What is the maximum possible value of a positive integer n,such that for any choice of seven distinct elements from {1,2,....n}, there will exist two numbers x and y satisfying 1<x/y≤2?
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1 Answers
2305n=126.
Partition the set as follows:
{1,2}
{3,4,5,6}
{7,8,9,...,14}
{15,16,...30}
{31,32,....62}
{63,64,.....,126}
If you want to take 7 elements from the above 6 sets,2 of them have to be taken from the same set.(Pigeonhole Principle)
Let these 2 elements be x and y with y≥x.
x/y has to be between 1 and 2 (as the sets have been constructed in that manner)