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 f(x) = ax+b and solve for a and b 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