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)

Glad to see the target coming into sight!  Is the method (even for 12 terms) still exhaustive search, or is there a more systematic way (beyond Broll's combinatorial formula for a negative result, which for 12 terms would show always two sum pairs equal for 12*11/2 or 66).  Also, since ALL of the numbers must be less than EN, the stated set from 1..72, would affect the case of EN=73, wouldn't it?

Posted by ed bottemiller on 2010-07-20 11:25:49