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 re: High-Low-Jack... by Dej Mar)

I think F(21) is a good surmise (so an excluded set of 20 would be F(2)=1 thru F(21)=10946). F(0)=0 is not positive, and F(1) and F(2) are both = 1 (can't repeat same number).  I was suggesting 17711 which is F(22)+1 to "play it safe" but no proof.

The wording of the puzzle still gives me pause.  We would both be setting an upper bound for EN such that not all sets of twenty with that EN limit have the requisite sums (by "construction" using the Fibonacci series).  But what would rule out a lower EN which still had the property that some set S might not in fact allow the equation for some subset of four members?  A set S could easily be verified, if found , but the exponential search space would seem to be prohibitive.

 

Posted by ed bottemiller on 2010-07-14 20:20:58