Difference between revisions of "1990 USAMO Problems/Problem 2"
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* [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=356624#p356624 Discussion on AoPS/MathLinks] | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=356624#p356624 Discussion on AoPS/MathLinks] | ||
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[[Category:Olympiad Algebra Problems]] | [[Category:Olympiad Algebra Problems]] |
Latest revision as of 19:14, 18 July 2016
Problem
A sequence of functions is defined recursively as follows:
(Recall that
is understood to represent the positive square root.) For each positive integer
, find all real solutions of the equation
.
Solution
We define . Then the recursive relation holds for
, as well.
Since for all nonnegative integers
, it suffices to consider nonnegative values of
.
We claim that the following set of relations hold true for all natural numbers and nonnegative reals
:
To prove this claim, we induct on
. The statement evidently holds for our base case,
.
Now, suppose the claim holds for . Then
The claim therefore holds by induction. It then follows that for all nonnegative integers
,
is the unique solution to the equation
.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See Also
1990 USAMO (Problems • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
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.