1989 USAMO Problems/Problem 1
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Problem
For each positive integer , let
Find, with proof, integers
such that
and
.
Solution
We note that for all integers ,
It then follows that
If we let , we see that
is a suitable solution.
Notice that it is also possible to use induction to prove the equations relating and
with
.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See also
1989 USAMO (Problems • Resources) | ||
Preceded by First question |
Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |