2024 USAJMO Problems/Problem 5
Contents
[hide]Problem
Find all functions that satisfy for all .
Solution 1
Plugging in as
-codemaster11
Solution 2
Let our equation be . We start by plugging in some initial values:
Plugging in into gives From , we get Substituting in what we have in gives Plugging in gives As a result, becomes .
Now, becomes and becomes Note that is a solution. Now, assume .
Claim: is injective over .
Let with . Plugging in and into gives us Subtracting, and using gives us , which implies that either or . Either way leads to contradiction. Thus, is injective.
As a result, becomes . Piecing everything yields .
It just remains to verify these solutions work, and doing so is quite trivial; all of which are obviously true.
~sml1809
See Also
2024 USAJMO (Problems • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |
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