1989 USAMO Problems/Problem 1
For each positive integer , let
Find, with proof, integers such that and .
Let us prove that . Expanding:
Grouping like terms, there are s, s, and so on:
which completes our proof. Thus, for , we have that , and so .
For the second part, use our previously derived identity to determine in terms of . The problem simplifies to:
Thus, we have . For , we get that and
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