Functional equation, dysfunctional mind?
by wu2481632, Jan 30, 2017, 3:59 AM
Darn I actually can't think straight today. Wasted way too much time on AoPS and spent too little time doing HW. To make up for that, here is a functional equation:
Find all functions
such that
for all
(Romania 2009)
Solution
Small Piece of Tasty Food
Find all functions



Solution
Setting
, we see that
; ergo the equation reduces to
so
is additive over the reals.
It follows that if
is a solution of the functional equation, then
for some constant
must be a solution of the functional equation. Therefore there exists some function satisfying the given conditions such that
, unless
.
We know that
, so therefore
. The LHS reduces to
. The RHS is
. Therefore we have
which reduces to
which is simply
After manipulations this reduces to
Now we put in
into the above equation, giving us
The RHS reduces to
and the LHS reduces to
Equating these gives
Now we can put
and
together by subtracting them, implying
. Therefore
for constant
is a solution.
In the case that
, we get
. The LHS is simply
and the RHS reduces to
. Equating these gives
Similarly to the above case, we can switch
out for
, which implies that
Putting the above two equations together gives
, so therefore the only solutions are of the form
for all real
.




It follows that if





We know that


















In the case that











Small Piece of Tasty Food
darn happy chinese new year! unfortunately, the party was not good.