2023 USAMO Problems/Problem 2
Problem 2
Let be the set of positive real numbers. Find all functions such that, for all ,
Solution 1
Make the following substitutions to the equation:
1.
2.
3.
It then follows from (2) and (3) that , so we know that this function is linear for . Substitute and solve for and in the functional equation; we find that .
Now, we can let and . Since , , so . It becomes clear then that as well, so is the only solution to the functional equation.
~jkmmm3
See Also
2023 USAMO (Problems • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |
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