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 kudos by Ady TZIDON)

Is this right?

1. A Golomb Ruler has 2 properties:      
a. Order (number of marks, including 0)      
b. length (maximum mark)      
2. A Justin Ruler (or Tzidon Set) has 3 properties:-      
a. order (j)      
b. length (l)      
c. range (R). The range is the sum of the last 2 marks      
3. For a given order, the formula 1/2n(n+1)+2 gives the theoretical minimum range for a Justin Ruler of that order. This range is the 'canonical range', Rc      
4. A ruler is a Justin Ruler if       
a. the sum of every pair of marks is distinct from that of every other pair.      
b. the range of the ruler is a minimum for a ruler of that order.      
5. A Justin Ruler is perfect if its range is Rc for a ruler of that order      
6. If the range, Rj of a Justin Ruler of a particular order exceeds Rc, then the difference is the looseness (Rj-Rc) or the fractional looseness (1- (Rc/Rj))*100%) of the Ruler      
7. For obvious reasons, the minimum order of a Justin Ruler is 4      

 

 

Edited on July 21, 2010, 7:00 am

Posted by broll on 2010-07-21 06:58:37