2018 USAMO Problems/Problem 2
Problem 2
Find all functions such that
for all
with
Solution
The only such function is .
Letting gives
, hence
. Now observe that even if we fix
,
is not fixed. Specifically,
This is continuous on the interval
and has an asymptote at
. Since it takes the value 2 when
, it can take on all values greater than or equal to 2. So for any
, we can find
such that
. Therefore,
for all
.
Now, for any , if we let
,
, and
, then
. Since
,
, hence
. Therefore,
for all
.
-- wzs26843645602