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Restoring the erased (Posted on 2010-07-14) Difficulty: 3 of 5
The following text represents a valid contest question
in which I have erased one number :

Let S be a set of 20 distinct positive integers,
each less than EN(=THE ERASED NUMBER).
Show that there exist four distinct elements a, b, c, d , all in S,
such that a + b = c + d.


What maximal value of EN guarantees the existence of these elements ?

No Solution Yet Submitted by Ady TZIDON    
Rating: 4.0000 (5 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Some Thoughts re(4): No Subject FOR ALL SOLVERS | Comment 14 of 21 |
(In reply to re(3): No Subject by ed bottemiller)

 

     

     

     

     

     

     

     

     

     

     Hi, everybody!

     

    I am addressing all those floobers that spent considerable amount of times trying to get the maximal value of EN. I did not realize that solving for n=20 is so much harder than, say, n=8. After private email exchange between me and Justin I became convinced that to climb to higher values necessitates solving for each new value "from the scratches" and  is  amazingly time-consuming.

    Since I do not have the correct answer, I would like

  1. To tell you the history of my post.
  2. To define what exactly we are looking for.
  3.  

    Let start with the "original" text from some math contest at

    high school level:

     

     

     

     

     Let S be a set of 20 distinct positive integers, all  less than 97.

    Show that there exist four distinct elements a, b, c, d in S such that
     a + b = c + d.

     

     

    SOLUTION: 

    Is based on the pigeonhole principle:  less  sums than pairs are needed.

    20*19*.5=190 PAIRS

    SMALLEST SUM =1+2=3

    LARGEST =96+95 =191

    3,4,5,.191

     so # of  distinct  sums* =191-2=189

      i.e. 189 SUMS FOR 190 PAIRS

    189 is less than  190  

         qed

     

    This  kind of question has nothing to do with the maximal value of  EN ,

    a   value-added  and challenging contribution by  my  humble self ,

    requesting  the following:

    1.Provide a   list  of 20 integers, the lowest =1 , highest EN-1 s..t.

    at least two distinct  pairs satisfy  a+b=c+d.

    2.    Prove, either analytically or by exaustive  run that fror EN all

    Combination of pairs  for any subset of  20 members  produce distinct

    Sums.

     

     

    If    it is too tiresome and no shortcuts are known

    let's  solve the problem for a  reduced set , say 12 numbers.

     

    PLEASE  COMMENT re: official solution.

     

     

     

     

     

     

     

     

     

     

     

     

     


  Posted by Ady TZIDON on 2010-07-18 12:41:15
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