1990 USAMO Problems/Problem 2
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 .
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.
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