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 Restoring the erased (Posted on 2010-07-14)
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 ?

Comments: ( Back to comment list | You must be logged in to post comments.)
 re: kudos Comment 21 of 21 |

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

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Edited on July 21, 2010, 7:00 am
 Posted by broll on 2010-07-21 06:58:37

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