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 ?

(In reply to solution for n=12 by Justin)

Justin,

Thanks for your post.

You say  "The "worse" set I found as far as failing at the lowest possible EN value was my previously stated {1, 2, 3, 8, 13, 23, 38, 41, 55, 64, 68, 72}.

To put the problem to bed we need to leave no stone unturned
i.e. we have  to be certain that no "bad" set exists for EN=71 or less.

Please let me know whether the searches for all lower values values were exaustive or not. If not, how do we now that there is no valid EN  lower than 72.

REGARDS

Ady

 

 

Posted by Ady TZIDON on 2010-07-20 11:22:19