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Kool Numbers (Posted on 2004-11-15) Difficulty: 4 of 5
A collection of positive integers (not necessarily distinct) is called Kool if the sum of all its elements equals their product.

For example, {2, 2, 2, 1, 1} is a Kool set.

a) Show that there exists a Kool set of n numbers for all n>1

b) Find all Kool sets with sums of 100

c) Find all Kool sets with 100 members.

No Solution Yet Submitted by SilverKnight    
Rating: 3.0000 (5 votes)

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Hints/Tips Third part | Comment 4 of 10 |

Let the k non-one numbers of the set be a1, a2,..., ak.  Let Pk be their product, and let Sk be their sum.  We then demand that f(a1,a2,...,ak) = 0, where f is a function from [2,infty)^k to (-infty, infty) such that

f(a1,a2,...,ak) = Pk - Sk + (k-100).

Now for k>2, it is clear that this function increases with respect to each of the ai, since letting ai->ai+1 inccreases Sk by 1 and increases Pk by more than 1.  But for k>6, then, we have:

f(a1,a2,...,ak) >= f(2,2,...,2)
                       = 2^k - 2k + (k-100)
                       = 2^k - k - 100
                       > 0.

This implies that the only solutions are with k<7.  We can do a brute force search through those possibilities, by running a simple program.  I don't have time to do this.  Maybe somebody else can do this.

  Posted by David Shin on 2004-11-15 17:36:28
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