1989 USAMO Problems/Problem 1
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.