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 running some numbers ... by Justin)

Welcome aboard!  Proving a failed set of twenty members does show that EN cannot be larger than the highest number in that set.    Since any set of 20 will give 190 sums to test for some duplication, Broll has shown this gives some lower limit of EN (i.e. all subsets of 20 below that limit must contain at least one repeated sum).  I agree the search space grows too large for a feasible brute force search.  If EN is 350 or larger, a single set such as yours shows that the test for EN is not satisfied. Good luck in the hunt!  

Posted by ed bottemiller on 2010-07-16 15:00:13