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1998 APMO – Problem #1

Let F be the set of all n-tuples (A1, A2,…,An) where each Ai, i = 1,2,…,n is a subset of {1,2,…,1998}. Let |A| denote the number of elements of the set A. Find the number



Solution #1 by Sasha Schwartz

Of the terms of the form A1 È A2 È … È An exactly of them don’t contain m, where m is an arbitrary integer between 1 and 1998, inclusive.

Therefore, terms contain the integer m.

Since this is true for each of the 1998 integers within the range, the total number of elements appearing in each of the terms, which is what was to be computed, is