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Minimal cardinality (Posted on 2016-03-16) Difficulty: 3 of 5
Let S be a set of n distinct real numbers.
Let AS be the set of numbers that occur as averages
of two distinct elements of S.

For a given n >= 2, what is the smallest possible number
of distinct elements in As?

See The Solution Submitted by Ady TZIDON    
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Solution Simplified version Comment 3 of 3 |

Any arithmetic sequence a1, a2,   ... can be mapped into sequence of first n integers 1,2, ...n. The set of the pairwise averages will be an arithmetic sequence A(s):  3/2, 2, 5/2, 3 ... n-1/2   i.e. all the numbers  between the smallest and the biggest, including the ends spaced by 1/2.

If their number is k then there are k-1 spaces  which should equal to (n-1/2-3/2)/(1/2)=2n-2

so k-1= 2n-2  and

k= 2n-3


Edited on March 18, 2016, 4:36 am
  Posted by Ady TZIDON on 2016-03-18 04:32:45

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