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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 proposed solution | Comment 1 of 3
It would seem intuitively that the minimum would be achieved when the original set, S, was evenly spaced, so that many of the averages would be multiple, so as not add to the cardinality of A(s).

When n = 2, the cardinality is 1.

When n=3 the cardinality is 3.

When n=4, the cardinality is 5.

These are the set of points midway between the successive points of S, in union with the set of points of S other than the two end points. They number n-1 and n-2 respectively.

They add to 2*n - 3.

Edited on March 16, 2016, 9:51 am
  Posted by Charlie on 2016-03-16 09:48:15

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