Baltic Way 1995 · Problem 11

Solution:

We construct the three subsets by adding the numbers successively, and disregard at first the condition that the sets must be non-empty. The numbers $1$ and $2$ must belong to two different subsets, say $A$ and $B$. We then have two choices for each of the numbers $3,4, \ldots, 1995$, and different choices lead to different partitions. Hence there are $2^{1993}$ such partitions, one of which has an empty part. The number of partitions satisfying the requirements of the problem is therefore $2^{1993}-1$.