problem may arise include data analysis, symmetry in mathematical objects, and algorithm development. Additionally, this problem can be generalized to other sets and properties, as long as the goal is to find sub
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al-mahed
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Given a positive whole number n, [tex]\exists[/tex] 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 [tex]\cap[/tex]{m+1, m+2,..., m+k}|>=k/2

[tex]\forall[/tex] k = 1, 2, …, n.
 
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Just look at the top half and the bottom half.
 
  • #3
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.
 
  • #4
this is an olympic problem, by the way
 

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

What is the significance of finding subgroups of size N/2 in {1, 2,..., N} with property m<=N-n?

Finding subgroups of size N/2 with this property can have various applications in mathematics, computer science, and other fields. It can help in understanding the structure of a group and its subgroups, as well as in developing algorithms for solving certain problems.

What is the relationship between N, m, and n in this problem?

N represents the total number of elements in the set {1, 2,..., N}, while m and n represent specific values within this set. The property m<=N-n implies that m and n are on opposite sides of the midpoint N/2, with m being less than or equal to N/2 and n being greater than or equal to N/2.

How can this problem be solved?

There is no one specific method for solving this problem as it depends on the specific values of N, m, and n. However, some common approaches include using algebraic techniques, graph theory, and combinatorial methods.

What are some real-life examples where this problem may arise?

This problem can arise in various situations, such as in the analysis of data sets, the study of symmetry in mathematical objects, and the development of efficient algorithms for sorting or searching.

Can this problem be generalized to other sets and properties?

Yes, this problem can be generalized to other sets with different sizes and properties. The key idea is to find subgroups of a certain size with a specific property, which can be applied to different sets and properties.

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