2017 IMO Problems/Problem 2
Revision as of 12:35, 9 December 2020 by Circlegeometrygang (talk | contribs)
Let be the set of real numbers , determine all functions
such that for any real numbers
and
=
Solution
Let , so the equation becomes
. Notice that if
,
, so
, meaning that there exists at least 1 real solution to
.
Let , so
.
Let , so
.
If , or
, then
, so
. The only way n can not equal 0 is if there is no solution to
, so
if
does not equal 0.
This means that the only possible values of is -1,0, and 1.
Go through the cases:
(Based on
)
...
(Based on
)
...
So the only solutions are ,
, and
.