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 Low 300s ?? by ed bottemiller)

Ed,

Some further thoughts:

1. Let S be the set of marks on a Golomb Ruler of length EN and order x, in which i1+i2=i3+i4   
2. Then it follows arithmetically that i1-i3=i4-i2   
3. But S would not then be a Golomb Ruler.
It seems to follow that for any order, x, (here, 20) the last mark on a Golomb Ruler of order (x+1) represents the  smallest possible number for which there is at least one (non-randomly selected) subset, S, having the required number of x elements, with no stipulated duplicate. Accordingly, since the last mark on a Golomb Ruler of order 21 is 333, 332 should be the largest number for which it is possible to be certain, without further verification, that each and every randomly selected S will surely contain the stipulated duplicate.

(I am sure that there are much more elegant ways of expressing this!)

Edited on July 17, 2010, 10:39 am

Posted by broll on 2010-07-17 09:05:23