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.
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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.

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

f(a₁,a₂,...,a_k) = P_k - S_k + (k-100).

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

f(a₁,a₂,...,a_k) >= 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